Some properties of a generalized multiplier transform on analytic $p$-valent functions
UDC 517.5 For a function  $$f(z)=z^{p}+\sum^{\infty}_{k=1} a_{k+p}z^{k+p},$$ where $p\in\mathbb{N},$ the authors investigate some properties of a more general multiplier transform on analytic $p$-valent functions in an open unit disk. The applications of the obtained...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512284853403648 |
|---|---|
| author | Hamzat, J. O. El-Ashwah, R. M. Hamzat, J. O. El-Ashwah, R. M. |
| author_facet | Hamzat, J. O. El-Ashwah, R. M. Hamzat, J. O. El-Ashwah, R. M. |
| author_sort | Hamzat, J. O. |
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UDC 517.5
For a function  $$f(z)=z^{p}+\sum^{\infty}_{k=1} a_{k+p}z^{k+p},$$ where $p\in\mathbb{N},$ the authors investigate some properties of a more general multiplier transform on analytic $p$-valent functions in an open unit disk. The applications of the obtained results to fractional calculus are pointed out, while several other corollaries follow as simple consequences. |
| doi_str_mv | 10.37863/umzh.v74i9.6173 |
| first_indexed | 2026-03-24T03:26:21Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i9.6173
UDC 517.5
J. O. Hamzat1 (Univ. Lagos, Nigeria),
R. M. El-Ashwah (Faculty of Science Damietta Univ., New Damietta, Egypt)
SOME PROPERTIES OF A GENERALIZED MULTIPLIER TRANSFORM
ON ANALYTIC \bfitp -VALENT FUNCTIONS
ДЕЯКI ВЛАСТИВОСТI УЗАГАЛЬНЕНОГО МУЛЬТИПЛIКАТИВНОГО
ПЕРЕТВОРЕННЯ НА АНАЛIТИЧНИХ \bfitp -ВАЛЕНТНИХ ФУНКЦIЯХ
For a function
f(z) = zp +
\infty \sum
k=1
ak+pz
k+p
where p \in \BbbN , the authors investigate some properties of a more general multiplier transform on analytic p-valent functions
in an open unit disk. The applications of the obtained results to fractional calculus are pointed out, while several other
corollaries follow as simple consequences.
Для функцiї
f(z) = zp +
\infty \sum
k=1
ak+pz
k+p,
де p \in \BbbN , дослiджено деякi властивостi бiльш загального мультиплiкативного перетворення на аналiтичних p-
валентних функцiях у вiдкритому одиничному колi. Розглянуто застосування отриманих результатiв до дробового
числення, а деякi iншi результати отримано як простi наслiдки.
1. Introduction and preliminaries. Let \Gamma denote the class of analytic functions f(z), having the
series representation
f(z) = z +
\infty \sum
k=2
akz
k (1)
and normalized by f \prime (0) - 1 = 0 = f(0) in the open unit disk U = \{ z \in \BbbC : | z| < 1\} .
Also let \Gamma p denote the class of analytic p-valent functions f(z) having the form
f(z) = zp +
\infty \sum
k=1
ak+pz
k+p, p \in \BbbN . (2)
A function U = u(x, y) is said to be harmonic if it is a real-valued function having continuous partial
derivatives of order one and two, and satisfying
\partial 2u
\partial x2
+
\partial 2u
\partial y2
= 0.
However, a continuous complex-valued function f(z) = u(x, y) + iv(x, y) is said to be harmonic
in a complex domain D say, if both the real and imaginary parts u(x, y) and v(x, y), respectively,
are harmonic in D. The geometric function theory is mostly interested in the survey of properties of
analytic functions (see [2, 5]). Given any simply connected region R \subset D, we can say that
1 Corresponding author, e-mail: jhamzat@unilag.edu.ng.
c\bigcirc J. O. HAMZAT, R. M. EL-ASHWAH, 2022
1274 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
SOME PROPERTIES OF A GENERALIZED MULTIPLIER TRANSFORM . . . 1275
f(z) = h(z) + g(z),
where h and g are analytic in the connected region R. Conventionally, we refer to h and g as the
analytic and co-analytic parts, respectively. Thus a necessary and sufficient condition for function f
to be locally univalent and orientation preserving is that\bigm| \bigm| h\prime (z)\bigm| \bigm| > | g\prime (z)| \in R,
see [1, 4] among others. Let \scrH denote the family of p-valent harmonic function in U. Then h and g
can be expressed as
h(z) = zp +
\infty \sum
k=1
ak+pz
k+p, g(z) =
\infty \sum
k=0
bk+pz
k+p
for p \in \BbbN and, in particular, 0 \leq | bp| < 1.
Therefore, we write that
f(z) = zp +
\infty \sum
k=1
ak+pz
k+p +
\infty \sum
k=0
bk+pzk+p. (3)
As a special case, if the co-analytic part of f is identically zero (i.e., g = 0), then the family
of orientation preserving, normalized harmonic univalent functions reduces to the usual class of
normalized analytic functions. For function f(z) of the form (1), Swamy [10], in 2012 introduced
and studied a multiplier differential operator In\alpha ,\beta f(z) given by
In\alpha ,\beta f(z) = z +
\infty \sum
k=2
\biggl(
\alpha + k\beta
\alpha + \beta
\biggr) n
akz
k,
see also [8].
Furthermore, we define for function f(z) of the form (2), a linear differential operator Ln,p
\alpha ,\beta ,\gamma f(z)
such that
L1,p
\alpha ,\beta ,\gamma f(z) =
\alpha f(z) + \beta zf \prime (z) + \gamma z(zf \prime (z))\prime
\alpha + \beta p+ \gamma p2
,
L2,p
\alpha ,\beta ,\gamma f(z) = L1,p
\alpha ,\beta ,\gamma f(z)
\Bigl(
L1,p
\alpha ,\beta ,\gamma f(z)
\Bigr)
,
L3,p
\alpha ,\beta ,\gamma f(z) = L1,p
\alpha ,\beta ,\gamma f(z)
\Bigl(
L2,p
\alpha ,\beta ,\gamma f(z)
\Bigr)
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ln,p
\alpha ,\beta ,\gamma f(z) = L1,p
\alpha ,\beta ,\gamma f(z)
\Bigl(
Ln - 1,p
\alpha ,\beta ,\gamma f(z)
\Bigr)
, (4)
where p \in \BbbN , n, \alpha , \beta \geq 0 and \alpha is real such that \alpha + \beta + \gamma > 0. It follows from (4) that
Ln,p
\alpha ,\beta ,\gamma f(z) = zp +
\infty \sum
k=1
\biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
ak+pz
k+p. (5)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1276 J. O. HAMZAT, R. M. EL-ASHWAH
Remark. Suppose that the function f(z) has the form (2), it is easily verified from (5) that
L0,p
\alpha ,0,0f(z) = f(z) \in \Gamma p and L0,1
\alpha ,0,0f(z) = f(z) \in \Gamma p.
It is obvious that the operator Ln,p
\alpha ,\beta ,\gamma f(z) generalizes many existing operators of this kind which
were introduced and studied by different authors. For instance,
(i) Ln,1
\alpha ,\beta ,\gamma f(z) = In\alpha ,\beta ,\gamma f(z) studied by Makinde et al. [8];
(ii) Ln,1
\alpha ,\beta ,0f(z) = In\alpha ,\beta f(z) studied by Swamy [10];
(iii) Ln,1
\alpha ,1,\gamma f(z) = In\alpha f(z), \alpha > - 1 studied by Cho and Srivastava [3];
(iv) Ln,1
1,\beta ,0f(z) = Nn
\beta f(z) studied by Swamy [10].
With reference to (5), we can write that
Ln,p
\alpha ,\beta ,\gamma f(z) = Hn,p
\alpha ,\beta ,\gamma f(z) +Gn,p
\alpha ,\beta ,\gamma f(z). (6)
Now using (6), we give the following definition.
Definition 1. Let f(z) be of the form (3), then f(z) \in \scrH n,p
\mu (\alpha , \beta , \gamma ) if it satisfies the condition
that
\Re
\left\{
z
\Bigl(
Hn,p
\alpha ,\beta ,\gamma f(z)
\Bigr) \prime
- z
\Bigl(
Gn,p
\alpha ,\beta ,\gamma f(z)
\Bigr) \prime
Hn,p
\alpha ,\beta ,\gamma f(z) +Gn,p
\alpha ,\beta ,\gamma f(z)
\right\} \geq \mu (7)
for p \in \BbbN , 0 \leq \mu < p, n, \beta , \gamma \geq 0 and \alpha is real such that \alpha + \beta + \gamma > 0.
In addition, suppose that
\scrV n,p
\scrH (\alpha , \beta , \gamma , \mu ) = \scrV n,p
\scrH \cap \scrH n,p
\mu (\alpha , \beta , \gamma ), (8)
where \scrV n,p
\scrH is the harmonic functions with varying arguments consists of functions f of the form (3)
in \scrH n,p
\mu for which there exists a real number \sigma such that
\psi k+p + k\sigma \equiv \pi (\mathrm{m}\mathrm{o}\mathrm{d} 2\pi ), \tau k+p + k\sigma \equiv 0(\mathrm{m}\mathrm{o}\mathrm{d} 2\pi ), k \geq 1, (9)
where
\psi k+p = \mathrm{a}\mathrm{r}\mathrm{g}(ak+p) and \tau k+p = \mathrm{a}\mathrm{r}\mathrm{g}(bk+p).
At this juncture, we shall obtain a sufficient coefficient condition for function f of the form (3) to be
in the aforementioned class \scrH n,p
\mu (\alpha , \beta , \gamma ). It is noted that this coefficient condition is also necessary
for functions belonging to the class \scrV n,p
\scrH (\alpha , \beta , \gamma ).
2. Necessary and sufficient coefficient for the class \bfscrH \bfitn ,\bfitp
\bfitmu (\bfitalpha , \bfitbeta , \bfitgamma ).
Theorem 2.1. Let f(z) be of the form (3). Then, for p \in \BbbN , 0 \leq \mu < p, \alpha > 0, \beta , \gamma \geq 0,
\alpha + \beta + \gamma > 0, and | bp| <
p - \mu
p+ \mu
, f \in \scrH n,p
\mu (\alpha , \beta , \gamma ) if
\infty \sum
k=1
\biggl[
k + p - \mu
p - \mu
| ak+p| +
k + p+ \mu
p - \mu
| bk+p|
\biggr]
\times
\times
\biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
\leq 1 - p+ \mu
p - \mu
| bp| . (10)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
SOME PROPERTIES OF A GENERALIZED MULTIPLIER TRANSFORM . . . 1277
Proof. We begin the prove by showing that the condition (7) is satisfied if the inequality (10)
holds true for the coefficient of f defined in (3). Using (6) and (7), we have
\omega (z) =
\left\{
z
\Bigl(
Hn,p
\alpha ,\beta ,\gamma f(z)
\Bigr) \prime
- z
\Bigl(
Gn,p
\alpha ,\beta ,\gamma f(z)
\Bigr) \prime
p
\Bigl(
Hn,p
\alpha ,\beta ,\gamma f(z) +Gn,p
\alpha ,\beta ,\gamma f(z)
\Bigr)
\right\} =
M(z)
N(z)
, (11)
where
M(z) =
z
p
\Bigl(
Hn,p
\alpha ,\beta ,\gamma f(z)
\Bigr) \prime
- z
p
\Bigl(
Gn,p
\alpha ,\beta ,\gamma f(z)
\Bigr) \prime
and
N(z) = Hn,p
\alpha ,\beta ,\gamma f(z) +Gn,p
\alpha ,\beta ,\gamma f(z).
Here, we recall that \Re (\omega ) > \mu
p
if and only if
| p - \mu + p\omega | \geq | p+ \mu - p\omega | .
Then, from (11), it suffices to show that
| M(z) + (p - \mu )N(z)| \geq | M(z) - (p+ \mu )N(z)|
and
| M(z) + (p - \mu )N(z)| - | M(z) - (p+ \mu )N(z)| \geq 0. (12)
Having substituted for the values of M(z) and N(z) in (12), we obtain
| M(z) + (p - \mu )N(z)| - | M(z) - (p+ \mu )N(z)| \geq
\geq 2(p - \mu )| z| p -
\infty \sum
k=1
2(k + p)
\biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
| ak+p| | z| k+p -
-
\infty \sum
k=0
2(k + p)
\biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
| bk+p| | z| k+p+
+2\mu
\infty \sum
k=1
\biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
| ak+p| | z| k+p -
- 2\mu
\infty \sum
k=0
\biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
| bk+p| | z| k+p,
that is,
| M(z) + (p - \mu )N(z)| - | M(z) - (p+ \mu )N(z)| \geq
\geq 2(p - \mu )| z| p -
\infty \sum
k=1
2(k + p - \mu )Y n| ak+p| | z| k+p -
\infty \sum
k=0
2(k + p+ \mu )Y n| bk+p| | z| k+p \geq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1278 J. O. HAMZAT, R. M. EL-ASHWAH
\geq 2(p - \mu )| z| p
\Biggl\{
1 -
\infty \sum
k=1
k + p - \mu
p - \mu
Y n| ak+p| -
\infty \sum
k=0
k + p+ \mu
p - \mu
Y n| bk+p|
\Biggr\}
\geq
\geq 2(p - \mu )| z| p
\Biggl\{
1 - p+ \mu
p - \mu
| bp| -
\infty \sum
k=1
\biggl[
k + p - \mu
p - \mu
| ak+p| -
k + p+ \mu
p - \mu
| bk+p|
\biggr]
Y n
\Biggr\}
\geq 0
by virtue of inequality (10) where Y =
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
.
This shows that f \in \scrH n,p
\mu (\alpha , \beta , \gamma ).
Theorem 2.1 is proved.
Corollary 2.1. Let f(z) \in \scrH n,1
\mu (\alpha , \beta , \gamma ). Then, for 0 \leq \mu < 1, \alpha > 0, \beta , \gamma \geq 0 and \alpha +\beta +\gamma >
> 0,
\infty \sum
k=2
\biggl[
k - \mu
1 - \mu
| ak| +
k + \mu
1 - \mu
| bk|
\biggr] \biggl(
\alpha + \beta k + \gamma k2
\alpha + \beta + \gamma
\biggr) n
\leq 1 - 1 + \mu
1 - \mu
| b1| . (13)
Corollary 2.2. Let f(z) \in \scrH n,1
0 (\alpha , \beta , \gamma ). Then, for \alpha > 0, \beta , \gamma \geq 0 and \alpha + \beta + \gamma > 0,
\infty \sum
k=2
[k| ak| + k| bk| ]
\biggl(
\alpha + \beta k + \gamma k2
\alpha + \beta + \gamma
\biggr) n
\leq 1 - | b1| .
Corollary 2.3. Let f(z) \in \scrH 0,1
0 (\alpha , \beta , \gamma ). Then, for \alpha > 0, \beta , \gamma \geq 0 and \alpha + \beta + \gamma > 0,
\infty \sum
k=2
[k| ak| + k| bk| ] \leq 1 - | b1| . (14)
Next we obtain both the necessary and sufficient condition for function f of the form (3) given
the condition (8).
Theorem 2.2. f \in \scrV n,p
\scrH (\alpha , \beta , \gamma , \mu ) if and only if
\infty \sum
k=1
\biggl[
k + p - \mu
p - \mu
| ak+p| +
k + p - 2\mu
p - \mu
| bk+p|
\biggr] \biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
\leq 1 - p+ \mu
p - \mu
| bp|
(15)
for p \in \BbbN , 0 \leq \mu < p, \alpha > 0, \beta , \gamma \geq 0 and \alpha + \beta + \gamma > 0.
Proof. Since \scrV n,p
\scrH (\alpha , \beta , \gamma , \mu ) \subset \scrH n,p
\mu (\alpha , \beta , \gamma ). Then the necessary condition part of the theorem
shall be established. Suppose that f \in \scrV n,p
\scrH (\alpha , \beta , \gamma , \mu ), then appealing to (6) and (7), we have that
\Re
\left\{
\left[ z
\Bigl(
Hn,p
\alpha ,\beta ,\gamma f(z)
\Bigr) \prime
- z
\Bigl(
Gn,p
\alpha ,\beta ,\gamma f(z)
\Bigr) \prime
Hn,p
\alpha ,\beta ,\gamma f(z) +Gn,p
\alpha ,\beta ,\gamma f(z)
\right] - \mu
\right\} \geq 0.
Equivalently, we can write that
\Re
\left\{ pz
p +
\sum \infty
k=1
(k + p)Y nak+pz
k+p -
\sum \infty
k=0
(k + p)Y nbk+pzk+p
zp +
\sum \infty
k=1
Y nak+pz
k+p +
\sum \infty
k=0
Y nbk+pzk+p
- \mu
\right\} \geq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
SOME PROPERTIES OF A GENERALIZED MULTIPLIER TRANSFORM . . . 1279
\geq
\left\{
[(p - \mu ) - (p+ \mu )| bp| ]
(1 + | bp| ) +
\sum \infty
k=1
Y n| ak+p| | zk| +
\bigm| \bigm| \bigm| \bigm| zz
\bigm| \bigm| \bigm| \bigm| p \infty \sum
k=1
Y n| bk+p|
\bigm| \bigm| zk\bigm| \bigm|
\right\} -
-
\left\{
-
\sum \infty
k=1
(k + p - \mu )Y n
\bigm| \bigm| ak+p
\bigm| \bigm| \bigm| \bigm| zk\bigm| \bigm|
(1 + | bp| ) +
\sum \infty
k=1
Y n| ak+p|
\bigm| \bigm| zk\bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| zz
\bigm| \bigm| \bigm| \bigm| p\sum \infty
k=1
Y n| bk+p|
\bigm| \bigm| zk\bigm| \bigm|
\right\} -
-
\left\{
-
\bigm| \bigm| \bigm| \bigm| zz
\bigm| \bigm| \bigm| \bigm| p\sum \infty
k=1
(k + p+ \mu )Y n| bk+p|
\bigm| \bigm| zk\bigm| \bigm|
(1 + | bp| ) +
\sum \infty
k=1
Y n| ak+p|
\bigm| \bigm| zk\bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| zz
\bigm| \bigm| \bigm| \bigm| p\sum \infty
k=1
Y n| bk+p|
\bigm| \bigm| zk\bigm| \bigm|
\right\} \geq 0
and Y is as earlier defined.
The above condition must hold for all the values of z such that | z| = r < 1. With \sigma as in (9),
we obtain
[(p - \mu ) - (p+ \mu )| bp| ] -
\sum \infty
k=1
[(k + p - \mu )| ak+p| - (k + p+ \mu )| bk+p| ]
(1 + | bp| ) +
\sum \infty
k=1
[| ak+p| + | bk+p| ]
\biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
rk
\times
\times
\biggl(
\alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
rk
(1 + | bp| ) +
\sum \infty
k=1
\bigl[ \bigm| \bigm| ak+p
\bigm| \bigm| + \bigm| \bigm| bk+p
\bigm| \bigm| \bigr] \biggl( \alpha + \beta (k + p) + \gamma (k + p)2
\alpha + \beta p+ \gamma p2
\biggr) n
rk
\geq 0. (16)
Suppose that (15) does not hold, then the numerator in (16) is negative for r sufficiently close to
1. Thus there exists point z0 = r0, 0 < r0 < 1 for which the quotient in (16) is negative and this
negates our assumption that f \in \scrV n,p
\scrH (\alpha , \beta , \gamma , \mu ). Therefore, we can conclude that it is necessary
as well as sufficient that (15) holds true whenever f \in \scrV n,p
\scrH (\alpha , \beta , \gamma , \mu ) and this ends the proof of
Theorem 2.2.
Next we obtain both the growth and distortion results.
Theorem 2.3. Let f \in \scrH n,p
\mu (\alpha , \beta , \gamma ). Then
\bigm| \bigm| f(z)\bigm| \bigm| \geq (1 - | bp| )rp -
\biggl[
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
| bp|
\biggr] \biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr) n
rp+1
or
| f(z)| \leq (1 + | bp| )rp +
\biggl[
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
| bp|
\biggr] \biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr) n
rp+1.
Proof. From (3), we have
f(z) = zp +
\infty \sum
k=1
ak+pz
k+p +
\infty \sum
k=0
bk+pzk+p.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1280 J. O. HAMZAT, R. M. EL-ASHWAH
Now \bigm| \bigm| f(z)\bigm| \bigm| \leq (1 + | bp| )rp +
p - \mu
1 + p - \mu
\biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr) n
\times
\times
\infty \sum
k=1
\biggl[
k + p - \mu
p - \mu
| ap+k| +
k + p - \mu
p - \mu
\bigm| \bigm| bp+k
\bigm| \bigm| \biggr] Y nrp+1 \leq
\leq (1 + | bp| )rp +
\biggl[
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
| bp|
\biggr] \biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr) n
rp+1,
where Y is as defined earlier.
Theorem 2.3 is proved.
Theorem 2.4. Let f \in \scrH n,p
\mu (\alpha , \beta , \gamma ). Then
\bigm| \bigm| f \prime (z)\bigm| \bigm| \geq p(1 - | bp| )rp - 1 -
\biggl[
(1 + p)(p - \mu )
1 + p - \mu
- (1 + p)(p+ \mu )
1 + p - \mu
| bp|
\biggr]
\times
\times
\biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr) n
rp
or \bigm| \bigm| f \prime (z)\bigm| \bigm| \leq p(1 + | bp| )rp - 1 +
\biggl[
(1 + p)(p - \mu )
1 + p - \mu
- (1 + p)(p+ \mu )
1 + p - \mu
| bp|
\biggr]
\times
\times
\biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr) n
rp.
Proof is much similar to that of Theorem 2.2.
3. Application of fractional calculus. Given function f(z) of the form (1). The fractional
integral of order \epsilon , 0 < \epsilon \leq 1, is defined such that
D - \epsilon
z f(z) =
1
\Gamma (\epsilon )
z\int
0
f(t)
(z - t)1 - \epsilon
dt, (17)
where f(z) is analytic function in a simply connected region of z-plane containing the origin and
the multiplicity of (z - t)\epsilon - 1 is removed by requiring \mathrm{l}\mathrm{o}\mathrm{g}(z - t) to be real when z - t > 0.
Similarly, the fractional derivative of order \epsilon , 0 \leq \epsilon < 1, is given by
D\epsilon
zf(z) =
1
\Gamma (1 - \epsilon )
d
dz
z\int
0
f(t)
(z - t)\epsilon
dt, (18)
where f(z) is as defined above and the multiplicity of (z - t) - \epsilon is removed by requiring \mathrm{l}\mathrm{o}\mathrm{g}(z - t)
to be real when z - t > 0. Interestingly both (17) and (18) have the series representations
D - \epsilon
z f(z) =
1
\Gamma (2 + \epsilon )
z\epsilon +1 +
\infty \sum
k=2
\Gamma (k + 1)
\Gamma (k + 1 + \epsilon )
akz
k+\epsilon
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
SOME PROPERTIES OF A GENERALIZED MULTIPLIER TRANSFORM . . . 1281
and
D\epsilon
zf(z) =
1
\Gamma (2 - \epsilon )
z1 - \epsilon +
\infty \sum
k=2
\Gamma (k + 1)
\Gamma (k + 1 - \epsilon )
akz
k+\epsilon ,
respectively (see [6, 7, 9, 11]).
Theorem 3.1. Let f(z) be of the form (3). If f \in \scrH n,p
\mu (\alpha , \beta , \gamma ), then
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \leq \Gamma (p+ 1)
\bigm| \bigm| z\bigm| \bigm| p+\epsilon
\Gamma (p+ 1 + \epsilon )
\biggl\{ \bigl(
1 +
\bigm| \bigm| bp\bigm| \bigm| \bigr) + p+ 1
p+ 1 + \epsilon
\biggl(
p - \mu
1 + p - \mu
- p+ \mu
p+ 1 + \epsilon
| bp|
\biggr)
Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\}
and
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \geq \Gamma (p+ 1)
\bigm| \bigm| z\bigm| \bigm| p+\epsilon
\Gamma (p+ 1 + \epsilon )
\biggl\{ \bigl(
1 -
\bigm| \bigm| bp\bigm| \bigm| \bigr) - p+ 1
p+ 1 + \epsilon
\biggl(
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
| bp|
\biggr)
Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\} ,
where
X =
\biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr)
.
Proof. Following the representation of D - \epsilon
z f(z), we have
D - \epsilon
z f(z) =
1
\Gamma (\epsilon )
z\int
0
(z - t) - (1 - \epsilon )[f(t) + g(z)]dt =
=
1
\Gamma (\epsilon )
\left\{
z\int
0
(z - t) - (1 - \epsilon )
\Biggl(
tp +
\infty \sum
k=1
ak+pt
k+p
\Biggr)
dt+
z\int
0
(z - t) - (1 - \epsilon )
\Biggl( \infty \sum
k=0
ak+ptk+p
\Biggr)
dt
\right\} =
=
\Gamma (p+ 1)
\Gamma (p+ 1 + \epsilon )
z\epsilon +p +
\infty \sum
k=1
\Gamma (k + p+ 1)
\Gamma (k + p+ 1 + \epsilon )
ak+pz
k+p+\epsilon +
\infty \sum
k=0
\Gamma (k + p+ 1)
\Gamma (k + p+ 1 + \epsilon )
bk+pz
k+p+\epsilon .
Then
\Gamma (p+ 1 + \epsilon )
\Gamma (p+ 1)
z - \epsilon D - \epsilon
z f(z) =
= zp +
\infty \sum
k=1
\Gamma (p+ 1 + \epsilon )\Gamma (k + p+ 1)
\Gamma (p+ 1)\Gamma (k + p+ 1 + \epsilon )
ak+pz
k+p +
\infty \sum
k=0
\Gamma (p+ 1 + \epsilon )\Gamma (k + p+ 1)
\Gamma (p+ 1)\Gamma (k + p+ 1 + \epsilon )
bk+pz
k+p.
Simple computation of the above yields\bigm| \bigm| \bigm| \bigm| \Gamma (p+ 1 + \epsilon )
\Gamma (p+ 1)
z - \epsilon D - \epsilon
z f(z)
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| z\bigm| \bigm| p + \bigm| \bigm| bp\bigm| \bigm| \bigm| \bigm| z\bigm| \bigm| p + p+ 1
p+ 1 + \epsilon
\biggl(
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
| bp|
\biggr)
Xn
\bigm| \bigm| z\bigm| \bigm| p+1
,
where X =
\biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr)
. Therefore,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
1282 J. O. HAMZAT, R. M. EL-ASHWAH
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \leq \Gamma (p+ 1)
\bigm| \bigm| z\bigm| \bigm| p+\epsilon
\Gamma (p+ 1 + \epsilon )
\biggl\{ \bigl(
1 +
\bigm| \bigm| bp\bigm| \bigm| \bigr) + p+ 1
p+ 1 + \epsilon
\biggl(
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
| bp|
\biggr)
Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\}
and
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \geq \Gamma (p+ 1)
\bigm| \bigm| z\bigm| \bigm| p+\epsilon
\Gamma (p+ 1 + \epsilon )
\biggl\{ \bigl(
1 -
\bigm| \bigm| bp\bigm| \bigm| \bigr) - p+ 1
p+ 1 + \epsilon
\biggl(
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
| bp|
\biggr)
Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\} .
Theorem 3.1 is proved.
Corollary 3.1. Let f(z) be of the form (3). If f(z) \in \scrH n,1
\mu (\alpha , \beta , \gamma ), then
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \leq \bigm| \bigm| z\bigm| \bigm| 1+\epsilon
\Gamma (2 + \epsilon )
\biggl\{ \bigl(
1 +
\bigm| \bigm| b1\bigm| \bigm| \bigr) + 2
2 + \epsilon
\biggl(
1 - \mu
2 - \mu
- 1 + \mu
2 - \mu
| b1|
\biggr)
Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\}
and
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \geq \bigm| \bigm| z\bigm| \bigm| 1+\epsilon
\Gamma (2 + \epsilon )
\biggl\{ \bigl(
1 -
\bigm| \bigm| b1\bigm| \bigm| \bigr) - 2
2 + \epsilon
\biggl(
1 - \mu
2 - \mu
- 1 + \mu
2 - \mu
| b1|
\biggr)
Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\} ,
where
X =
\biggl(
\alpha + \beta + \gamma
\alpha + 2\beta + 4\gamma
\biggr)
.
Corollary 3.2. Let f(z) be of the form (3). If f(z) \in \scrH n,1
0 (\alpha , \beta , \gamma ), then
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \leq \bigm| \bigm| z\bigm| \bigm| 1+\epsilon
\Gamma (2 + \epsilon )
\biggl\{ \bigl(
1 +
\bigm| \bigm| b1\bigm| \bigm| \bigr) + 1
2 + \epsilon
(1 - | b1| )Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\}
and
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \geq \bigm| \bigm| z\bigm| \bigm| 1+\epsilon
\Gamma (2 + \epsilon )
\biggl\{ \bigl(
1 -
\bigm| \bigm| b1\bigm| \bigm| \bigr) - 1
2 + \epsilon
(1 - | b1| )Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\} ,
where
X =
\biggl(
\alpha + \beta + \gamma
\alpha + 2\beta + 4\gamma
\biggr)
.
Corollary 3.3. Let f(z) be of the form (3). If f(z) \in \scrH 0,1
0 (\alpha , \beta , \gamma ), then
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \leq \bigm| \bigm| z\bigm| \bigm| 1+\epsilon
\Gamma (2 + \epsilon )
\biggl\{ \bigl(
1 +
\bigm| \bigm| b1\bigm| \bigm| \bigr) + 1
2 + \epsilon
(1 - | b1| )z
\bigm| \bigm| \biggr\}
and
\bigm| \bigm| D - \epsilon
z f(z)
\bigm| \bigm| \geq \bigm| \bigm| z\bigm| \bigm| 1+\epsilon
\Gamma (2 + \epsilon )
\biggl\{ \bigl(
1 -
\bigm| \bigm| b1\bigm| \bigm| \bigr) - 1
2 + \epsilon
(1 - | b1| )
\bigm| \bigm| z\bigm| \bigm| \biggr\} .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
SOME PROPERTIES OF A GENERALIZED MULTIPLIER TRANSFORM . . . 1283
Theorem 3.2. Let f(z) be of the form (3). If f \in \scrH n,p
\mu (\alpha , \beta , \gamma ), then
| D\epsilon
zf(z)| \leq
\Gamma (p+ 1)| z| p - \epsilon
\Gamma (p+ 1 - \epsilon )
\biggl\{
(1 + | bp| ) +
p+ 1
p+ 1 - \epsilon
\biggl(
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
bp
\biggr)
Xn| z|
\biggr\}
and
\bigm| \bigm| D\epsilon
zf(z)
\bigm| \bigm| \geq \Gamma (p+ 1)
\bigm| \bigm| z\bigm| \bigm| p - \epsilon
\Gamma (p+ 1 - \epsilon )
\biggl\{ \bigl(
1 -
\bigm| \bigm| bp\bigm| \bigm| \bigr) - p+ 1
p+ 1 - \epsilon
\biggl(
p - \mu
1 + p - \mu
- p+ \mu
1 + p - \mu
| bp|
\biggr)
Xn
\bigm| \bigm| z\bigm| \bigm| \biggr\} ,
where
X =
\biggl(
\alpha + \beta p+ \gamma p2
\alpha + \beta (1 + p) + \gamma (1 + p)2
\biggr)
.
Proof is similar to that of Theorem 3.1.
References
1. Y. Avici, E. Zlotkiewicz, On harmonic univalent mappings, Ann. Univ. Marie Curie-Sklodowska Sect. A, 44, 1 – 7
(1990).
2. A. K. Bakhtin, I. V. Denega, Extremal decomposition of the complex plane with free ploes, J. Math. Sci., 246, № 1,
1 – 17 (2020).
3. N. E. Cho, H. M. Srivastava, Argument estimates of certain analytic functions defined by a class of multiplier
transformations, Math. Comput. Modeling, 37, № 1-2, 39 – 49 (2003).
4. J. Clunie, T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Fenn. Math., 9, 3 – 25 (1984).
5. I. V. Denega, Estimate of the inner radii of non-overlapping domains, J. Math. Sci., 242, № 6, 787 – 795 (2019).
6. J. O. Hamzat, M. O. Olayiwola, Application of fractional calculus on certain new subclasses of analytic function,
Int. J. Sci. Tech., 3, Issue 10, 235 – 245 (2015).
7. Y. Komatu, On analytic prolongation family of integral operators, Mathematics (Cluj), 32(55), 141 – 145 (1990).
8. D. O. Makinde, J. O. Hamzat, A. M. Gbolagade, A generalized multiplier transform on a univalent integral operator,
J. Contemp. Appl. Math., 9, № 1, 24 – 31 (2019).
9. H. M. Srivastava, S. Owa, Current topics in analytic function, World Sci., Singapore (1992).
10. S. R. Swamy, Inclusion properties of certain subclasses of analytic functions, Int. Math. Forum, 7, № 36, 1751 – 1760
(2012).
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Received 19.06.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 9
|
| id | umjimathkievua-article-6173 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:26:21Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f3/254a48e31b00cbab251048ca6d8022f3.pdf |
| spelling | umjimathkievua-article-61732023-01-07T13:45:32Z Some properties of a generalized multiplier transform on analytic $p$-valent functions Some properties of a generalized multiplier transform on analytic $p$-valent functions Hamzat, J. O. El-Ashwah, R. M. Hamzat, J. O. El-Ashwah, R. M. Analytic, Univalent, Multiplier transform, Differential operator, Growth and Distortion Theorem. MSC[2010]: 30C45 UDC 517.5 For a function&nbsp;&nbsp;$$f(z)=z^{p}+\sum^{\infty}_{k=1} a_{k+p}z^{k+p},$$ where $p\in\mathbb{N},$ the authors investigate some properties of a more general multiplier transform on analytic $p$-valent functions in an open unit disk. The applications of the obtained results to fractional calculus are pointed out, while several other corollaries follow as simple consequences. УДК 517.5 Деякі властивості узагальненого мультиплікативного перетворення на аналітичних $p$-валентних функціях Для функції&nbsp;&nbsp;$$f(z)=z^{p}+\sum\limits^{\infty}_{k=1} a_{k+p}z^{k+p},$$&nbsp;де $p\in\mathbb{N},$ досліджено деякі властивості більш загального мультиплікативного перетворення на аналітичних $p$-валентних функціях у відкритому одиничному колі.&nbsp;Розглянуто застосування отриманих результатів до дробового числення, а деякі інші результати отримано як прості наслідки. Institute of Mathematics, NAS of Ukraine 2022-11-08 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6173 10.37863/umzh.v74i9.6173 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 9 (2022); 1274 - 1283 Український математичний журнал; Том 74 № 9 (2022); 1274 - 1283 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6173/9324 Copyright (c) 2022 Jamiu Olusegun Hamzat |
| spellingShingle | Hamzat, J. O. El-Ashwah, R. M. Hamzat, J. O. El-Ashwah, R. M. Some properties of a generalized multiplier transform on analytic $p$-valent functions |
| title | Some properties of a generalized multiplier transform on analytic $p$-valent functions |
| title_alt | Some properties of a generalized multiplier transform on analytic $p$-valent functions |
| title_full | Some properties of a generalized multiplier transform on analytic $p$-valent functions |
| title_fullStr | Some properties of a generalized multiplier transform on analytic $p$-valent functions |
| title_full_unstemmed | Some properties of a generalized multiplier transform on analytic $p$-valent functions |
| title_short | Some properties of a generalized multiplier transform on analytic $p$-valent functions |
| title_sort | some properties of a generalized multiplier transform on analytic $p$-valent functions |
| topic_facet | Analytic Univalent Multiplier transform Differential operator Growth and Distortion Theorem. MSC[2010]: 30C45 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6173 |
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