A note on the coefficient estimates for some classes of $p$ -valent functions
We obtain estimates of the Taylor – Maclaurin coefficients of some classes of p-valent functions. This problem was initially studied by Aouf in the paper “Coefficient estimates for some classes of p-valent functions” (Internat. J. Math. and Math. Sci. – 1988. – 11. – P. 47 – 54). The proof given by...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507382962978816 |
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| author | Sharma, N. L. Шарма, Н. Л. |
| author_facet | Sharma, N. L. Шарма, Н. Л. |
| author_sort | Sharma, N. L. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:19:04Z |
| description | We obtain estimates of the Taylor – Maclaurin coefficients of some classes of p-valent functions. This problem was initially
studied by Aouf in the paper “Coefficient estimates for some classes of p-valent functions” (Internat. J. Math. and Math.
Sci. – 1988. – 11. – P. 47 – 54). The proof given by Aouf was found to be partially erroneous. We propose the correct proof
of this result. |
| first_indexed | 2026-03-24T02:08:26Z |
| format | Article |
| fulltext |
UDC 517.5
N. L. Sharma (Indian Institute of Technology Indore)
A NOTE ON THE COEFFICIENT ESTIMATES
FOR SOME CLASSES OF \bfitp -VALENT FUNCTIONS*
ЗАУВАЖЕННЯ ЩОДО КОЕФIЦIЄНТНИХ ОЦIНОК
ДЛЯ ДЕЯКИХ КЛАСIВ \bfitp -ВАЛЕНТНИХ ФУНКЦIЙ
We obtain estimates of the Taylor – Maclaurin coefficients of some classes of p-valent functions. This problem was initially
studied by Aouf in the paper “Coefficient estimates for some classes of p-valent functions” (Internat. J. Math. and Math.
Sci. – 1988. – 11. – P. 47 – 54). The proof given by Aouf was found to be partially erroneous. We propose the correct proof
of this result.
Отримано оцiнки для коефiцiєнтiв Тейлора – Маклорена для деяких класiв p-валентних функцiй. Ця задача була
вперше розглянута Ауфом у роботi “Coefficient estimates for some classes of p-valent functions” (Internat. J. Math. and
Math. Sci. – 1988. – 11. – P. 47 – 54). Доведення, наведене Ауфом, виявилось частково помилковим. Ми пропонуємо
коректне доведення цього результату.
1. Introduction. The concept of univalence has a natural extension as described in p-valent function
theory. A functions
f(z) =
\infty \sum
n=1
anz
n (1.1)
is said to be p-valent in the open unit disk \BbbD := \{ z \in \BbbC : | z| < 1\} . if it is analytic and assumes no
value more than p times in \BbbD and there is some w such that f(z) = w has exactly p solutions in
\BbbD , when roots are counted in accordance with their multiplicities. We let \scrS p denote the class of all
functions that are analytic and p-valent in \BbbD .
By definition, the function f is said to be p-valent (or multivalent of order p) in \BbbD if
f(z1) = f(z2) = . . . = f(zp+1), z1, z2, . . . , zp+1 \in \BbbD ,
imply that zr = zs for some pair such that r \not = s, and if there is some w such that the equation
f(z) = w has p roots (counted in accordance with their multiplicities) in \BbbD . For example, f(z) = z2
is a 2-valent in \BbbD .
Let \scrS \ast
p denote the class of functions, which are analytic and p-valent starlike in \BbbD . A function
f \in \scrS p is said to be p-valent starlike functions in \BbbD , if there exists a \rho > 0 such that for \rho < | z| < 1,
\mathrm{R}\mathrm{e}
\biggl\{
zf \prime (z)
f(z)
\biggr\}
> 0 (1.2)
and
2\pi \int
0
\mathrm{R}\mathrm{e}
\biggl\{
zf \prime (z)
f(z)
\biggr\}
d\theta = 2p\pi
* This paper was supported by the National Board for Higher Mathematics, Department of Atomic Energy, India (grant
No. 2/39(20)/2010-R\&D-II).
c\bigcirc N. L. SHARMA, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4 549
550 N. L. SHARMA
for z = rei\theta . Goodman [10] has studied the class \scrS \ast
p and shown that a function in \scrS \ast
p has exactly p
zeros and it is p-valent in \BbbD .
Let \scrC p denote the class of functions, which are analytic and p-valent convex in \BbbD . A function
f \in \scrS p is said to be in \scrC p, if there exists a \rho > 0 such that for \rho < | z| < 1,
\mathrm{R}\mathrm{e}
\biggl\{
1 +
zf \prime \prime (z)
f \prime (z)
\biggr\}
> 0 (1.3)
and
2\pi \int
0
\mathrm{R}\mathrm{e}
\biggl\{
1 +
zf \prime \prime (z)
f \prime (z)
\biggr\}
d\theta = 2p\pi
for z = rei\theta . Goodman [10] proved that a function in \scrC p is at most p-valent and f \prime has exactly p - 1
zeros in \BbbD , multiple zeros being counted in accordance with their multiplicities. There is a closely
analytic relation between \scrS \ast
p and \scrC p in the same way as Alexander theorem. Namely,
f \in \scrC p \Leftarrow \Rightarrow zf \prime
p
\in \scrS \ast
p .
For p = 1, the classes \scrS \ast
p and \scrC p are the usually classes of univalent starlike and convex, respectively.
An analytic function f is said to be subordinate to an analytic function g if f(z) = g(\phi (z)),
z \in \BbbD , for some analytic function \phi in \BbbD with \phi (0) = 0 and | \phi (z)| < 1, z \in \BbbD . We write this
subordination relation by f(z) \prec g(z) (see [7, 11, 18]). The relations (1.2) and (1.3) are respectively
equivalent to
zf \prime (z)
pf(z)
\prec 1 + z
1 - z
and
1
p
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
\prec 1 + z
1 - z
.
In 1948, Goodman [9] has conjectured that if f \in \scrS p, then
| an| \leq
p\sum
k=1
2k(p+ n) !
(p+ k) !(p - k) !(n - p - 1) !(n2 - k2)
| ak| (1.4)
for n > p. For p = 2 and n = 3, this gives the conjecture that
| a3| \leq 5| a1| + 4| a2| . (1.5)
For p = 1, inequality (1.4) reduces to the well-known Bieberbach conjecture | an| \leq n. For instance,
Goodman [10] showed that (1.5) is valid for f in \scrS \ast
2 has the form (1.1) with all real coefficients an
and this bound is sharp for all pairs | a1| , | a2| , not both zero. In the same paper, Goodman suggested
the similar conjecture as (1.4) for f \in \scrC p . For n = p+1, he proved the inequality (1.4) for the classes
\scrS \ast
p and \scrC p , respectively, when f has the form (1.1) with the conditions a1 = a2 = . . . = ap - 2 = 0
and all the coefficients an are real. Umezawa [31] obtained the coefficient bound | an| for function
belongs to the class of p-valent close-to-convex functions. In 1969, Livingston [17] proved inequality
(1.4) for functions of the class p-valent close-to-convex, in case a1 = a2 = . . . = ap - 2 = 0 and the
remaining the coefficients being complex.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
A NOTE ON THE COEFFICIENT ESTIMATES FOR SOME CLASSES OF p-VALENT FUNCTIONS 551
In addition, let \scrA p denote the class of functions of the form
f(z) = zp +
\infty \sum
n=1
an+pz
n+p, p \in \BbbN , (1.6)
which are analytic and p-valent in \BbbD . Denote by A1 := \scrA , the class of all analytic functions of
the form f(z) = z +
\sum \infty
n=2
anz
n in \BbbD and \scrS denotes the usual class of functions in \scrA which are
univalent in \BbbD .
For the special subclass \scrA p of \scrS p, Hayman in [14] has showed that | ap+1| \leq 2p and Jenkins in
[15] has showed | ap+2| \leq p(2p+1). Both of these results are consistent with (1.4). Both inequalities,
for p-valent functions, are the analogues of the coefficient bounds | a2| \leq 2 and | a3| \leq 3, known for
univalent functions. Goluzina [8], Patil and Thakare [22], Aouf [2], and several other authors also
proved the coefficient bounds for certain subclasses of p-valent functions.
Recently the authors in [29] obtained the correct form of the coefficient bounds for the class
\scrS \ast
p (A,B, \beta ) :=
\left\{ f \in \scrA p :
zf \prime (z)
f(z)
\prec
p+
\Bigl(
pB + (A - B)(p - \beta )
\Bigr)
z
1 +Bz
, z \in \BbbD
\right\} ,
where \beta , 0 \leq \beta < p and - 1 \leq B < A \leq 1. Here, we solve the coefficient bounds involving the
Taylor – Maclaurin coefficients | an| for n \geq p+1, for functions belonging to the classes \scrF p(\alpha , \beta , \lambda )
and \scrC p(b, \lambda ). These classes are defined below (see Definitions 1.2 and 1.3).
In [21], Padmanabhan introduced the class of starlike functions of order \lambda , 0 < \lambda \leq 1, defined
as follows:
Definition 1.1. A function f \in \scrA is said to be in \scrT (\lambda ), if\bigm| \bigm| \bigm| \bigm| \biggl( zf \prime (z)
f(z)
- 1
\biggr) \Big/ \biggl( zf \prime (z)
f(z)
+ 1
\biggr) \bigm| \bigm| \bigm| \bigm| < \lambda ,
equivalently,
zf \prime (z)
f(z)
\prec 1 + \lambda z
1 - \lambda z
or
zf \prime (z)
f(z)
\prec 1 - \lambda z
1 + \lambda z
for all z \in \BbbD and 0 < \lambda \leq 1.
A function f \in \scrA p is said to be p-valent \alpha -spiral-like function of order \beta in \BbbD , if it is analytic
and if there exists a \rho > 0 such that for \rho < | z| < 1
\mathrm{R}\mathrm{e}
\biggl\{
ei\alpha
zf \prime (z)
f(z)
\biggr\}
> \beta \mathrm{c}\mathrm{o}\mathrm{s}\alpha
and
2\pi \int
0
\mathrm{R}\mathrm{e}
\biggl\{
ei\alpha
zf \prime (z)
f(z)
\biggr\}
d\theta = 2p\pi
for z = rei\theta , | \alpha | < \pi /2 and 0 \leq \beta < p. The class of p-valent \alpha -spiral-like of order \beta is denoted
by \scrS \alpha ,p(\beta ). In [22], Patil and Thakare introduced the class \scrS \alpha ,p(\beta ). The subordination form of the
definition of p-valent \alpha -spiral-like function of order \beta defined follows: f \in \scrS \alpha ,p(\beta ) if and only if
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
552 N. L. SHARMA
ei\alpha
zf \prime (z)
f(z)
\prec
\biggl(
p+ (p - 2\beta )z
1 - z
\biggr)
\mathrm{c}\mathrm{o}\mathrm{s}\alpha + ip \mathrm{s}\mathrm{i}\mathrm{n}\alpha .
Two subclasses \scrF p(\alpha , \beta , \lambda ) and \scrC p(b, \lambda ) of p-valent functions in \BbbD were acquainted by Aouf in
[2] which are defined as follows:
Definition 1.2. A function f \in \scrA p is said to belong to the class \scrF p(\alpha , \beta , \lambda ), if it satisfies the
condition \bigm| \bigm| \bigm| \bigm| H(f(z)) - 1
H(f(z)) + 1
\bigm| \bigm| \bigm| \bigm| < \lambda , z \in \BbbD ,
where
H(f(z)) =
ei\alpha zf \prime (z)
f(z) - \beta \mathrm{c}\mathrm{o}\mathrm{s}\alpha - ip \mathrm{s}\mathrm{i}\mathrm{n}\alpha
(p - \beta ) \mathrm{c}\mathrm{o}\mathrm{s}\alpha
.
By subordination property, equivalently, it can be written as
ei\alpha
zf \prime (z)
f(z)
\prec
\biggl(
p+ (p - 2\beta )\lambda z
1 - \lambda z
\biggr)
\mathrm{c}\mathrm{o}\mathrm{s}\alpha + ip \mathrm{s}\mathrm{i}\mathrm{n}\alpha (1.7)
for 0 < \lambda \leq 1, 0 \leq \beta < p, p \in \BbbN and | \alpha | < \pi /2.
Definition 1.3. Let b be a non-zero complex number. For 0 < \lambda \leq 1 and p \in \BbbN , let \scrC p(b, \lambda )
denote the class of functions f(z) \in \scrA p satisfying the relation\bigm| \bigm| \bigm| \bigm| H(f(z)) - 1
H(f(z)) + 1
\bigm| \bigm| \bigm| \bigm| < \lambda for z \in \BbbD ,
where
H(f(z)) = 1 +
1
pb
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
- p
\biggr)
.
By subordination relation,
1 +
zf \prime \prime (z)
f \prime (z)
\prec p(1 + (2b - 1)\lambda z)
1 - \lambda z
. (1.8)
We note that a number of subclasses have been studied by several authors and the subclasses can
be obtain by putting for different values of p, \alpha , \beta , \lambda and b. We list some of them here.
(1) \scrF p(0, 0, 1) =: \scrS \ast
p and \scrC p(1, 1) =: \scrC p are respectively the classes of p-valent starlike and p-
valent convex functions recognized by Goodman [10], and the class \scrF p(0, \beta , 1) =: \scrS \ast
p (\beta ), p-valent
starlike functions of order \beta was investigated by Goluzina [8]. \scrC p((1 - \beta /p), 1) =: \scrC p(\beta ), 0 \leq \beta < p,
the class of p-valent functions g(z) for which zg\prime (z)/p is in the class \scrS \ast
p (\beta ).
(2) \scrF p(\alpha , 0, 1) =: \scrS \alpha ,p and \scrF p(\alpha , \beta , 1) =: \scrS \alpha ,p(\beta ), respectively define the class of p-valent
\alpha -spirallike functions and p-valent \alpha -spirallike functions of order \beta .
(3) \scrC p(e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha , 1) and \scrC p(e - i\alpha (1 - \beta /p) \mathrm{c}\mathrm{o}\mathrm{s}\alpha , 1), 0 \leq \beta < p, | \alpha | < \pi /2, are the class
of p-valent functions g(z) for which zg\prime (z)/p are p-valent \alpha -spirallike functions and p-valent \alpha -
spirallike functions of order \beta respectively.
(4) The class \scrF 1(\alpha , \beta , \lambda ) =: \scrF (\alpha , \beta , \lambda ) was studied by Gopalakrishna and Umarani [13].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
A NOTE ON THE COEFFICIENT ESTIMATES FOR SOME CLASSES OF p-VALENT FUNCTIONS 553
(5) \scrC p(b, 1) is the class of p-valent functions g(z) \in \scrA p satisfying
\mathrm{R}\mathrm{e}
\biggl\{
p+
1
b
\biggl(
1 +
zg\prime \prime (z)
g\prime (z)
- p
\biggr) \biggr\}
> 0 for z \in \BbbD .
This class was considered by Aouf in [1].
(6) \scrF 1(0, 0, 1) =: \scrS \ast and \scrC 1(1, 1) =: \scrC are respectively the usual classes of starlike and convex
functions; \scrF 1(0, \beta , 1) =: \scrS \ast (\beta ) and \scrC 1(1 - \beta , 1) =: \scrC (\beta ), 0 \leq \beta < 1, are respectively the classes
of starlike and convex functions of order \beta were introduced by Robertson [24]; \scrF 1(0, 0, \lambda ) =: \scrT (\lambda )
(see Definition 1.1) and \scrC 1(1, \lambda ) =: \scrC (\lambda ) is the class of functions g(z) for which zg\prime (z) \in \scrS (\lambda ).
(7) \scrF 1(\alpha , 0, 1) =: \scrS \alpha and \scrC 1(e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha , 1), | \alpha | < \pi /2, respectively define the class of \alpha -
spirallike functions familiarized by Špaček [30] and the class of functions g(z) for which zg\prime (z) is
\alpha -spirallike introduced by Robertson [25]; \scrF 1(\alpha , \beta , 1) =: \scrS \alpha (\beta ) and \scrC 1
\bigl(
e - i\alpha (1 - \beta ) \mathrm{c}\mathrm{o}\mathrm{s}\alpha , 1
\bigr)
=:
=: \scrC \alpha (\beta ), 0 \leq \beta < 1, | \alpha | < \pi /2, are respectively the class of \alpha -spirallike functions of order \beta
introduced by Libra [16] and the class of functions g(z) for which zg\prime (z) is \alpha -spirallike of order \beta
studied by Chichra [4] and Sizuk [28].
(8) \scrC 1(b, 1) =: \scrC (b) is the class of functions g(z) \in \scrA satisfying
\mathrm{R}\mathrm{e}
\biggl(
1 +
1
b
zg\prime \prime (z)
g\prime (z)
\biggr)
> 0 for z \in \BbbD
introduced by Wiatrowski [32] and studied in [19, 20].
2. Main results. Aouf evaluated the coefficient bounds for the functions from the classes
\scrF p(\alpha , \beta , \lambda ) and \scrC p(b, \lambda ) in [2] in which the proofs are found to be incorrect. In the present paper,
we provide their correct proofs. The following theorems were mistakenly proven by Aouf in [2].
Theorem A ([2], Theorem 2). Let 0 < \lambda \leq 1, 0 \leq \beta < p, p \in \BbbN and | \alpha | < \pi /2. If f(z) =
= zp +
\sum \infty
n=p+1
anz
n \in \scrF p(\alpha , \beta , \lambda ), then
| an| \leq
n - p - 1\prod
j=0
\lambda
\bigm| \bigm| j + 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha
\bigm| \bigm|
j + 1
for n \geq p+ 1, and these bounds are sharp for all admissible \alpha , \beta , \lambda and for each n.
Theorem B ([2], Theorem 3). Let 0 < \lambda \leq 1, p \in \BbbN and b \not = 0 be any complex number. If
f(z) = zp +
\sum \infty
n=p+1
anz
n \in \scrC p(b, \lambda ), then
| an| \leq
n - p - 1\prod
j=0
\lambda | j + 2bp|
j + 1
for n \geq p+ 1, and these bounds are sharp for all admissible \alpha , \beta , \lambda and for each n.
First, we provide the correct form of the coefficients bounds for f \in \scrF p(\alpha , \beta , \lambda ) as stated in
Theorem A and its proof.
Theorem 2.1. Let 0 < \lambda \leq 1, 0 \leq \beta < p, p \in \BbbN and | \alpha | < \pi /2. If f(z) \in \scrF p(\alpha , \beta , \lambda ) is in
the form (1.6), then
| ap+1| \leq 2\lambda (p - \beta ) \mathrm{c}\mathrm{o}\mathrm{s}\alpha ; (2.1)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
554 N. L. SHARMA
for \lambda 2
\bigl(
2p - 2\beta + (n - p - 1)
\bigr) 2 \leq (n - p - 1)2
\bigl(
\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha
\bigr)
,
| an| \leq
2\lambda (p - \beta )
n - p
\mathrm{c}\mathrm{o}\mathrm{s}\alpha , n \geq p+ 2; (2.2)
and for \lambda 2
\bigl(
2p - 2\beta + (n - p - 1)
\bigr) 2
> (n - p - 1)2
\bigl(
\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha
\bigr)
,
| an| \leq
n - p\prod
j=1
\lambda
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm|
j
, n \geq p+ 2. (2.3)
The equality signs in (2.1), (2.2) and (2.3) are attained.
Proof. Let f(z) \in \scrF p(\alpha , \beta , \lambda ). It follows from (1.7) that
ei\alpha
zf \prime (z)
f(z)
=
\Biggl(
p+ (p - 2\beta )\lambda \phi (z)
1 - \lambda \phi (z)
\Biggr)
\mathrm{c}\mathrm{o}\mathrm{s}\alpha + ip \mathrm{s}\mathrm{i}\mathrm{n}\alpha
for some analytic function \phi (z) in \BbbD with \phi (0) = 0 and | \phi (z)| < 1. We divide the expansion by
\mathrm{c}\mathrm{o}\mathrm{s}\alpha on both sides and get
ei\alpha \mathrm{s}\mathrm{e}\mathrm{c}\alpha zf \prime (z) - (p+ ip \mathrm{t}\mathrm{a}\mathrm{n}\alpha )f(z) = \lambda
\Bigl(
ei\alpha \mathrm{s}\mathrm{e}\mathrm{c}\alpha zf \prime (z) + (p - 2\beta - ip \mathrm{t}\mathrm{a}\mathrm{n}\alpha )f(z)
\Bigr)
\phi (z).
Substituting this in the series expansion (1.6), of f(z), we find that
\infty \sum
k=0
\Bigl(
ei\alpha (k + p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha - p - ip \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\Bigr)
ak+pz
k+p =
= \lambda
\Biggl( \infty \sum
k=0
\Bigl(
ei\alpha (k + p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha + p - 2\beta - i \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\Bigr)
ak+pz
k+p
\Biggr)
\phi (z),
where ap = 1 and \phi (z) =
\sum \infty
k=0
wk+pz
k+p. Rewriting it, we obtain
m\sum
k=0
\Bigl(
ei\alpha (k + p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha - p - ip \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\Bigr)
ak+pz
k+p +
\infty \sum
k=m+1
Ckz
k+p =
= \lambda
\Biggl(
m - 1\sum
k=0
\Bigl(
ei\alpha (k + p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha + p - 2\beta - i \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\Bigr)
ak+pz
k+p
\Biggr)
\phi (z)
for certain coefficients Ck. Since | \phi (z)| < 1 in \BbbD , then by Parseval – Gutzmer formula (see also
Clunie’s method [5] and [26, 27]), we get
m\sum
k=0
\bigm| \bigm| \bigm| ei\alpha (k + p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha - p - ip \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\bigm| \bigm| \bigm| 2| ak+p| 2r2p+2k +
\infty \sum
k=m+1
| Ck| 2r2p+2k \leq
\leq \lambda 2
\Biggl(
m - 1\sum
k=0
\bigm| \bigm| \bigm| ei\alpha (k + p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha + p - 2\beta - i \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\bigm| \bigm| \bigm| 2| ak+p| 2r2p+2k
\Biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
A NOTE ON THE COEFFICIENT ESTIMATES FOR SOME CLASSES OF p-VALENT FUNCTIONS 555
Table 1
k p \alpha \beta \lambda T
all 1 all all 1 positive
2 1 \pm \pi /4 0.9 0.9 - 0.0236
3 2 \pm \pi /3 1 0.6 - 5.92
3 2 \pm \pi /3 1 0.8 1.92
(This is the place where the incorrectness of Aouf’s proof is found!)
Letting r \rightarrow 1, the above inequality can be written as
\bigm| \bigm| \bigm| ei\alpha (m+ p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha - p - ip \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\bigm| \bigm| \bigm| 2| am+p| 2 \leq
m - 1\sum
k=0
\Biggl(
\lambda 2
\bigm| \bigm| \bigm| ei\alpha (k + p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha + p - 2\beta - i \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\bigm| \bigm| \bigm| 2 -
-
\bigm| \bigm| \bigm| ei\alpha (k + p) \mathrm{s}\mathrm{e}\mathrm{c}\alpha - p - ip \mathrm{t}\mathrm{a}\mathrm{n}\alpha
\bigm| \bigm| \bigm| 2\Biggr) | ak+p| 2.
Simplification of the above inequality leads
m2 \mathrm{s}\mathrm{e}\mathrm{c}2 \alpha | am+p| 2 \leq
m - 1\sum
k=0
\Bigl(
\lambda 2(k + 2p - 2\beta )2 - k2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
\Bigr)
| ak+p| 2
or
| am+p| 2 \leq
\mathrm{c}\mathrm{o}\mathrm{s}2 \alpha
m2
\Biggl(
4\lambda 2(p - \beta )2 +
m\sum
k=2
\Bigl(
\lambda 2(k - 1 + 2p - 2\beta )2 -
- (k - 1)2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
\Bigr) \Biggr)
| ak+p - 1| 2.
Above inequality can be rewritten by replacing m+ p by n as
| an| 2 \leq
\mathrm{c}\mathrm{o}\mathrm{s}2 \alpha
(n - p)2
\Biggl(
4\lambda 2(p - \beta )2 +
n - p\sum
k=2
\Bigl(
\lambda 2(k - 1 + 2p - 2\beta )2 -
- (k - 1)2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
\Bigr) \Biggr)
| ak+p - 1| 2 for n \geq p+ 1. (2.4)
Note that the terms under the summation in the right-hand side of (2.4) may be positive as well as
negative. We verify it by including here a table (see Table 1) for values of
T := \lambda 2(k - 1 + 2p - 2\beta )2 - (k - 1)2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
for various choices for k, p, \alpha , \beta and \lambda . So, we can not apply direct principle of mathematical
induction in (2.4) to establish the desired bounds for | an| . Therefore, we are considering different
cases for this.
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556 N. L. SHARMA
First, for n = p+ 1, we readily see that (2.4) reduces to
| ap+1| \leq 2\lambda (p - \beta ) \mathrm{c}\mathrm{o}\mathrm{s}\alpha ,
which is equivalent to (2.1).
Secondly, \lambda 2
\bigl(
2p - 2\beta +(n - p - 1)
\bigr) 2 \leq (n - p - 1)2
\bigl(
\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha
\bigr)
for n \geq p+2. Since
all the terms under the summation in (2.4) are negative, we get
| an| \leq
2\lambda (p - \beta )
n - p
\mathrm{c}\mathrm{o}\mathrm{s}\alpha .
This gives the bound for | an| as asserted in (2.2). The equality holds in (2.1) and (2.2) for the
rotation of the functions
kn,p,\alpha ,\beta ,\lambda ,(z) =
zp
(1 + \lambda zn - 1
\bigr) \zeta n .
Here \zeta n := 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha /(n - 1).
Finally, we consider the case \lambda 2
\bigl(
2p - 2\beta +(n - p - 1)
\bigr) 2
> (n - p - 1)2
\bigl(
\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha
\bigr)
for
n \geq p+ 2 and obtain bound for | an| stated in (2.3). We see that all the terms under the summation
in (2.4) are nonnegative. We prove the inequality by the usual induction principle. Fix n, n \geq p+ 2
and suppose that (2.3) holds for k = 3, 4, . . . , n - p. Then by (2.4), we obtain
| an| 2 \leq
\mathrm{c}\mathrm{o}\mathrm{s}2 \alpha
(n - p)2
\Biggl(
4\lambda 2(p - \beta )2 +
n - p\sum
k=2
\Bigl(
\lambda 2(2p - 2\beta + k - 1)2 -
- (k - 1)2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
\Bigr) \Biggr) k - 1\prod
j=1
\lambda 2
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm| 2
j2
. (2.5)
It is now sufficient to prove that the square of the right-hand side of (2.3) is equal to the right-hand
side of (2.5), that is to show
m - p\prod
j=1
\lambda 2
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm| 2
j2
=
\mathrm{c}\mathrm{o}\mathrm{s}2 \alpha
(m - p)2
\Biggl(
4\lambda 2(p - \beta )2 +
m - p\sum
k=2
\Bigl(
\lambda 2(2p - 2\beta + k - 1)2 -
- (k - 1)2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
\Bigr) \Biggr) k - 1\prod
j=1
\lambda 2
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm| 2
j2
, (2.6)
when \lambda 2
\bigl(
2p - 2\beta + (m - p - 1)
\bigr) 2
> (m - p - 1)2
\bigl(
\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha
\bigr)
for m \geq p+ 2.
The equation (2.6) is valid for m = p+2. Suppose that (2.6) is true for all m, p+2 < m \leq n - p.
Then by (2.5), we have
| an| 2 \leq
\mathrm{c}\mathrm{o}\mathrm{s}2 \alpha
(n - p)2
\Biggl\{
4\lambda 2(p - \beta )2 +
n - p - 1\sum
k=2
\Bigl(
\lambda 2(2p - 2\beta + k - 1)2 -
- (k - 1)2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
\Bigr) k - 1\prod
j=1
\lambda 2
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm| 2
j2
+
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A NOTE ON THE COEFFICIENT ESTIMATES FOR SOME CLASSES OF p-VALENT FUNCTIONS 557
+
\Biggl(
\lambda 2(2p - 2\beta + n - p - 1)2 - (n - p - 1)2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
\Biggr)
\times
\times
n - p - 1\prod
j=1
\lambda 2
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm| 2
j2
\Biggr\}
.
By induction hypothesis for m = n - 1, we get
| an| 2 \leq
\mathrm{c}\mathrm{o}\mathrm{s}2 \alpha
(n - p)2
\Biggl\{
(n - p - 1)2
\mathrm{c}\mathrm{o}\mathrm{s}2 \alpha
n - p - 1\prod
j=1
\lambda 2
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm| 2
j2
+
+
\Biggl(
\lambda 2(2p - 2\beta + n - p - 1)2 - (n - p - 1)2(\mathrm{s}\mathrm{e}\mathrm{c}2 \alpha - \lambda 2 \mathrm{t}\mathrm{a}\mathrm{n}2 \alpha )
\Biggr)
\times
\times
n - p - 1\prod
j=1
\lambda 2
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm| 2
j2
\Biggr\}
,
i.e.,
| an| 2 \leq
\lambda 2
(n - p)2
\Biggl(
(2p - 2\beta + n - p - 1)2 \mathrm{c}\mathrm{o}\mathrm{s}2 \alpha + (n - p - 1)2 \mathrm{s}\mathrm{i}\mathrm{n}2 \alpha )
\Biggr)
\times
\times
n - p - 1\prod
j=1
\lambda 2
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm| 2
j2
.
On simplification, the above inequality leads to
| an| \leq
n - p\prod
j=1
\lambda
\bigm| \bigm| 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha + j - 1
\bigm| \bigm|
j
.
It is easy to prove that the bounds are sharp as can be seen by the rotation of the function
kp,\alpha ,\beta ,\lambda (z) =
zp
(1 + \lambda z)\zeta
.
Here \zeta := 2(p - \beta )e - i\alpha \mathrm{c}\mathrm{o}\mathrm{s}\alpha .
Theorem 2.1 is proved.
Table 2
kn p \alpha \beta \lambda
k1 2 \pi /4 1 0.5
k2 2 \pi /4 1.5 0.9
k3 3 - \pi /3 2 0.8
k4 3 - \pi /3 0.5 0.2
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558 N. L. SHARMA
-1 0 1 2
-1.5
-1.0
- 0.5
0.0
0.5
1.0
1.5
2.0
(a)
-1 0 1 2 3
- 2
-1
0
1
2
(b)
Fig. 1. Images of the unit disk under k1 (a) and k2 (b).
- 3 - 2 -1 0 1 2
- 2
-1
0
1
2
3
(a)
-1.0 - 0.5 0.0 0.5 1.0 1.5
-1.5
-1.0
- 0.5
0.0
0.5
1.0
1.5
(b)
Fig. 2. Images of the unit disk under k3 (a) and k4 (b).
Remark 2.1. Letting the different values of p, \alpha , \beta and \lambda in Theorem 2.1, we obtain results
which were proved in [8 – 10, 13, 16, 22 – 24, 33].
For different values of p, \alpha , \beta and \lambda (see Table 2), the images of the unit disk under the extremal
functions kn := kp,\alpha ,\beta ,\lambda (z) are described in Figures 1 and 2.
We now give the correct form of the statement stated in Theorem B and its proof.
Theorem 2.2. Let 0 < \lambda \leq 1, p \in \BbbN and b \not = 0 be any complex number. If f(z) \in \scrC p(b, \lambda ) is
of the form (1.6), then
| ap+1| \leq
2\lambda p2| b|
1 + p
; (2.7)
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A NOTE ON THE COEFFICIENT ESTIMATES FOR SOME CLASSES OF p-VALENT FUNCTIONS 559
for | 2bp+ n - p - 1| \leq n - p - 1 (equivalently | 1 + 2bp| \leq 1),
| an| \leq
2\lambda p2| b|
n(n - p)
, n \geq p+ 2; (2.8)
and for | 2bp+ n - p - 1| > n - p - 1,
| an| \leq
p
n
n - p - 1\prod
j=0
\lambda | j + 2bp|
j + 1
, n \geq p+ 2. (2.9)
The equality signs in (2.7), (2.8) and (2.9) are attained.
Proof. Let f(z) \in \scrC p(b, \lambda ). By the equation (1.8), we see that there is an analytic function \phi :
\BbbD \rightarrow \BbbD with \phi (0) = 0 such that
1 +
zf \prime \prime (z)
f \prime (z)
=
p(1 + (2b - 1)\lambda \phi (z))
1 - \lambda \phi (z)
,
or
zf \prime \prime (z) - (p - 1)f \prime (z) = - \lambda
\Bigl(
(p - 2bp - 1)f \prime (z) - zf \prime \prime (z)
\Bigr)
\phi (z).
Using the representation (1.6), we observe that
\infty \sum
k=1
k(k + p)ak+pz
k = \lambda
\Biggl(
2p2b+
\infty \sum
k=1
(k + p)(k + 2bp)ak+pz
k
\Biggr)
\phi (z).
We apply Clunie’s method [5] for m \in \BbbN (see also [26, 27]) and obtain
m\sum
k=1
k2(k + p)2| ak+p| 2 \leq \lambda 2
\Biggl(
4p4| b| 2 +
m - 1\sum
k=1
(k + p)2| k + 2bp| 2| ak+p| 2
\Biggr)
.
The above inequality yields
| am+p| 2 \leq
1
m2(m+ p)2
\Biggl(
4\lambda 2p4| b| 2 +
m - 1\sum
k=1
(k + p)2
\bigl(
\lambda 2| k + 2bp| 2 - k2
\bigr)
| ak+p| 2
\Biggr)
.
Replacing m+ p by n, we get
| an| 2 \leq
1
n2(n - p)2
\Biggl(
4\lambda 2p4| b| 2 +
n - p - 1\sum
k=1
(k + p)2
\bigl(
\lambda 2| k + 2bp| 2 - k2
\bigr)
| ak+p| 2
\Biggr)
(2.10)
for n \geq p+ 1.
Note that the terms under the summation in the right-hand side of (2.10) may be positive as well
as negative. We inspect it by including here a table (see Table 3) for values of
U := \lambda 2| k + 2bp| 2 - k2
for different choices of k, p, b and \lambda . So, we can not apply direct mathematical induction in (2.10)
to prove the required coefficients bounds for f \in \scrC p(b, \lambda ). Therefore, we are taking different cases
for this.
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560 N. L. SHARMA
Table 3
k p b \lambda V
2 1 1 0.1 - 3.998
2 1 1 0.6 1.76
4 2 3 - 2i 0.2 - 3.2
4 2 3 - 2i 0.3 12.8
(This is the place where the in correctness of Aouf’s proof is found!)
First, for n = p+ 1, (2.10) reduces to
| ap+1| \leq
2\lambda p2| b|
1 + p
.
This proves (2.7).
Secondly, we consider the case | 2bp+ n - p - 1| \leq n - p - 1 (equivalently | 1 + 2bp| \leq 1) for
n \geq p+ 2. Since all the terms under the summation in (2.10) are nonpositive, we get
| an| \leq
2\lambda p2| b|
n(n - p)
,
which establishes (2.8). The equality holds in (2.7) and (2.8) for the rotation of the functions
kn,p,b,\lambda (z) \in \scrC p(b, \lambda ) given by
k\prime n,p,b,\lambda (z) =
pzp - 1
(1 + \lambda zn - 1)2bp/(n - 1)
.
Finally, we prove (2.9) when | 1+2bp| \geq | 2bp+n - p - 1| > n - p - 1 for n \geq p+2. We see that all
the terms under the summation in (2.10) are positive. We prove the inequality by the mathematical
induction. We consider that (2.9) holds for k = 3, 4, . . . , n - p. Then from (2.10), we obtain
| an| 2 \leq
1
n2(p - n)2
\left( 4\lambda 2p4| b| 2 +
n - p - 1\sum
k=1
p2
\Bigl(
\lambda 2| k + 2bp| 2 - k2
\Bigr) k - 1\prod
j=0
\lambda 2| j + 2bp| 2
(j + 1)2
\right) . (2.11)
We now prove that the square of the right-hand side of (2.9) is equal to the right-hand side of (2.11),
that is
m - p - 1\prod
j=0
\lambda 2| j + 2bp| 2
(j + 1)2
=
1
(p - m)2
\Biggl(
4\lambda 2p2| b| 2 +
n - p - 1\sum
k=1
\Bigl(
\lambda 2| k + 2bp| 2 - k2
\Bigr)
\times
\times
k - 1\prod
j=0
\lambda 2| j + 2bp| 2
(j + 1)2
\Biggr)
(2.12)
when | 2bm+ p - p - 1| > m - p - 1, m \geq p+ 2.
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A NOTE ON THE COEFFICIENT ESTIMATES FOR SOME CLASSES OF p-VALENT FUNCTIONS 561
- 30 - 20 -10 0 10
- 20
-10
0
10
20
(a)
-1500 -1000 - 500 0 500 1000
-1000
- 500
0
500
1000
1500
(b)
Fig. 3. Images of the unit disk under g1 (a) and g2 (b).
- 2 -1 0 1
- 2
-1
0
1
(a) (b)
Fig. 4. Images of the unit disk under g3 (a) and g4 (b).
For m = p+ 2, the equation (2.12) is recognized. Suppose that (2.12) is true for all m, p+ 2 <
< m \leq n - p. Then from (2.11), we have
| an| 2 \leq
1
n2(p - n)2
\Biggl(
4\lambda 2p4| b| 2 +
n - p - 2\sum
k=1
p2
\Bigl(
\lambda 2| k + 2bp| 2 - k2
\Bigr) k - 1\prod
j=0
\lambda 2| j + 2bp| 2
(j + 1)2
+
+p2
\Bigl(
\lambda 2| n - p - 1 + 2bp| 2 - (n - p - 1)2
\Bigr) n - p - 2\prod
j=0
\lambda 2| j + 2bp| 2
(j + 1)2
\Biggr)
.
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562 N. L. SHARMA
Table 4
gn p b \lambda
g1 2 1 + i 0.4
g2 2 2 - 3i 0.4
g3 3 1 - 2i 0.7
g4 3 3 - 2i 0.7
Using the relation (2.12) for m = n - 1, we find that
| an| 2 \leq
1
n2(p - n)2
\Biggl(
p2(p - n+ 1)2
n - p - 2\prod
j=0
\lambda 2| j + 2bp| 2
(j + 1)2
+
+p2
\Bigl(
\lambda 2| n - p - 1 + 2bp| 2 - (n - p - 1)2
\Bigr) n - p - 2\prod
j=0
\lambda 2| j + 2bp| 2
(j + 1)2
\Biggr)
.
It is equivalent to
| an| \leq
p\lambda | j + 2bp|
n(p - n)
n - p - 2\prod
j=0
\lambda | j + 2bp|
(j + 1)
which establishes (2.9).
The bounds are sharp for the rotation of the function kp,b,\lambda (z) \in \scrC p(b, \lambda ) given by
k\prime p,b,\lambda (z) =
pzp - 1
(1 + \lambda z)2bp
.
Theorem 2.2 is proved.
Remark 2.2. Letting the different values of p, b and \lambda in Theorem 2.2, we obtain results which
were proved in [1, 10, 24, 32].
For different values of p, b and \lambda (see Table 4), the images of the unit disk under the extremal
functions gn := k\prime p,b,\lambda (z) are described in Figures 3 and 4.
Acknowledgements. The author would like to thank Dr. Swadesh Kumar Sahoo for careful
reading of the manuscript and helpful guidance.
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Received 06.01.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 4
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| id | umjimathkievua-article-1575 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:08:26Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/05/3402e09db0b7655184f1e6bc7c37ba05.pdf |
| spelling | umjimathkievua-article-15752019-12-05T09:19:04Z A note on the coefficient estimates for some classes of $p$ -valent functions Зауваження щодо коефiцiєнтних оцiнок для деяких класiв $p$ -валентних функцiй Sharma, N. L. Шарма, Н. Л. We obtain estimates of the Taylor – Maclaurin coefficients of some classes of p-valent functions. This problem was initially studied by Aouf in the paper “Coefficient estimates for some classes of p-valent functions” (Internat. J. Math. and Math. Sci. – 1988. – 11. – P. 47 – 54). The proof given by Aouf was found to be partially erroneous. We propose the correct proof of this result. Отримано оцiнки для коефiцiєнтiв Тейлора – Маклорена для деяких класiв p-валентних функцiй. Ця задача була вперше розглянута Ауфом у роботi “Coefficient estimates for some classes of p-valent functions” (Internat. J. Math. and Math. Sci. – 1988. – 11. – P. 47 – 54). Доведення, наведене Ауфом, виявилось частково помилковим. Ми пропонуємо коректне доведення цього результату. Institute of Mathematics, NAS of Ukraine 2018-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1575 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 4 (2018); 549-563 Український математичний журнал; Том 70 № 4 (2018); 549-563 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1575/557 Copyright (c) 2018 Sharma N. L. |
| spellingShingle | Sharma, N. L. Шарма, Н. Л. A note on the coefficient estimates for some classes of $p$ -valent functions |
| title | A note on the coefficient estimates for some classes of $p$ -valent functions |
| title_alt | Зауваження щодо коефiцiєнтних оцiнок
для деяких класiв $p$ -валентних функцiй |
| title_full | A note on the coefficient estimates for some classes of $p$ -valent functions |
| title_fullStr | A note on the coefficient estimates for some classes of $p$ -valent functions |
| title_full_unstemmed | A note on the coefficient estimates for some classes of $p$ -valent functions |
| title_short | A note on the coefficient estimates for some classes of $p$ -valent functions |
| title_sort | note on the coefficient estimates for some classes of $p$ -valent functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1575 |
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