Radii of starlikeness and convexity of Bessel function derivatives
UDC 517.5 In this paper, our aim is to find the radii of starlikeness andconvexity of Bessel function derivatives for three different kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for $n$th derivative of Bessel function andproperties of real z...
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| author | Deniz, E. Kazımoğlu, S. Çağlar, M. Deniz, E. Kazımoğlu, S. Çağlar, M. S. |
| author_facet | Deniz, E. Kazımoğlu, S. Çağlar, M. Deniz, E. Kazımoğlu, S. Çağlar, M. S. |
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| description | UDC 517.5
In this paper, our aim is to find the radii of starlikeness andconvexity of Bessel function derivatives for three different kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for $n$th derivative of Bessel function andproperties of real zeros of it. In addition, by using the Euler–Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized $n$th derivative of Bessel function. The main results of the paper are natural extensions of some known results on classical Bessel functions of the first kind.
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| doi_str_mv | 10.37863/umzh.v73i11.1014 |
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DOI: 10.37863/umzh.v73i11.1014
UDC 517.5
E. Deniz, S. Kazımoğlu (Kafkas Univ., Kars, Turkey),
M. Çağlar (Erzurum Techn. Univ., Turkey)
RADII OF STARLIKENESS AND CONVEXITY
OF BESSEL FUNCTION DERIVATIVES*
РАДIУСИ ЗIРЧАСТОСТI ТА ОПУКЛОСТI
ПОХIДНИХ ФУНКЦIЇ БЕССЕЛЯ
In this paper, our aim is to find the radii of starlikeness and convexity of Bessel function derivatives for three different
kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for nth derivative
of Bessel function and properties of real zeros of it. In addition, by using the Euler – Rayleigh inequalities we obtain some
tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized nth derivative of
Bessel function. The main results of the paper are natural extensions of some known results on classical Bessel functions
of the first kind.
Знайдено радiуси зiрчастостi та опуклостi похiдних функцiї Бесселя для трьох рiзних видiв нормалiзацiї. Ключовими
iнструментами доведення основних результатiв є розклад Мiттаг-Леффлера для n-ї похiдної функцiї Бесселя та
властивостi його дiйсних нулiв. Крiм того, за допомогою нерiвностей Ейлера – Релея отримано деякi точнi нижнi й
верхнi межi для радiусiв зiрчастостi та опуклостi нульового порядку для нормованої n-ї похiдної функцiї Бесселя.
Основними результатами роботи є природнi розширення деяких вiдомих результатiв щодо класичних функцiй
Бесселя першого роду.
1. Introduction. Denote by \BbbD r = \{ z \in \BbbC : | z| < r\} (r > 0) the disk of radius r and let \BbbD =
= \BbbD 1. Let \scrA be the class of analytic functions f in the open unit disk \BbbD which satisfy the usual
normalization conditions f(0) = f \prime (0) - 1 = 0. Traditionally, the subclass of \scrA consisting of
univalent functions is denoted by \scrS . We say that the function f \in \scrA is starlike in the disk \BbbD r if f
is univalent in \BbbD r, and f(\BbbD r) is a starlike domain in \BbbC with respect to the origin. Analytically, the
function f is starlike in \BbbD r if and only if \mathrm{R}\mathrm{e}
\biggl(
zf \prime (z)
f(z)
\biggr)
> 0, z \in \BbbD r. For \beta \in [0, 1) we say that the
function f is starlike of order \beta in \BbbD r if and only if \mathrm{R}\mathrm{e}
\biggl(
zf \prime (z)
f(z)
\biggr)
> \beta , z \in \BbbD r. We define by the
real number
r\ast \beta (f) = \mathrm{s}\mathrm{u}\mathrm{p}
\biggl\{
r \in (0, rf ) : \mathrm{R}\mathrm{e}
\biggl(
zf \prime (z)
f(z)
\biggr)
> \beta for all z \in \BbbD r
\biggr\}
the radius of starlikeness of order \beta of the function f. Note that r\ast (f) = r\ast 0(f) is the largest radius
such that the image region f(\BbbD r\ast \beta (f)
) is a starlike domain with respect to the origin.
The function f \in \scrA is convex in the disk \BbbD r if f is univalent in \BbbD r, and f(\BbbD r) is a convex
domain in \BbbC . Analytically, the function f is convex in \BbbD r if and only if \mathrm{R}\mathrm{e}
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
> 0,
z \in \BbbD r. For \beta \in [0, 1) we say that the function f is convex of order \beta in \BbbD r if and only if
\mathrm{R}\mathrm{e}
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
> \beta , z \in \BbbD r. The radius of convexity of order \beta of the function f is defined by
* This research was supported by Kafkas University Scientific Research Projects Coordination Unit (Project Number
2018-FEF-15).
c\bigcirc E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11 1461
1462 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
the real number
rc\beta (f) = \mathrm{s}\mathrm{u}\mathrm{p}
\biggl\{
r \in (0, rf ) : \mathrm{R}\mathrm{e}
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
> \beta for all z \in \BbbD r
\biggr\}
.
Note that rc(f) = rc0(f) is the largest radius such that the image region f(\BbbD rc\beta (f)
) is a convex
domain.
The first kind of Bessel function of order \nu is defined by [18, p. 217]
J\nu (z) =
\infty \sum
m=0
( - 1)m
m!\Gamma (m+ \nu + 1)
\Bigl( z
2
\Bigr) 2m+\nu
, z \in \BbbC .
Now, we consider the nth derivative of Bessel function of the first kind by
J (n)
\nu (z) =
\infty \sum
m=0
( - 1)m\Gamma (2m+ \nu + 1)
m!2n\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
\Bigl( z
2
\Bigr) 2m - n+\nu
, z \in \BbbC .
Here, it is important mentioning that for n = 0 the J (n)
\nu reduce to classical Bessel function J\nu . Since
the function J (n)
\nu is not belongs to \scrA , firstly, we form and focus on the following normalized forms:
f\nu ,n(z) =
\Bigl[
2\nu \Gamma (\nu - n+ 1)J (n)
\nu (z)
\Bigr] 1
\nu - n
,
g\nu ,n(z) = 2\nu \Gamma (\nu - n+ 1)z1+n - \nu J (n)
\nu (z),
h\nu ,n(z) = 2\nu \Gamma (\nu - n+ 1)z1+
n - \nu
2 J (n)
\nu (
\surd
z),
(1.1)
where \nu > n - 1.
The first studies on geometric properties of Bessel functions of first kind was conducted in 1960
by Brown, Kreyszig and Todd [10, 16]. They determined the radius of starlikeness of the functions
f\nu ,0(z) and g\nu ,0(z) for the case \nu > 0. Recently, in 2014, Baricz et al. [3] and Baricz and Szász [4]
obtained, respectively, the radius of starlikeness of order \beta and the radius of convexity of order \beta
for the functions f\nu ,0(z), g\nu ,0(z) and h\nu ,0(z) in the case when \nu > - 1. On the other hand, we know
that if \nu \in ( - 2, - 1), then the Bessel function has exactly two purely imaginary conjugate complex
zeros, and all the other zeros are real [21, p. 483]. In 2015, Szász [20] investigated the radius of
starlikeness of order \beta for the functions g\nu (z) and h\nu (z) in the case when \nu \in ( - 2, - 1) by using
some inequalities. In the same year, Baricz and Szász [5] obtained the radius of convexity of order
\beta for the functions g\nu (z) and h\nu (z) in the case when \nu \in ( - 2, - 1). Later, in 2016, Baricz et al. [7]
determined the radius of \alpha -convexity of the same three functions for \nu > - 1. After a year, Çağlar
et al. [11] extended it for the case when \nu \in ( - 2, - 1). In 2017, Deniz and Szász [12] determined
the radius of uniform convexity of f\nu ,0(z), g\nu ,0(z) and h\nu ,0(z) for \nu > - 1. They also determined
necessary and sufficient conditions on the parameters of these three normalized functions such that
they are uniformly convex in the unit disk. Moreover, in [1, 2] authors determined tight lower and
upper bounds for the radii of starlikeness and convexity of the functions g\nu ,0(z) and h\nu ,0(z). The key
tools in their proofs were some new Mittag-Leffler expansions for quotients of Bessel functions of
the first kind, special properties of the zeros of Bessel functions of the first kind and their derivatives,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1463
Euler – Rayleigh inequalities and the fact that the smallest positive zeros of some Dini functions are
less than the first positive zero of the Bessel function of first kind.
Another study on Bessel functions investigate the properties of derivatives and the zeros of
these derivatives. In the last three decades the zeros of the nth derivative of Bessel functions of
the first kind for n \in \{ 1, 2, 3\} have been also studied by researchers like Elbert, Ifantis, Ismail,
Kokologiannaki, Laforgia, Landau, Lorch, Mercer, Muldoon, Petropoulou, Siafarikas and Szegö; for
more details see the papers [13, 15] and the references therein. Very recently in 2018, Baricz et al. [8]
obtained some results for the zeros of the nth derivative of Bessel functions of the first kind for all
n \in \BbbN by using the Laguerre – Pólya class of entire functions and the so-called Laguerre inequalities.
Motivated by the above results in this paper, we deal with the radii of starlikeness and convexity
of order \beta for the functions f\nu ,n(z), g\nu ,n(z) and h\nu ,n(z) in the case when \nu > n - 1 for n \in \BbbN .
Also, we determined tight lower and upper bounds for the radii of starlikeness and convexity of these
functions.
2. Preliminaries. In order to prove the main results we need the following preliminary results.
Lemma 2.1 [8]. The following assertions are valid:
(a) If \nu > n - 1, then z \mapsto \rightarrow J
(n)
\nu (z) has infinitely many zeros, which are all real and simple,
expect the origin.
(b) If \nu > n, then the positive zeros of the nth and (n+ 1)th derivative of J\nu are interlacing.
(c) If \nu > n - 1, then all zeros of z \mapsto \rightarrow (n - \nu )J
(n)
\nu (z) + zJ
(n+1)
\nu (z) are real and interlace with
the zeros of z \mapsto \rightarrow J
(n)
\nu (z).
The lemma below (see [9, 19]) is also required for our work.
Lemma 2.2. Let f(x) =
\sum \infty
n=0
anx
n, an \in \BbbR , and g(x) =
\sum \infty
n=0
bnx
n, bn > 0, for all
n \geq 0, converge on an interval ( - r, r) for some r > 0. If the sequence \{ an\diagup bn\} n\geq 0 is decreasing
(increasing), then the function x\rightarrow f(x)\diagup g(x) is decreasing (increasing) too on (0, r). So the same
result holds for the following:
f(x) =
\infty \sum
n=0
anx
2n and g(x) =
\infty \sum
n=0
bnx
2n.
2.1. Zeros of hyperbolic polynomials and the Laguerre – Pólya class of entire functions. In
this subsection, some necessary knowledge about polynomials and entire functions with real zeros
are given. An algebraic polynomial is named hyperbolic if its all zeros are real. We will be using the
following lemma given in [6] and obtain new results.
Lemma 2.3. Assume
p(x) = 1 - a1x+ a2x
2 - a3x
3 + . . .+ ( - 1)nanx
n = (1 - x/x1) . . . (1 - x/xn)
is a hyperbolic polynomial with positive zeros 0 < x1 \leq x2 \leq . . . \leq xn, and it is normalized by
p(0) = 1. Then the polynomial q(x) = Cp(x) - xp\prime (x) is hyperbolic for any constant C. Also, the
smallest zero \eta 1 is in (0, x1) if and only if C < 0.
Clearly, a real entire function \psi is in the Laguerre – Pólya class \scrL \scrP if it is in the form
\psi (x) = cxme - ax2+\beta x
\prod
k\geq 1
\biggl(
1 +
x
xk
\biggr)
e
- x
xk ,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1464 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
with c, \beta , xk \in \BbbR , a \geq 0, m \in \BbbN \cup \{ 0\} and
\sum
x - 2
k < \infty . Similarly, we say \phi is of type \scrI in the
Laguerre – Pólya class, denoted by \phi \in \scrL \scrP \scrI , if \phi (x) or \phi ( - x) is written as
\phi (x) = cxme\sigma x
\prod
k\geq 1
\biggl(
1 +
x
xk
\biggr)
,
with c \in \BbbR , \sigma \geq 0, m \in \BbbN \cup \{ 0\} , xk > 0 and
\sum
x - 1
k < \infty . The complement of the space of
hyperbolic polynomials in the topology induced by the uniform convergence on the compact sets of
the complex plane is the class \scrL \scrP if the complement of the hyperbolic polynomials whose zeros
possess a preassigned constant sign is \scrL \scrP \scrI . For any entire function \varphi in the form
\varphi (x) =
\sum
k\geq 0
\mu k
xk
k!
,
its Jensen polynomials are given by
Pm(\varphi ;x) = Pm(x) =
m\sum
k=0
\Biggl(
m
k
\Biggr)
\mu kx
k.
The following lemma is a well-known characterization of functions in the class \scrL \scrP (see [14]).
Lemma 2.4. \varphi is in the class \scrL \scrP (\scrL \scrP \scrI , respectively) if and only if all the polynomials
Pm(\varphi ;x), m = 1, 2, . . . , are hyperbolic such that they are hyperbolic with zeros of equal sign.
Also, the sequence Pm(\varphi ; z\diagup n) is locally uniformly convergent to \varphi (z).
The following lemma is necessary for the proof of main results.
Lemma 2.5. Let \nu > n - 1 and a < 0. Then the functions z \mapsto - \rightarrow (2a - n + \nu )J
(n)
\nu (z) -
- zJ
(n+1)
\nu (z) are written in the form
2n - 1\Gamma (\nu + 1 - n)
\Bigl(
(2a - n+ \nu )J (n)
\nu (z) - zJ (n+1)
\nu (z)
\Bigr)
=
\Bigl( z
2
\Bigr) \nu - n
W\nu ,n(z),
where W\nu ,n is entire functions belonging to the Laguerre – Pólya class \scrL \scrP . Moreover, the smallest
positive zero of W\nu ,n cannot exceed the first positive zero j(n)\nu ,1 , where j(n)\nu ,m is the mth positive zero
of J (n)
\nu (z), m \in \BbbN , n \in \BbbN 0.
Proof. It is obvious from the infinite product representation of z \mapsto - \rightarrow \scrJ (n)
\nu (z) = 2\nu \Gamma (\nu + 1 -
- n)(z)n - \nu J
(n)
\nu (z) that this function is in the class \scrL \scrP . This shows that the function z \mapsto - \rightarrow \BbbJ (n)\nu (z) =
= \scrJ (n)
\nu (2
\surd
z ) is in the class \scrL \scrP \scrI . Then, due to Lemma 2.4, its Jensen polynomials
Pm(\BbbJ (n)\nu ; \varsigma ) =
m\sum
k=0
\Biggl(
m
k
\Biggr)
\mu kx
k
are all hyperbolic. However, it can be seen that the Jensen polynomials of \widetilde W\nu ,n(z) = W\nu ,n(2
\surd
z )
are clearly
Pm
\bigl( \widetilde W\nu ,n; \varsigma
\bigr)
= aPm
\bigl(
\BbbJ (n)\nu ; \varsigma
\bigr)
- \varsigma P \prime
m
\bigl(
\BbbJ (n)\nu ; \varsigma
\bigr)
.
Moreover, Lemma 2.3 tells us that all zeros of Pm
\bigl( \widetilde W\nu ,n; \varsigma
\bigr)
are real and positive and that the smallest
one precedes the first zero of Pm
\bigl(
\BbbJ (n)\nu ; \varsigma
\bigr)
. From Lemma 2.4, the latter result immediately implies
that \widetilde W\nu ,n \in \scrL \scrP \scrI and that its first zero precedes j(n)\nu ,1 . Finally, the first part of the statement of the
lemma follows after we go back from \widetilde W\nu ,n to W\nu ,n by setting \varsigma =
z2
4
.
Lemma 2.5 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1465
2.2. Euler – Rayleigh sums for positive zeros of \bfitJ (\bfitn )
\bfitnu (\bfitz ). Baricz et al. [8] proved the Weier-
strassian decomposition of J (n)
\nu (z) as follows:
J (n)
\nu (z) =
z\nu - n
2\nu \Gamma (\nu + 1 - n)
\prod
m\geq 1
\left( 1 - z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
\right) , (2.1)
where j(n)\nu ,m is the mth positive zero of J (n)
\nu (z), m \in \BbbN , n \in \BbbN 0. Therefore we can write
g\nu ,n(z) = 2\nu \Gamma (\nu - n+ 1)z1+n - \nu J (n)
\nu (z) = z
\prod
m\geq 1
\left( 1 - z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
\right) . (2.2)
On the other hand, the series representation of g\nu ,n(z)
g\nu ,n(z) =
\infty \sum
m=0
( - 1)m\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)
m!4m\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
z2m+1. (2.3)
Now, we would like to mention that by using the equations (2.2) and (2.3) we can obtain the
following Euler – Rayleigh sums for the positive zeros of the function g\nu ,n. From the equality (2.3)
we have
g\nu ,n(z) = z - \nu + 2
4(\nu - n+ 2)(\nu - n+ 1)
z3 +
+
(\nu + 4)(\nu + 3)
32(\nu - n+ 4)(\nu - n+ 3)(\nu - n+ 2)(\nu - n+ 1)
z5 - . . . . (2.4)
Now, if we consider (2.2), then some calculations yield that
g\nu ,n(z) = z -
\sum
m\geq 1
1\Bigl(
j
(n)
\nu ,m
\Bigr) 2 z3 + 1
2
\left(
\left( \sum
m\geq 1
1\Bigl(
j
(n)
\nu ,m
\Bigr) 2
\right)
2
-
\sum
m\geq 1
1\Bigl(
j
(n)
\nu ,m
\Bigr) 4
\right) z5 - . . . . (2.5)
By equating the first few coefficients with the same degrees in equations (2.4) and (2.5), we get\sum
m\geq 1
1\Bigl(
j
(n)
\nu ,m
\Bigr) 2 =
\nu + 2
4(\nu - n+ 2)(\nu - n+ 1)
(2.6)
and \sum
m\geq 1
1\Bigl(
j
(n)
\nu ,m
\Bigr) 4 =
1
16(\nu - n+ 2)(\nu - n+ 1)
\times
\times
\biggl(
(\nu + 2)2
(\nu - n+ 2)(\nu - n+ 1)
- (\nu + 4)(\nu + 3)
(\nu - n+ 4)(\nu - n+ 3)
\biggr)
. (2.7)
Here, it is important mentioning that for n = 0 the equations (2.6) and (2.7) reduce to
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1466 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
\sum
m\geq 1
1
(j\nu ,m)2
=
1
4(\nu + 1)
and
\sum
m\geq 1
1
(j\nu ,m)4
=
1
16(\nu + 2)(\nu + 1)2
,
respectively, where j\nu ,m denotes the mth zero of classical Bessel function J\nu .
Another special case for n = 1, 2 the equations (2.6) and (2.7) reduce to
\sum
m\geq 1
1\bigl(
j\prime \nu ,m
\bigr) 2 =
\nu + 2
4\nu (\nu + 1)
,
\sum
m\geq 1
1\bigl(
j\prime \nu ,m
\bigr) 4 =
\nu 2 + 8\nu + 8
16\nu 2(\nu + 1)2(\nu + 2)
and \sum
m\geq 1
1\bigl(
j\prime \prime \nu ,m
\bigr) 2 =
\nu + 2
4(\nu - 1)\nu
,
\sum
m\geq 1
1\bigl(
j\prime \prime \nu ,m
\bigr) 4 =
13\nu 3 + 19\nu 2 + 26\nu + 8
16(\nu - 1)2\nu 2(\nu + 1)(\nu + 2)
,
where j\prime \nu ,m and j\prime \prime \nu ,m denotes the mth zeros of function J \prime
\nu and J \prime \prime
\nu , respectively.
3. Main results. 3.1. Radii of starlikeness and convexity of the functions \bfitf \bfitnu ,\bfitn , \bfitg \bfitnu ,\bfitn and
\bfith \bfitnu ,\bfitn . The first principal result we established concerns the radii of starlikeness and reads as follows.
Here and in the sequel I\nu denotes the modified Bessel function of the first kind and order \nu . Note
that I\nu (z) = i - \nu J\nu (iz).
Theorem 3.1. The followings hold:
(a) If \nu > n and \beta \in [0, 1), then r\ast \beta (f\nu ,n) = x
(n)
\nu ,1 , where x(n)\nu ,1 is the smallest positive root of the
equation
rJ
(n+1)
\nu (r)
(\nu - n)J
(n)
\nu (r)
- \beta = 0.
Besides, if n - 1 < \nu < n and \beta \in [0, 1), then we have r\ast \beta (f\nu ,n) = x
(n)
\nu ,2 , where x(n)\nu ,2 is the smallest
positive root of the equation
rI
(n+1)
\nu (r)
(\nu - n)I
(n)
\nu (r)
- \beta = 0.
(b) If \nu > n - 1 and \beta \in [0, 1), then r\ast \beta (g\nu ,n) = y
(n)
\nu ,1 , where y(n)\nu ,1 is the smallest positive root
of the equation
rJ
(n+1)
\nu (r)
J
(n)
\nu (r)
+ n+ 1 - \nu - \beta = 0.
(c) If \nu > n - 1 and \beta \in [0, 1), then r\ast \beta (h\nu ,n) = z
(n)
\nu ,1 , where z(n)\nu ,1 is the smallest positive root
of the equation \surd
rJ
(n+1)
\nu (
\surd
r)
J
(n)
\nu (
\surd
r)
+ n+ 2 - \nu - 2\beta = 0.
Proof. Firstly, we prove part (a) for \nu > n and parts (b) and (c) for \nu > n - 1. We need to
show that the following inequalities:
\mathrm{R}\mathrm{e}
\biggl(
zf \prime \nu ,n(z)
f\nu ,n(z)
\biggr)
> \beta , \mathrm{R}\mathrm{e}
\biggl(
zg\prime \nu ,n(z)
g\nu ,n(z)
\biggr)
> \beta and \mathrm{R}\mathrm{e}
\biggl(
zh\prime \nu ,n(z)
h\nu ,n(z)
\biggr)
> \beta (3.1)
are valid for z \in \BbbD r\ast \beta (f\nu ,n)
, z \in \BbbD r\ast \beta (g\nu ,n)
and z \in \BbbD r\ast \beta (h\nu ,n), respectively, and each inequality above
cannot holds in larger disks.
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RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1467
When we write the equation (2.1) in definition of the functions f\nu ,n(z), g\nu ,n(z) and h\nu ,n(z), we
get by using logarithmic derivation
zf \prime \nu ,n(z)
f\nu ,n(z)
=
1
\nu - n
zJ
(n+1)
\nu (z)
J
(n)
\nu (z)
= 1 - 1
\nu - n
\sum
m\geq 1
2z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
, \nu > n,
zg\prime \nu ,n(z)
g\nu ,n(z)
= n+ 1 - \nu +
zJ
(n+1)
\nu (z)
J
(n)
\nu (z)
= 1 -
\sum
m\geq 1
2z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
, \nu > n - 1,
zh\prime \nu ,n(z)
h\nu ,n(z)
= 1 +
n - \nu
2
+
1
2
\surd
zJ
(n+1)
\nu (
\surd
z)
J
(n)
\nu (
\surd
z)
= 1 -
\sum
m\geq 1
z\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z
, \nu > n - 1.
It is known [4] that if z \in \BbbC and \lambda \in \BbbR are such that \lambda > | z| , then
| z|
\lambda - | z|
\geq \mathrm{R}\mathrm{e}
\biggl(
z
\lambda - z
\biggr)
. (3.2)
Then the inequality
| z| 2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- | z| 2
\geq \mathrm{R}\mathrm{e}
\left( z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
\right)
holds for every \nu > n - 1. Therefore,
\mathrm{R}\mathrm{e}
\biggl(
zf \prime \nu ,n(z)
f\nu ,n(z)
\biggr)
= 1 - 1
\nu - n
\sum
m\geq 1
\mathrm{R}\mathrm{e}
\left( 2z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
\right) \geq
\geq 1 - 1
\nu - n
\sum
m\geq 1
2| z| 2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- | z| 2
=
| z| f \prime \nu ,n(| z| )
f\nu ,n(| z| )
,
\mathrm{R}\mathrm{e}
\biggl(
zg\prime \nu ,n(z)
g\nu ,n(z)
\biggr)
= 1 -
\sum
m\geq 1
\mathrm{R}\mathrm{e}
\left( 2z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
\right) \geq
\geq 1 -
\sum
m\geq 1
2| z| 2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- | z| 2
=
| z| g\prime \nu ,n(| z| )
g\nu ,n(| z| )
,
\mathrm{R}\mathrm{e}
\biggl(
zh\prime \nu ,n(z)
h\nu ,n(z)
\biggr)
= 1 -
\sum
m\geq 1
\mathrm{R}\mathrm{e}
\left( z\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z
\right) \geq
\geq 1 -
\sum
m\geq 1
| z| \Bigl(
j
(n)
\nu ,m
\Bigr) 2
- | z|
=
| z| h\prime \nu ,n(| z| )
h\nu ,n(| z| )
,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1468 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
where equalities are obtained only if z = | z| = r. From the latest inequalities and the minimum
principle for harmonic functions, we conclude that the corresponding inequalities in (3.1) hold if and
only if | z| < x
(n)
\nu ,1 , | z| < y
(n)
\nu ,1 and | z| < z
(n)
\nu ,1 , respectively, where x(n)\nu ,1 , y
(n)
\nu ,1 and z(n)\nu ,1 is the smallest
positive roots of the equations
rf \prime \nu ,n(r)
f\nu ,n(r)
= \beta ,
rg\prime \nu ,n(r)
g\nu ,n(r)
= \beta and
rh\prime \nu ,n(r)
h\nu ,n(r)
= \beta ,
which are equivalent to
rJ
(n+1)
\nu (r)
(\nu - n)J
(n)
\nu (r)
- \beta = 0,
rJ
(n+1)
\nu (r)
J
(n)
\nu (r)
+ n+ 1 - \nu - \beta = 0
and \surd
rJ
(n+1)
\nu (
\surd
r)
J
(n)
\nu (
\surd
r)
+ n+ 2 - \nu - 2\beta = 0.
The result follows from Lemma 2.5 by taking instead of a the values
(\beta - 1)(\nu - n)
2
,
\beta - 1
2
and
\beta - 1, respectively. In other words, Lemma 2.5 show that all the zeros of the above three functions
are real and their first positive zeros do not exceed the first positive zeros j(n)v,1 and
\sqrt{}
j
(n)
v,1 . This
guarantees that the above inequalities hold. This completes the proof of part (a) if \nu > n and
parts (b) and (c) if \nu > n - 1.
Now, to prove the statement for part (a) when \nu \in (n - 1, n), we use the counterpart of (3.2),
that is,
\mathrm{R}\mathrm{e}
\biggl(
z
\lambda - z
\biggr)
\geq - | z|
\lambda + | z|
, (3.3)
which holds for all z \in \BbbC and \lambda \in \BbbR are such that \lambda > | z| (see [3]). If in the inequality (3.3), we
replace z by z2 and \lambda by
\Bigl(
j
(n)
\nu ,m
\Bigr) 2
, it follows that
\mathrm{R}\mathrm{e}
\left( z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
\right) \geq - | z| 2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
+ | z| 2
,
provided that | z| < j
(n)
\nu ,1 . Thus, for n - 1 < \nu < n, we obtain
\mathrm{R}\mathrm{e}
\biggl(
zf \prime \nu ,n(z)
f\nu ,n(z)
\biggr)
= 1 - 1
\nu - n
\sum
m\geq 1
\mathrm{R}\mathrm{e}
\left( 2z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
\right) \geq
\geq 1 +
1
\nu - n
\sum
m\geq 1
2| z| 2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
+ | z| 2
=
i| z| f \prime \nu ,n(i| z| )
f\nu ,n(i| z| )
.
In this case equality is obtained when z = i| z| = ir. Also, the latter inequality tells us that
\mathrm{R}\mathrm{e}
\biggl(
zf \prime \nu ,n(z)
f\nu ,n(z)
\biggr)
> \beta
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1469
if and only if | z| < x
(n)
v,2 , where x(n)v,2 denotes the smallest positive root of the equations
irf \prime \nu ,n(ir)
f\nu ,n(ir)
= \beta ,
which is equivalent to
irJ
(n+1)
\nu (ir)
(\nu - n)J
(n)
\nu (ir)
- \beta = 0 or
rI
(n+1)
\nu (r)
(\nu - n)I
(n)
\nu (r)
- \beta = 0
for n - 1 < \nu < n. It follows from Lemma 2.5 that the first positive zero of z \mapsto \rightarrow irJ
(n+1)
\nu (ir) -
- \beta (\nu - n)J (n)
\nu (ir) cannot exceed j(n)\nu ,1 so the above inequalities are verified. So we would only need
to prove that the above function has actually only one zero in (0,\infty ). Note that, due to Lemma 2.2,
the function
r \mapsto \rightarrow irJ
(n+1)
\nu (ir)
J
(n)
\nu (ir)
=
Q1
Q2
,
where
Q1 =
\infty \sum
m=0
(2m - n+ \nu )\Gamma (2m+ \nu + 1)
m!22m+\nu \Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
r2m,
Q2 =
\infty \sum
m=0
\Gamma (2m+ \nu + 1)
m!22m+\nu \Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
r2m,
is increasing on (0,\infty ) as a quotient of two power series whose positive coefficients form the
increasing “quotient sequence” \{ 2m - n + \nu \} m\geq 0. On the other hand, the above function tends to
\nu - n when r \rightarrow 0, so that its graph can intersect the horizontal line y = \beta (\nu - n) > \nu - n only
once. Thus, proof for part (a) of the theorem is completed if \nu \in (n - 1, n).
Theorem 3.1 is proved.
With regards to Theorem 3.1, we tabulate the radius of starlikeness for f\nu ,n, g\nu ,n and h\nu ,n
for a fixed \nu = 2.5, n = 0, 1, 2, 3 and, respectively, \beta = 0 and \beta = 0.5. These are given in
Table 3.1. Also, in Table 3.1, we see that radius of starlikeness is decreasing according to the order
of derivative and the order of starlikeness. On the other words, from all these results we concluded
that r\ast \beta (f\nu ,0) > r\ast \beta (f\nu ,1) > r\ast \beta (f\nu ,2) > . . . > r\ast \beta (f\nu ,n) > . . . for \beta \in [0, 1) and \nu > n - 1, n \in \BbbN 0.
In addition to, we can write r\ast \beta 1
(f\nu ,n) < r\ast \beta 0
(f\nu ,n) for 0 \leq \beta 0 < \beta 1 < 1 and \nu > n - 1, n \in \BbbN 0.
Same inequalities is also true for r\ast \beta (g\nu ,n) and r\ast \beta (h\nu ,n).
For n = 0 in the Theorem 3.1 we obtain the results of Baricz et al. [3]. Our results is a common
generalization of these results.
Table 3.1
n
r\ast \beta (f2.5,n) r\ast \beta (g2.5,n) r\ast \beta (h2.5,n)
\beta = 0 \beta = 0.5 \beta = 0 \beta = 0.5 \beta = 0 \beta = 0.5
0 3.6328 2.7569 2.5011 1.8192 11.1696 6.2556
1 2.1056 1.5926 1.7975 1.3307 5.4265 3.2312
2 0.8512 0.6229 1.1285 0.8512 2.0284 1.2735
3 0.4586 0.3051 0.4819 0.3703 0.3543 0.2323
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1470 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
The second principal result we established concerns the radii of convexity and reads as follows.
Theorem 3.2. The following statements hold:
(a) If \nu > n and \beta \in [0, 1), then the radius rc\beta (f\nu ,n) is the smallest positive root of the equation
1 - \beta +
rJ
(n+2)
\nu (r)
J
(n+1)
\nu (r)
+
\biggl(
1
\nu - n
- 1
\biggr)
rJ
(n+1)
\nu (r)
J
(n)
\nu (r)
= 0.
Moreover, rc\beta (f\nu ,n) < j
(n+1)
\nu ,1 < j
(n)
\nu ,1 .
(b) If \nu > n - 1 and \beta \in [0, 1), then the radius rc\beta (g\nu ,n) is the smallest positive root of the
equation
n+ 1 - \nu - \beta +
(n - \nu + 2)rJ
(n+1)
\nu (r) + r2J
(n+2)
\nu (r)
(n - \nu + 1)J
(n)
\nu (r) + rJ
(n+1)
\nu (r)
= 0.
(c) If \nu > n - 1 and \beta \in [0, 1), then the radius rc\beta (h\nu ,n) is the smallest positive root of the
equation
n+ 2 - \nu - 2\beta
2
+
\surd
r
2
(n - \nu + 3)J
(n+1)
\nu (
\surd
r) +
\surd
rJ
(n+2)
\nu (
\surd
r)
(n - \nu + 2)J
(n)
\nu (
\surd
r) +
\surd
rJ
(n+1)
\nu (
\surd
r)
= 0.
Proof. (a) Since
1 +
zf \prime \prime \nu ,n(z)
f \prime \nu ,n(z)
= 1 +
zJ
(n+2)
\nu (z)
J
(n+1)
\nu (z)
+
\biggl(
1
\nu - n
- 1
\biggr)
zJ
(n+1)
\nu (z)
J
(n)
\nu (z)
and by means of (2.1) we have
zJ
(n+1)
\nu (z)
J
(n)
\nu (z)
= \nu - n -
\sum
m\geq 1
2z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
,
it follows that
1 +
zf \prime \prime \nu ,n(z)
f \prime \nu ,n(z)
= 1 -
\biggl(
1
\nu - n
- 1
\biggr) \sum
m\geq 1
2z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
-
\sum
m\geq 1
2z2\Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
- z2
.
Now, suppose that \nu \in (n, n + 1]. If we use the inequality (3.2), for all z \in \BbbD
j
(n)
\nu ,1
we get the
inequality
\mathrm{R}\mathrm{e}
\biggl(
1 +
zf \prime \prime \nu ,n(z)
f \prime \nu ,n(z)
\biggr)
= 1 -
\biggl(
1
\nu - n
- 1
\biggr) \sum
m\geq 1
\mathrm{R}\mathrm{e}
\left( 2z2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- z2
\right) -
-
\sum
m\geq 1
\mathrm{R}\mathrm{e}
\left( 2z2\Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
- z2
\right) \geq
\geq 1 -
\biggl(
1
\nu - n
- 1
\biggr) \sum
m\geq 1
2r2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- r2
-
\sum
m\geq 1
2r2\Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
- r2
,
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RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1471
where | z| = r. Also, observe that if we use the inequality [4] (Lemma 2.1)
\mu \mathrm{R}\mathrm{e}
\biggl(
z
a - z
\biggr)
- \mathrm{R}\mathrm{e}
\biggl(
z
b - z
\biggr)
\geq \mu
| z|
a - | z|
- | z|
b - | z|
,
where a > b > 0, \mu \in [0, 1] and z \in \BbbC such that | z| < b, then we get that the above inequality is
also valid when \nu > n + 1. Here we used that the zeros of the nth and (n + 1)th derivative of Jv
are interlacing according to Lemma 2.1. The above inequality implies for r \in (0, j
(n)
\nu ,1 )
\mathrm{i}\mathrm{n}\mathrm{f}
z\in \BbbD r
\biggl\{
\mathrm{R}\mathrm{e}
\biggl(
1 +
zf \prime \prime \nu ,n(z)
f \prime \nu ,n(z)
\biggr) \biggr\}
= 1 +
rf \prime \prime \nu ,n(r)
f \prime \nu ,n(r)
.
On the other hand, we define the function \varphi \nu ,n : (n, j(n)\nu ,1 ) \rightarrow \BbbR ,
\varphi \nu ,n(r) = 1 +
rf \prime \prime \nu ,n(r)
f \prime \nu ,n(r)
.
Since the zeros of the nth and (n + 1)th derivative of Jv are interlacing according to Lemma 2.1
and r < j
(n+1)
\nu ,1 < j
(n)
\nu ,1
\biggl(
or r <
\sqrt{}
j
(n)
\nu ,1 j
(n+1)
\nu ,1
\biggr)
for all \nu > n, we have
\Bigl(
j(n)\nu ,m
\Bigr) \biggl( \Bigl(
j(n+1)
\nu ,m
\Bigr) 2
- r2
\biggr)
-
\Bigl(
j(n+1)
\nu ,m
\Bigr) \biggl( \Bigl(
j(n)\nu ,m
\Bigr) 2
- r2
\biggr)
< 0.
Thus, the inequality
d\varphi \nu ,n(r)
dr
= -
\biggl(
1
\nu - n
- 1
\biggr) \sum
m\geq 1
4r
\Bigl(
j
(n)
\nu ,m
\Bigr) 2
\biggl( \Bigl(
j
(n)
\nu ,m
\Bigr) 2
- r2
\biggr) 2 -
\sum
m\geq 1
4r
\Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
\biggl( \Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
- r2
\biggr) 2 <
<
\sum
m\geq 1
4r
\Bigl(
j
(n)
\nu ,m
\Bigr) 2
\biggl( \Bigl(
j
(n)
\nu ,m
\Bigr) 2
- r2
\biggr) 2 -
\sum
m\geq 1
4r
\Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
\biggl( \Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
- r2
\biggr) 2 =
= 4r
\sum
m\geq 1
\Bigl(
j
(n)
\nu ,m
\Bigr) 2\biggl( \Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
- r2
\biggr) 2
-
\Bigl(
j
(n+1)
\nu ,m
\Bigr) 2\biggl( \Bigl(
j
(n)
\nu ,m
\Bigr) 2
- r2
\biggr) 2
\biggl( \Bigl(
j
(n)
\nu ,m
\Bigr) 2
- r2
\biggr) 2\biggl( \Bigl(
j
(n+1)
\nu ,m
\Bigr) 2
- r2
\biggr) 2 < 0
is satisfied. Consequently, the function \varphi \nu ,n is strictly decreasing. Observe also that \mathrm{l}\mathrm{i}\mathrm{m}r\searrow 0 \varphi \nu ,n(r) =
= 1 > \beta and \mathrm{l}\mathrm{i}\mathrm{m}
r\nearrow j
(n)
\nu ,1
\varphi \nu ,n(r) = - \infty , which means that for z \in \BbbD r1 we have
\mathrm{R}\mathrm{e}
\biggl(
1 +
zf \prime \prime \nu ,n(z)
f \prime \nu ,n(z)
\biggr)
> \beta
if and only if r1 is the unique root of
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1472 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
1 +
rf \prime \prime \nu ,n(r)
f \prime \nu ,n(r)
= \beta ,
situated in (0, j
(n)
\nu ,1 ).
(b) Observe that
1 +
zg\prime \prime \nu ,n(z)
g\prime \nu ,n(z)
= (n - \nu + 1) +
(n - \nu + 2)zJ
(n+1)
\nu (z) + z2J
(n+2)
\nu (z)
(n - \nu + 1)J
(n)
\nu (z) + zJ
(n+1)
\nu (z)
.
By using (1.1) and (2.1), we have
g\prime \nu ,n(z) = 2\nu \Gamma (\nu - n+ 1)zn - \nu
\Bigl[
(n - \nu + 1)J (n)
\nu (z) + zJ (n+1)
\nu (z)
\Bigr]
=
=
\infty \sum
m=0
( - 1)m(2m+ 1)\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)
m!\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
\Bigl( z
2
\Bigr) 2m
(3.4)
and
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
m \mathrm{l}\mathrm{o}\mathrm{g}m
\lambda (m,n, \nu )
=
1
2
,
where
\lambda (m,n, \nu ) = [2m \mathrm{l}\mathrm{o}\mathrm{g} 2 + \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (m+ 1) + \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (2m - n+ \nu + 1) + \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (m+ \nu + 1) -
- \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (2m+ \nu + 1) - \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (\nu - n+ 1) - \mathrm{l}\mathrm{o}\mathrm{g}(2m+ 1)].
Here, we used m! = \Gamma (m + 1) and \mathrm{l}\mathrm{i}\mathrm{m}m\rightarrow \infty
\mathrm{l}\mathrm{o}\mathrm{g} \Gamma (am+ b)
m \mathrm{l}\mathrm{o}\mathrm{g}m
= a, where a and b are positive
constants. So, by applying Hadamard’s theorem [17, p. 26], we can write the infinite product repre-
sentation of g\prime \nu ,n(z) as follows:
g\prime \nu ,n(z) =
\prod
m\geq 1
\left( 1 - z2\Bigl(
\gamma
(n)
\nu ,m
\Bigr) 2
\right) , (3.5)
where \gamma (n)\nu ,m denotes the mth positive zero of the function g\prime \nu ,n. From Lemma 2.5 for \nu > n - 1 the
function g\prime \nu ,n \in \scrL \scrP , and the smallest positive zero of g\prime \nu ,n does not exceed the first positive zero
of J (n)
\nu .
By means of (3.5) we have
1 +
zg\prime \prime \nu ,n(z)
g\prime \nu ,n(z)
= 1 -
\sum
m\geq 1
2z2\Bigl(
\gamma
(n)
\nu ,m
\Bigr) 2
- z2
.
If we use the inequality (3.2), for all z \in \BbbD
\gamma
(n)
\nu ,m
, we get the inequality
\mathrm{R}\mathrm{e}
\biggl(
1 +
zg\prime \prime \nu ,n(z)
g\prime \nu ,n(z)
\biggr)
\geq 1 -
\sum
m\geq 1
2r2\Bigl(
\gamma
(n)
\nu ,m
\Bigr) 2
- r2
,
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RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1473
where | z| = r. Thus, for r \in (0, \gamma
(n)
\nu ,1 ), we have
\mathrm{i}\mathrm{n}\mathrm{f}
z\in \BbbD r
\biggl\{
\mathrm{R}\mathrm{e}
\biggl(
1 +
zg\prime \prime \nu ,n(z)
g\prime \nu ,n(z)
\biggr) \biggr\}
= 1 +
rg\prime \prime \nu ,n(r)
g\prime \nu ,n(r)
.
The function G\nu ,n : (0, \gamma (n)\nu ,1 ) \rightarrow \BbbR , defined by
G\nu ,n(r) = 1 +
rg\prime \prime \nu ,n(r)
g\prime \nu ,n(r)
,
is strictly decreasing and \mathrm{l}\mathrm{i}\mathrm{m}r\searrow 0G\nu ,n(r) = 1 > \beta and \mathrm{l}\mathrm{i}\mathrm{m}
r\nearrow \gamma
(n)
\nu ,1
G\nu ,n(r) = - \infty . Herewith, in view
of the minimum principle for harmonic functions for z \in \BbbD r2 , we get that
\mathrm{R}\mathrm{e}
\biggl(
1 +
zg\prime \prime \nu ,n(z)
g\prime \nu ,n(z)
\biggr)
> \beta
if and only if r2 is the unique root of
1 +
rg\prime \prime \nu ,n(r)
g\prime \nu ,n(r)
= \beta ,
situated in (0, \gamma
(n)
\nu ,1 ).
(c) Observe that
1 +
zh\prime \prime \nu ,n(z)
h\prime \nu ,n(z)
=
n - \nu + 2
2
+
\surd
z
2
(n - \nu + 3)J
(n+1)
\nu (
\surd
z) +
\surd
zJ
(n+2)
\nu (
\surd
z)
(n - \nu + 2)J
(n)
\nu (
\surd
z) +
\surd
zJ
(n+1)
\nu (
\surd
z)
.
By using (1.1) and (2.1), we have
h\prime \nu ,n(z) = 2\nu - 1\Gamma (\nu - n+ 1)z
n - \nu
2
\Bigl[
(n - \nu + 2)J (n)
\nu (
\surd
z) +
\surd
zJ (n+1)
\nu (
\surd
z)
\Bigr]
=
=
\infty \sum
m=0
( - 1)m(m+ 1)\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)
m!\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
\Bigl( z
4
\Bigr) m
(3.6)
and
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
m \mathrm{l}\mathrm{o}\mathrm{g}m
\tau (m,n, \nu )
=
1
2
,
where
\tau (m,n, \nu ) = [2m \mathrm{l}\mathrm{o}\mathrm{g} 2 + \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (m+ 1) + \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (2m - n+ \nu + 1) + \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (m+ \nu + 1) -
- \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (2m+ \nu + 1) - \mathrm{l}\mathrm{o}\mathrm{g} \Gamma (\nu - n+ 1) - \mathrm{l}\mathrm{o}\mathrm{g}(m+ 1)].
So, by applying Hadamard’s theorem [17, p. 26] we can write the infinite product representation of
h\prime \nu ,n(z) as follows:
h\prime \nu ,n(z) =
\prod
m\geq 1
\Biggl(
1 - z
\delta
(n)
\nu ,m
\Biggr)
, (3.7)
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1474 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
where \delta (n)\nu ,m denotes the mth positive zero of the function h\prime \nu ,n. From Lemma 2.5 for \nu > n - 1 the
function h\prime \nu ,n \in \scrL \scrP , and the smallest positive zero of h\prime \nu ,n does not exceed the first positive zero
of J (n)
\nu .
By means of (3.5) we have
1 +
zh\prime \prime \nu ,n(z)
h\prime \nu ,n(z)
= 1 -
\sum
m\geq 1
z
\delta
(n)
\nu ,m - z
.
By using the inequality (3.2), for all z \in \BbbD
\delta
(n)
\nu ,m
, we get the inequality
\mathrm{R}\mathrm{e}
\biggl(
1 +
zh\prime \prime \nu ,n(z)
h\prime \nu ,n(z)
\biggr)
\geq 1 -
\sum
m\geq 1
r
\delta
(n)
\nu ,m - r
,
where | z| = r. Thus, for r \in
\bigl(
0, \delta
(n)
\nu ,1
\bigr)
, we have
\mathrm{i}\mathrm{n}\mathrm{f}
z\in \BbbD r
\biggl\{
\mathrm{R}\mathrm{e}
\biggl(
1 +
zh\prime \prime \nu ,n(z)
h\prime \nu ,n(z)
\biggr) \biggr\}
= 1 +
rh\prime \prime \nu ,n(r)
h\prime \nu ,n(r)
.
The function H\nu ,n :
\bigl(
0, \delta
(n)
\nu ,1
\bigr)
\rightarrow \BbbR , defined by
H\nu ,n(r) = 1 +
rh\prime \prime \nu ,n(r)
h\prime \nu ,n(r)
,
is strictly decreasing and \mathrm{l}\mathrm{i}\mathrm{m}r\searrow 0H\nu ,n(r) = 1 > \beta and \mathrm{l}\mathrm{i}\mathrm{m}
r\nearrow \delta
(n)
\nu ,1
H\nu ,n(r) = - \infty . As a result, in
view of the minimum principle for harmonic functions for z \in \BbbD r3 we obtain that
\mathrm{R}\mathrm{e}
\biggl(
1 +
zh\prime \prime \nu ,n(z)
h\prime \nu ,n(z)
\biggr)
> \beta
if and only if r3 is the unique root of
1 +
rh\prime \prime \nu ,n(r)
h\prime \nu ,n(r)
= \beta ,
situated in
\bigl(
0, \delta
(n)
\nu ,1
\bigr)
.
Theorem 3.2 is proved.
With regards to Theorem 3.2, we tabulate the radius of convexity for f\nu ,n, g\nu ,n and h\nu ,n for a
fixed \nu = 3.5, n = 0, 1, 2, 3 and, respectively, \beta = 0 and \beta = 0.5. These are given in Table 3.2.
Also, in Table 3.2, we see that radius of convexity is decreasing according to the order of derivative
and the order of convexity. On the other words, from all these results we concluded that rc\beta (f\nu ,0) >
> rc\beta (f\nu ,1) > rc\beta (f\nu ,2) > . . . > rc\beta (f\nu ,n) > . . . for \beta \in [0, 1) and \nu > n - 1, n \in \BbbN 0. In addition to,
we can write rc\beta 1
(f\nu ,n) < rc\beta 0
(f\nu ,n) for 0 \leq \beta 0 < \beta 1 < 1 and \nu > n - 1, n \in \BbbN 0. Same inequalities
is also true for rc\beta (g\nu ,n) and rc\beta (h\nu ,n).
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RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1475
Table 3.2
n
rc\beta (f3.5,n) rc\beta (g3.5,n) rc\beta (h3.5,n)
\beta = 0 \beta = 0.5 \beta = 0 \beta = 0.5 \beta = 0 \beta = 0.5
0 2.7183 2.0865 0.5234 1.1461 6.2189 3.7194
1 1.8179 1.3998 1.2017 0.9084 3.7394 2.2873
2 1.0592 0.8123 0.8833 0.6715 1.9450 1.2190
3 0.4141 0.3131 0.5683 0.4350 0.7726 0.4968
For n = 0 in the Theorem 3.2 we obtain the results of Baricz and Szász [4]. Our results is a
common generalization of these results.
3.2. Bounds for radii of starlikeness and convexity of the functions \bfitg \bfitnu ,\bfitn and \bfith \bfitnu ,\bfitn . In this
subsection, we consider two different functions g\nu ,n and h\nu ,n which are normalized forms of the
Bessel function derivatives of the first kind given by (1.1). Here, firstly, our aim is to show that the
radii of univalence of these functions correspond to the radii of starlikeness.
Theorem 3.3. The following inequalities hold:
(a) If \nu > n - 1, then r\ast (g\nu ,n) satisfies the inequalities
r\ast (g\nu ,n) <
\surd
2
\sqrt{}
a - 1
\nu ,n,
2
\surd
3
3
\sqrt{}
a - 1
\nu ,n < r\ast (g\nu ,n) < 2
\surd
3
\sqrt{}
1
9a\nu ,n - 5b\nu ,n
.
(b) If \nu > n - 1, then r\ast (h\nu ,n) satisfies the inequalities
r\ast (h\nu ,n) < 2a - 1
\nu ,n,
2a - 1
\nu ,n < r\ast (h\nu ,n) <
8
4a\nu ,n - 3b\nu ,n
,
where a\nu ,n =
\nu + 2
(\nu - n+ 2)(\nu - n+ 1)
and b\nu ,n =
(\nu + 4)(\nu + 3)
(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
.
Proof. (a) By using the first Rayleigh sum (2.6) and the implict relation for r\ast (g\nu ,n), obtained
by Kreyszing and Todd [16], we get, for all \nu > n - 1,
1
(r\ast (g\nu ,n))
2 =
\sum
m\geq 1
2\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- (r\ast (g\nu ,n))
2
>
\sum
m\geq 1
2\Bigl(
j
(n)
\nu ,m
\Bigr) 2 =
\nu + 2
2(\nu - n+ 2)(\nu - n+ 1)
.
Now, by using the Euler – Rayleigh inequalities it is possible to have more tight bounds for the radius
of univalence (and starlikeness) r\ast (g\nu ,n). We define the function \Psi \nu ,n(z) = g\prime \nu ,n(z), where g\prime \nu ,n
defined by (3.5). Now, taking logarithmic derivative of both sides of (3.5) for | z| < \gamma
(n)
\nu ,1 , we have
\Psi \prime
\nu ,n(z)
\Psi \nu ,n(z)
= -
\sum
m\geq 1
2z\Bigl(
\gamma
(n)
\nu ,m
\Bigr) 2
- z2
=
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1476 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
= - 2
\sum
m\geq 1
\sum
k\geq 0
1\Bigl(
\gamma
(n)
\nu ,m
\Bigr) 2(k+1)
z2k+1 = - 2
\sum
k\geq 0
\sigma k+1z
2k+1, (3.8)
where \sigma k =
\sum
m\geq 1
\Bigl(
\gamma (n)\nu ,m
\Bigr) - k
is Euler – Rayleigh sum for the zeros of \Psi \nu ,n. Also, using (3.4) from
the infinite sum representation of \Psi \nu ,n, we obtain
\Psi \prime
\nu ,n(z)
\Psi \nu ,n(z)
=
\sum
m\geq 0
Umz
2m+1\sum
m\geq 0
Vmz
2m
, (3.9)
where
Um =
2( - 1)m+1\Gamma (2m+ \nu + 3)\Gamma (\nu - n+ 1)(2m+ 3)
m!4m+1\Gamma (2m - n+ \nu + 3)\Gamma (m+ \nu + 2)
and
Vm =
( - 1)m\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)(2m+ 1)
m!4m\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
.
By comparing the coefficients with the same degrees of (3.8) and (3.9), we obtain the Euler – Rayleigh
sums
\sigma 1 =
3(\nu + 2)
4(\nu - n+ 2)(\nu - n+ 1)
and
\sigma 2 =
3(\nu + 2)
16(\nu - n+ 2)(\nu - n+ 1)
\times
\times
\biggl(
3(\nu + 2)
(\nu - n+ 2)(\nu - n+ 1)
- 5(\nu + 4)(\nu + 3)
3(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
\biggr)
.
By using the Euler – Rayleigh inequalities
\sigma
- 1
k
k <
\Bigl(
\gamma
(n)
\nu ,1
\Bigr) 2
<
\sigma k
\sigma k+1
for \nu > n - 1, k \in \BbbN and k = 1, we get the following inequality:
4(\nu - n+ 2)(\nu - n+ 1)
3(\nu + 2)
< (r\ast (g\nu ,n))
2 <
<
4
3(\nu + 2)
(\nu - n+ 2)(\nu - n+ 1)
- 5(\nu + 4)(\nu + 3)
3(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
,
and it is possible to have more tighter bounds for other values of k \in \BbbN .
(b) By using the first Rayleigh sum (2.6) and the implict relation for r\ast (h\nu ,n), obtained by
Kreyszing and Todd [16], we get, for all \nu > n - 1,
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RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1477
1
r\ast (h\nu ,n)
=
\sum
m\geq 1
1\Bigl(
j
(n)
\nu ,m
\Bigr) 2
- r\ast (h\nu ,n)
>
\sum
m\geq 1
1\Bigl(
j
(n)
\nu ,m
\Bigr) 2 =
\nu + 2
2(\nu - n+ 2)(\nu - n+ 1)
.
Now, by using the Euler – Rayleigh inequalities it is possible to have more tight bounds for the radius
of univalence (and starlikeness) r\ast (h\nu ,n). We define the function \Phi \nu ,n(z) = h\prime \nu ,n(z), where h\prime \nu ,n
defined by (3.6) or (3.7). Now, taking logarithmic derivative of both sides of (3.7), we have
\Phi \prime
\nu ,n(z)
\Phi \nu ,n(z)
= -
\sum
m\geq 1
1
\delta
(n)
\nu ,m - z
= -
\sum
m\geq 1
\sum
k\geq 0
1\Bigl(
\delta
(n)
\nu ,m
\Bigr) k+1
zk =
= -
\sum
k\geq 0
\rho k+1z
k, | z| < \delta
(n)
\nu ,1 , (3.10)
where \rho k =
\sum
m\geq 1
\Bigl(
\delta (n)\nu ,m
\Bigr) - k
is Euler – Rayleigh sum for the zeros of \Phi \nu ,n. Also, using (3.6) from
the infinite sum representation of \Phi \nu ,n, we obtain
\Phi \prime
\nu ,n(z)
\Phi \nu ,n(z)
=
\sum
m\geq 0
Kmz
m\sum
m\geq 0
Lmz
m
, (3.11)
where
Km =
( - 1)m+1\Gamma (2m+ \nu + 3)\Gamma (\nu - n+ 1)(m+ 2)
m!4m+1\Gamma (2m - n+ \nu + 3)\Gamma (m+ \nu + 2)
and
Lm =
( - 1)m\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)(m+ 1)
m!4m\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
.
By comparing the coefficients with the same degrees of (3.10) and (3.11), we get the Euler – Rayleigh
sums
\rho 1 =
\nu + 2
2(\nu - n+ 2)(\nu - n+ 1)
and
\rho 2 =
\nu + 2
4(\nu - n+ 2)(\nu - n+ 1)
\times
\times
\biggl(
\nu + 2
(\nu - n+ 2)(\nu - n+ 1)
- 3(\nu + 4)(\nu + 3)
4(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
\biggr)
.
If we use the Euler – Rayleigh inequalities
\rho
- 1
k
k < \delta
(n)
\nu ,1 <
\rho k
\rho k+1
for \nu > n - 1, k \in \BbbN and k = 1, then we obtain the following inequality:
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1478 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
2(\nu - n+ 2)(\nu - n+ 1)
\nu + 2
< r\ast (h\nu ,n) <
<
2
\nu + 2
(\nu - n+ 2)(\nu - n+ 1)
- 3(\nu + 4)(\nu + 3)
4(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
and it is possible to have more tighter bounds for other values of k \in \BbbN .
Theorem 3.3 is proved.
If we take n = 0 in the Theorem 3.3 we obtain the results of Aktaş et al. [1]. Our results is
a common generalization of these results. For special cases of parameters \nu and n, Theorem 3.3
reduces tight lower and upper bounds for the radii of starlikeness and convexity of many elemanter
functions. For example, for \nu =
3
2
and n = 2 in Theorem 3.3, we have
\sqrt{}
2
7
< r\ast
\Bigl(
g 3
2
,2(z) = 4 \mathrm{s}\mathrm{i}\mathrm{n} z - 4z \mathrm{c}\mathrm{o}\mathrm{s} z
\Bigr)
<
\sqrt{}
3
7
and
3
7
< r\ast
\Bigl(
h 3
2
,2(z) = 4
\surd
z \mathrm{s}\mathrm{i}\mathrm{n}
\surd
z - 4z \mathrm{c}\mathrm{o}\mathrm{s}
\surd
z
\Bigr)
<
2940
5969
.
The next result concerning bounds for radii of convexity of functions g\nu ,n and h\nu ,n.
Theorem 3.4. The following statements hold:
(a) If \nu > n - 1, then rc(g\nu ,n) satisfies the inequalities
2
3
\sqrt{}
a - 1
\nu ,n < rc(g\nu ,n) < 6
\sqrt{}
1
81a\nu ,n - 25b\nu ,n
.
(b) If \nu > n - 1, then rc(h\nu ,n) satisfies the inequalities
a - 1
\nu ,n < rc(h\nu ,n) <
16
16a\nu ,n - 9b\nu ,n
,
where a\nu ,n and b\nu ,n given by in Theorem 3.3.
Proof. (a) By using the Alexander duality theorem for starlike and convex functions we can
say that the function g\nu ,n(z) is convex if and only if zg\prime \nu ,n(z) is starlike. But, the smallest positive
zero of z \mapsto \rightarrow z(zg\prime \nu ,n(z))
\prime is actually the radius of starlikeness of z \mapsto \rightarrow (zg\prime \nu ,n(z)), according to
Theorems 3.1 and 3.2. Therefore, the radius of convexity rc(g\nu ,n) is the smallest positive root of the
equation (zg\prime \nu ,n(z))
\prime = 0. Therefore, from (3.4), we have
\Delta \nu ,n(z) = (zg\prime \nu ,n(z))
\prime =
\infty \sum
m=0
( - 1)m(2m+ 1)2\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)
m!4m\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
z2m.
Since the function g\nu ,n(z) belongs to the Laguerre – Pólya class of entire functions and \scrL \scrP is
closed under differentiation, we can say that the function \Delta \nu ,n(z) \in \scrL \scrP . Therefore, the zeros of the
function \Delta \nu ,n are all real. Suppose that d(n)\nu ,m are the zeros of the function \Delta \nu ,n. Then the function
\Delta \nu ,n has the infinite product representation as follows:
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RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1479
\Delta \nu ,n(z) =
\prod
m\geq 1
\left( 1 - z2\Bigl(
d
(n)
\nu ,m
\Bigr) 2
\right) . (3.12)
By taking the logarithmic derivative of (3.12), we get
\Delta \prime
\nu ,n(z)
\Delta \nu ,n(z)
= - 2
\sum
m\geq 1
z\Bigl(
d
(n)
\nu ,m
\Bigr) 2
- z2
=
= - 2
\sum
m\geq 1
\sum
k\geq 0
1\Bigl(
d
(n)
\nu ,m
\Bigr) 2(k+1)
z2k+1 = - 2
\sum
k\geq 0
\kappa k+1z
2k+1, | z| < d
(n)
\nu ,1 , (3.13)
where \kappa k =
\sum
m\geq 1
\Bigl(
d(n)\nu ,m
\Bigr) - k
is Euler – Rayleigh sum for the zeros of \Delta \nu ,n. On the other hand, by
considering infinite sum representation of \Delta \nu ,n(z), we obtain
\Delta \prime
\nu ,n(z)
\Delta \nu ,n(z)
=
\sum
m\geq 0
Xmz
2m+1\sum
m\geq 0
Ymz
2m
, (3.14)
where
Xm =
2( - 1)m+1\Gamma (2m+ \nu + 3)\Gamma (\nu - n+ 1)(2m+ 3)2
m!4m+1\Gamma (2m - n+ \nu + 3)\Gamma (m+ \nu + 2)
and
Ym =
( - 1)m\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)(2m+ 1)2
m!4m\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
.
By comparing the coefficients of (3.13) and (3.14), we have
\kappa 1 =
9(\nu + 2)
4(\nu - n+ 2)(\nu - n+ 1)
and
\kappa 2 =
9(\nu + 2)
16(\nu - n+ 2)(\nu - n+ 1)
\times
\times
\biggl(
9(\nu + 2)
(\nu - n+ 2)(\nu - n+ 1)
- 25(\nu + 4)(\nu + 3)
9(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
\biggr)
.
By using the Euler – Rayleigh inequalities
\kappa
- 1
k
k <
\Bigl(
d
(n)
\nu ,1
\Bigr) 2
<
\kappa k
\kappa k+1
for \nu > n - 1, k \in \BbbN and k = 1, we get the inequality
4(\nu - n+ 2)(\nu - n+ 1)
9(\nu + 2)
< (rc(g\nu ,n))
2 <
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1480 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
<
4
9(\nu + 2)
(\nu - n+ 2)(\nu - n+ 1)
- 25(\nu + 4)(\nu + 3)
9(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
and it is possible to have more tighter bounds for other values of k \in \BbbN .
(b) By using the same procedure as in the previous proof, we can say that the radius of convexity
rc(h\nu ,n) is the smallest positive root of the equation
\bigl(
zh\prime \nu ,n(z)
\bigr) \prime
= 0 according to Theorem 3.2. From
(3.6), we get
\Theta \nu ,n(z) =
\bigl(
zh\prime \nu ,n(z)
\bigr) \prime
=
\infty \sum
m=0
( - 1)m(m+ 1)2\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)
m!4m\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
zm. (3.15)
Moreover, we know h\nu ,n(z) belongs to the Laguerre – Pólya class of entire functions and \scrL \scrP , con-
sequently, \Theta \nu ,n(z) \in \scrL \scrP . On the other words, the zeros of the function \Theta \nu ,n are all real. Assume
that l(n)\nu ,m are the zeros of the function \Theta \nu ,n. In this case, the function \Theta \nu ,n has the infinite product
representation as follows:
\Theta \nu ,n(z) =
\prod
m\geq 1
\left( 1 - z2\Bigl(
l
(n)
\nu ,m
\Bigr) 2
\right) . (3.16)
By taking the logarithmic derivative of both sides of (3.16) for | z| < l
(n)
\nu ,1 , we have
\Theta \prime
\nu ,n(z)
\Theta \nu ,n(z)
= -
\sum
m\geq 1
1
l
(n)
\nu ,m - z
= -
\sum
m\geq 1
\sum
k\geq 0
1\Bigl(
l
(n)
\nu ,m
\Bigr) k+1
zk = -
\sum
k\geq 0
\omega k+1z
k, (3.17)
where \omega k =
\sum
m\geq 1
\Bigl(
l(n)\nu ,m
\Bigr) - k
. In addition, by using the derivative of infinite sum representation
considering infinite sum representation of (3.15), we obtain
\Theta \prime
\nu ,n(z)
\Theta \nu ,n(z)
=
\sum
m\geq 0
Tmz
m\diagup
\sum
m\geq 0
Smz
m, (3.18)
where
Tm =
( - 1)m+1\Gamma (2m+ \nu + 3)\Gamma (\nu - n+ 1)(m+ 2)2
m!4m+1\Gamma (2m - n+ \nu + 3)\Gamma (m+ \nu + 2)
and
Sm =
( - 1)m\Gamma (2m+ \nu + 1)\Gamma (\nu - n+ 1)(m+ 1)2
m!4m\Gamma (2m - n+ \nu + 1)\Gamma (m+ \nu + 1)
.
By comparing the coefficients of (3.17) and (3.18), we get
\omega 1 =
\nu + 2
(\nu - n+ 2)(\nu - n+ 1)
and
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
RADII OF STARLIKENESS AND CONVEXITY OF BESSEL FUNCTION DERIVATIVES 1481
\omega 2 =
\nu + 2
(\nu - n+ 2)(\nu - n+ 1)
\times
\times
\biggl(
\nu + 2
(\nu - n+ 2)(\nu - n+ 1)
- 9(\nu + 4)(\nu + 3)
16(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
\biggr)
.
By using the Euler – Rayleigh inequalities
\omega
- 1
k
k < l
(n)
\nu ,1 <
\omega k
\omega k+1
for \nu > n - 1, k \in \BbbN and k = 1, we have the following inequality:
(\nu - n+ 2)(\nu - n+ 1)
\nu + 2
< rc(h\nu ,n) <
<
1
\nu + 2
(\nu - n+ 2)(\nu - n+ 1)
- 9(\nu + 4)(\nu + 3)
16(\nu - n+ 4)(\nu - n+ 3)(\nu + 2)
and it is possible to have more tighter bounds for other values of k \in \BbbN .
Theorem 3.4 is proved.
If we take n = 0 in the Theorem 3.4 we obtain the results of Aktaş et al. [2]. For special cases
n = 1, 2, 3, we obtain following result.
Corollary 3.1. The following inequalities hold:
2
3
\sqrt{}
a - 1
\nu ,1 < rc(g\nu ,1) < 6
\sqrt{}
1
81a\nu ,1 - 25b\nu ,1
, \nu > 0,
a - 1
\nu ,1 < rc(h\nu ,1) <
16
16a\nu ,1 - 9b\nu ,1
, \nu > 0,
2
3
\sqrt{}
a - 1
\nu ,2 < rc(g\nu ,2) < 6
\sqrt{}
1
81a\nu ,2 - 25b\nu ,2
, \nu > 1,
a - 1
\nu ,2 < rc(h\nu ,2) <
16
16a\nu ,2 - 9b\nu ,2
, \nu > 1,
2
3
\sqrt{}
a - 1
\nu ,3 < rc(g\nu ,3) < 6
\sqrt{}
1
81a\nu ,3 - 25b\nu ,3
, \nu > 2,
a - 1
\nu ,3 < rc(h\nu ,3) <
16
16a\nu ,3 - 9b\nu ,3
, \nu > 2,
where a\nu ,n and b\nu ,n for n = 1, 2, 3 given by in Theorem 3.3.
References
1. İ. Aktaş, Á. Baricz, N. Yağmur, Bounds for the radii of univalence of some special functions, Math. Inequal. Appl.,
20, 825 – 843 (2017).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1482 E. DENIZ, S. KAZIMOĞLU, M. ÇAĞLAR
2. İ Aktaş, Á. Baricz, H. Orhan, Bounds for radii of starlikeness and convexity of some special functions, Turkish J.
Math., 42, 211 – 226 (2018).
3. Á. Baricz, P. A. Kupán, R. Szász, The radius of starlikeness of normalized Bessel functions of the first kind, Proc.
Amer. Math. Soc., 142, № 5, 2019 – 2025 (2011).
4. Á. Baricz, R. Szász, The radius of convexity of normalized Bessel functions of the first kind, Anal. Appl., 12, № 5,
485 – 509 (2014).
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6. Á. Baricz, D. K. Dimitrov, H. Orhan, N. Yağmur, Radii of starlikeness of some special functions, Proc. Amer. Math.
Soc., 144, № 8, 3355 – 3367 (2016).
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Method Funct. Theory, 16, № 1, 93 – 103 (2016).
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209 – 222 (2018).
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Received 21.07.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
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| id | umjimathkievua-article-1014 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:21Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a9/fba0cd11efeb05496f85f2d3b13e6ea9.pdf |
| spelling | umjimathkievua-article-10142025-03-31T08:46:33Z Radii of starlikeness and convexity of Bessel function derivatives Radii of starlikeness and convexity of Bessel function derivatives Deniz, E. Kazımoğlu, S. Çağlar, M. Deniz, E. Kazımoğlu, S. Çağlar, M. S. Normalized Bessel functions of the first kind convex function starlike function zeros of Bessel function derivative Radius Normalized Bessel functions of the first kind , convex function starlike function zeros of Bessel function derivative Radius UDC 517.5 In this paper, our aim is to find the radii of starlikeness andconvexity of Bessel function derivatives for three different kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for $n$th derivative of Bessel function andproperties of real zeros of it. In addition, by using the Euler–Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized $n$th derivative of Bessel function. The main results of the paper are natural extensions of some known results on classical Bessel functions of the first kind. &nbsp; УДК 517.5 Радiуси зiрчастостi та опуклостi похiдних функцiї Бесселя Знайдено радіуси зірчастості та опуклості похідних функції Бесселя для трьох різних видів нормалізації. Ключовими інструментами доведення основних результатів є розклад Міттаг-Леффлера для $n$-ї похідної функції Бесселя та властивості його дійсних нулів. Крім того, за допомогою нерівностей Ейлера–Релея отримано деякі точні нижні й верхні межі для радіусів зірчастості та опуклості нульового порядку для нормованої $n$-ї похідної функції Бесселя. Основними результатами роботи є природні розширення деяких відомих результатів щодо класичних функцій Бесселя першого роду. Institute of Mathematics, NAS of Ukraine 2021-11-23 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1014 10.37863/umzh.v73i11.1014 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 11 (2021); 1461 - 1482 Український математичний журнал; Том 73 № 11 (2021); 1461 - 1482 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1014/9145 Copyright (c) 2021 E. Deniz, S. Kazımoglu, M. Çağlar |
| spellingShingle | Deniz, E. Kazımoğlu, S. Çağlar, M. Deniz, E. Kazımoğlu, S. Çağlar, M. S. Radii of starlikeness and convexity of Bessel function derivatives |
| title | Radii of starlikeness and convexity of Bessel function derivatives |
| title_alt | Radii of starlikeness and convexity of Bessel function derivatives |
| title_full | Radii of starlikeness and convexity of Bessel function derivatives |
| title_fullStr | Radii of starlikeness and convexity of Bessel function derivatives |
| title_full_unstemmed | Radii of starlikeness and convexity of Bessel function derivatives |
| title_short | Radii of starlikeness and convexity of Bessel function derivatives |
| title_sort | radii of starlikeness and convexity of bessel function derivatives |
| topic_facet | Normalized Bessel functions of the first kind convex function starlike function zeros of Bessel function derivative Radius Normalized Bessel functions of the first kind convex function starlike function zeros of Bessel function derivative Radius |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1014 |
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