Mapping properties for convolution involving hypergeometric series
We introduce sufficient conditions of (Gaussian) hypergeometric functions to be in a subclass of analytic functions. In addition, we investigate several mapping properties for convolution and integral convolution involving hypergeometric functions.
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| author | Aouf, M. K. Mostafa, A. O. Zayed, H. M. Ауф, М. К. Мустафа, А. О. Заїд, Г. М. |
| author_facet | Aouf, M. K. Mostafa, A. O. Zayed, H. M. Ауф, М. К. Мустафа, А. О. Заїд, Г. М. |
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| description | We introduce sufficient conditions of (Gaussian) hypergeometric functions to be in a subclass of analytic functions. In
addition, we investigate several mapping properties for convolution and integral convolution involving hypergeometric
functions. |
| first_indexed | 2026-03-24T02:09:54Z |
| format | Article |
| fulltext |
UDC 517.5
M. K. Aouf, A. O. Mostafa (Mansoura Univ., Egypt),
H. M. Zayed (Menofia Univ., Egypt)
MAPPING PROPERTIES FOR CONVOLUTION
INVOLVING HYPERGEOMETRIC SERIES
ВЛАСТИВОСТI ВIДОБРАЖЕННЯ ДЛЯ ЗГОРТКИ,
ЩО ВКЛЮЧАЄ ГIПЕРГЕОМЕТРИЧНI РЯДИ
We introduce sufficient conditions of (Gaussian) hypergeometric functions to be in a subclass of analytic functions. In
addition, we investigate several mapping properties for convolution and integral convolution involving hypergeometric
functions.
Введено достатнi умови того, що (гауссовi) гiпергеометричнi функцiї є пiдкласом аналiтичних функцiй. Крiм того,
розглянуто деякi властивостi вiдображення для згортки та iнтегральної згортки, що включають гiпергеометричнi
функцiї.
1. Introduction. Let \scrA (p) denote the class of functions of the form
f(z) = zp +
\infty \sum
n=1
ap+nz
p+n, p \in \BbbN = \{ 1, 2, . . .\} , (1.1)
which are analytic and p-valent in the open unit disc \BbbU = \{ z : z \in \BbbC and | z| < 1\} . We note that
\scrA (1) = \scrA . Also, for g(z) \in \scrA (p) given by
g(z) = zp +
\infty \sum
n=1
gp+nz
p+n,
the Hadamard product (or convolution) of two power series f(z) and g(z) is given by (see [4])
(f \ast g)(z) = zp +
\infty \sum
n=1
ap+ngp+nz
p+n = (g \ast f)(z)
and the integral convolution is defined by (see [4])
(f \circledast g)(z) = zp +
\infty \sum
n=1
ap+ngp+n
p+ n
zp+n = (g \circledast f)(z).
We recall some definitions which will be used in our paper.
Definition 1.1. For two functions f(z) and g(z), analytic in \BbbU , we say that the function f(z) is
subordinate to g(z) in \BbbU , and written f(z) \prec g(z), if there exists a Schwarz function w(z), analytic
in \BbbU with w(0) = 0 and | w(z)| < 1 such that f(z) = g(w(z)), z \in \BbbU . Furthermore, if the function
g(z) is univalent in \BbbU , then we have the following equivalence (see [7]):
f(z) \prec g(z) \leftrightarrow f(0) = g(0) and f(\BbbU ) \subset g(\BbbU ).
c\bigcirc M. K. AOUF, A. O. MOSTAFA, H. M. ZAYED, 2018
1466 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
MAPPING PROPERTIES FOR CONVOLUTION INVOLVING HYPERGEOMETRIC SERIES 1467
Definition 1.2 [10]. For 0 \leq \alpha < p, f(z) \in \scrA (p) is said to be in the class of p-valent starlike
of order \alpha , denoted by \scrS \ast
p (\alpha ), if
\mathrm{R}\mathrm{e}
\biggl\{
zf \prime (z)
f(z)
\biggr\}
> \alpha , z \in \BbbU , (1.2)
and in the class of p-valent convex of order \alpha , denoted by \scrK p(\alpha ), if
\mathrm{R}\mathrm{e}
\biggl\{
1 +
zf \prime \prime (z)
f \prime (z)
\biggr\}
> \alpha , z \in \BbbU . (1.3)
From (1.2) and (1.3) we can see that
f(z) \in \scrK p(\alpha ) \Leftarrow \Rightarrow zf \prime (z)
p
\in \scrS \ast
p (\alpha ).
We denote by \scrS \ast = \scrS \ast
1 (0) and \scrK = \scrK 1(0), where \scrS \ast and \scrK are the classes of starlike and convex
functions, respectively (see Robertson [11]).
We also define for \sigma > 0, the classes
\scrS \ast (p, \sigma ) =
\biggl\{
f(z) \in \scrA (p) :
\bigm| \bigm| \bigm| \bigm| zf \prime (z)
f(z)
- p
\bigm| \bigm| \bigm| \bigm| < \sigma , z \in \BbbU
\biggr\}
and
\scrK (p, \sigma ) =
\biggl\{
f(z) \in \scrA (p) :
\bigm| \bigm| \bigm| \bigm| 1 + zf \prime \prime (z)
f \prime (z)
- p
\bigm| \bigm| \bigm| \bigm| < \sigma , z \in \BbbU
\biggr\}
.
It is a known fact that a sufficient condition for f(z) \in \scrA (p) to be in the class \scrS \ast
p (\alpha ) is that\sum \infty
n=p+1
(n - \alpha ) | an| \leq p - \alpha . A simple extension of this result is (see [8])
\infty \sum
n=p+1
(n - p+ \sigma ) | an| \leq \sigma \Rightarrow f(z) \in \scrS \ast (p, \sigma ) .
Since f(z) \in \scrK (p, \sigma ) \Leftarrow \Rightarrow zf \prime (z)
p
\in \scrS \ast (p, \sigma ) , we have a corresponding result for \scrK (p, \sigma ) ,
\infty \sum
n=p+1
n
p
(n - p+ \sigma ) | an| \leq \sigma \Rightarrow f(z) \in \scrK (p, \sigma ) .
Definition 1.3 [1]. For - 1 \leq A < B \leq 1, | \lambda | < \pi
2
and 0 \leq \alpha < p, we define \scrR \lambda (A,B, p, \alpha )
which consists of functions f(z) of the form (1.1) and satisfying the analytic criterion
ei\lambda
f \prime (z)
zp - 1
\prec \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
(p - \alpha )
1 +Az
1 +Bz
+ \alpha
\biggr]
+ ip \mathrm{s}\mathrm{i}\mathrm{n}\lambda , z \in \BbbU .
According to the principle of subordination, f(z) \in \scrR \lambda (A,B, p, \alpha ) if and only if there exists function
w(z) satisfying w(0) = 0 and | w(z)| < 1, z \in \BbbU , such that
ei\lambda
f \prime (z)
zp - 1
= \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
(p - \alpha )
1 +Aw(z)
1 +Bw(z)
+ \alpha
\biggr]
+ ip \mathrm{s}\mathrm{i}\mathrm{n}\lambda , z \in \BbbU ,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1468 M. K. AOUF, A. O. MOSTAFA, H. M. ZAYED
or, equivalently, \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
ei\lambda
\biggl(
f \prime (z)
zp - 1
- p
\biggr)
Bei\lambda
f \prime (z)
zp - 1
- [pBei\lambda + (A - B)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda ]
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < 1, z \in \BbbU .
For suitable choices of A, B and \alpha , we obtain the following subclass: \scrR 0( - \beta , \beta , p, \alpha ) =
= \scrR (\beta , p, \alpha ) (see [9]).
Also, we note that:
(i) \scrR \lambda ( - \beta , \beta , p, \alpha ) = \scrR \lambda (\beta , p, \alpha ) =
=
\left\{ f(z) \in \scrA (p) :
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
f \prime (z)
zp - 1
- p
f \prime (z)
zp - 1
- [p - 2(p - \alpha )e - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda ]
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| < \beta , 0 < \beta \leq 1; z \in \BbbU
\right\} ,
(ii) \scrR \lambda ( - 1, 1, p, \alpha ) = \scrR \lambda (p, \alpha ) =
=
\biggl\{
f(z) \in \scrA (p) : \mathrm{R}\mathrm{e}
\biggl\{
ei\lambda
\biggl(
f \prime (z)
zp - 1
- \alpha
\biggr) \biggr\}
> 0, | \lambda | < \pi
2
; 0 \leq \alpha < p; z \in \BbbU
\biggr\}
.
Let 2F1(a, b; c; z) be the (Gaussian) hypergeometric function defined by
2F1(a, b; c; z) =
\infty \sum
n=0
(a)n(b)n
(c)n(1)n
zn,
where c \not = 0, - 1, - 2, . . . , and
(\gamma )n =
\Biggl\{
1, if n = 0,
\gamma (\gamma + 1)(\gamma + 2) . . . (\gamma + n - 1), if n \in \BbbN .
We note that 2F1(a, b; c; 1) converges for \mathrm{R}\mathrm{e}(c - a - b) > 0 and is related to Gamma function by
2F1(a, b; c; 1) =
\Gamma (c)\Gamma (c - a - b)
\Gamma (c - a)\Gamma (c - b)
.
Also, we define the functions
gp(a, b; c; z) = zp2F1(a, b; c; z) = zp +
\infty \sum
n=1
(a)n(b)n
(c)n(1)n
zp+n (1.4)
and
hp,\mu (a, b; c; z) = (1 - \mu ) (zp2F1(a, b; c; z)) + \mu
z
p
(zp2F1(a, b; c; z))
\prime
=
= zp +
\infty \sum
n=1
\biggl(
1 + \mu
n
p
\biggr)
(a)n(b)n
(c)n(1)n
zp+n, \mu \geq 0. (1.5)
Corresponding to 2F1(a, b; c; z), we define Ipa,b,c : \scrA (p) \rightarrow \scrA (p) by\bigl[
Ipa,b,c(f)
\bigr]
(z) = gp(a, b; c; z) \ast f(z) =
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
MAPPING PROPERTIES FOR CONVOLUTION INVOLVING HYPERGEOMETRIC SERIES 1469
=
\Gamma (c)
\Gamma (b)\Gamma (c - b)
1\int
0
tb - 1(1 - t)c - b - 1 f(tz)
t
dt \ast zp
(1 - z)a
=
= zp +
\infty \sum
n=1
(a)n(b)n
(c)n(1)n
ap+nz
p+n
and Lp,\mu : \scrA (p) \rightarrow \scrA (p) by
[Lp,\mu (f)] (z) = hp,\mu (a, b; c; z) \ast f(z) =
= zp +
\infty \sum
n=1
\biggl(
1 +
\mu n
p
\biggr)
(a)n(b)n
(c)n(1)n
ap+nz
p+n.
For p = 1, the operators Ia,b,c and L\mu were introduced by Hohlov (see [5]) and Kim and Shon (see
[6]), respectively.
Further, we define \scrM p
a,b,c : \scrA (p) \rightarrow \scrA (p) by\bigl[
\scrM p
a,b,c(f)
\bigr]
(z) = gp(a, b; c; z)\circledast f(z) =
=
\Gamma (c)
\Gamma (b)\Gamma (c - b)
1\int
0
tb - 1(1 - t)c - b - 1 f(tz)
t
dt\circledast
zp
(1 - z)a
=
= zp +
\infty \sum
n=1
(a)n(b)n
(c)n(1)n
ap+nz
p+n
(p+ n)
,
and \scrN p,\mu : \scrA (p) \rightarrow \scrA (p) by
[\scrN p,\mu (f)] (z) = hp,\mu (a, b; c; z)\circledast f(z) =
= zp +
\infty \sum
n=1
\biggl(
1 +
\mu n
p
\biggr)
(a)n(b)n
(c)n(1)n
ap+nz
p+n
(p+ n)
.
For p = 1, the operators \scrM a,b,c and \scrN \mu were introduced and studied by Aouf et al. (see [2]).
2. Main results. Unless otherwise mentioned, we assume throughout this paper that - 1 \leq A <
< B \leq 1, | \lambda | < \pi
2
, 0 \leq \alpha < p, p \in \BbbN and \BbbC \ast = \BbbC \setminus \{ 0\} . To establish our results, we need the
following lemmas.
Lemma 2.1 ([1], Theorem 4). A sufficient condition for f(z) defined by (1.1) to be in the class
\scrR \lambda (A,B, p, \alpha ) is
\infty \sum
n=1
(1 + | B| ) (p+ n) | ap+n| \leq (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda .
Lemma 2.2 ([1], Theorem 1). A function f(z) defined by (1.1) is in the class \scrR \lambda (A,B, p, \alpha ) if
| ap+n| \leq
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
(p+ n)
, n \geq 1.
The estimate is sharp.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1470 M. K. AOUF, A. O. MOSTAFA, H. M. ZAYED
By using Lemmas 2.1 and 2.2, we get the following results.
Theorem 2.1. Let a, b \in \BbbC \ast and c > | a| + | b| +1. Then the sufficient condition for gp(a, b; c; z)
to be in the class \scrR \lambda (A,B, p, \alpha ) is that
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl[
p+
| ab|
(c - | a| - | b| - 1)
\biggr]
\leq p+
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
1 + | B|
. (2.1)
Proof. According to Lemma 2.1 and (1.4), we need only to show that
\infty \sum
n=1
(1 + | B| ) (p+ n)
\bigm| \bigm| \bigm| \bigm| (a)n(b)n(c)n(1)n
\bigm| \bigm| \bigm| \bigm| \leq (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda . (2.2)
Since
| (d)n| \leq (| d| )n , (2.3)
then the left-hand side of (2.2) is less than or equal to
\infty \sum
n=1
(1 + | B| ) (p+ n)
(| a| )n(| b| )n
(c)n(1)n
= T0.
Now
T0 = p (1 + | B| )
\infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n
+ (1 + | B| )
\infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n - 1
=
= p (1 + | B| )
\biggl[
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
- 1
\biggr]
+
+(1 + | B| ) | ab|
c
\Gamma (c+ 1)\Gamma (c - | a| - | b| - 1)
\Gamma (c - | a| )\Gamma (c - | b| )
=
= (1 + | B| ) \Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl[
p+
| ab|
(c - | a| - | b| - 1)
\biggr]
- p (1 + | B| ) .
But this last expression is bounded above by (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda if (2.1) holds.
Theorem 2.1 is proved.
Remark 2.1. Putting p = 1 in Theorem 2.1, we obtain the result of Aouf et al. [3] (Theorem 2.1).
Theorem 2.2. Let a, b \in \BbbC \ast and c > | a| +| b| +2. Then the sufficient condition for hp,\mu (a, b; c; z)
to be in the class \scrR \lambda (A,B, p, \alpha ) is that
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl[
p+
\biggl(
1 + \mu +
\mu
p
\biggr)
| ab|
(c - | a| - | b| - 1)
+
\mu
p
(| a| )2 (| b| )2
(c - | a| - | b| - 2)2
\biggr]
\leq
\leq p+
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
1 + | B|
. (2.4)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
MAPPING PROPERTIES FOR CONVOLUTION INVOLVING HYPERGEOMETRIC SERIES 1471
Proof. According to Lemma 2.1 and (1.5), we need only to show that
\infty \sum
n=1
(1 + | B| ) (p+ n)
\bigm| \bigm| \bigm| \bigm| \biggl( 1 + \mu
n
p
\biggr)
(a)n(b)n
(c)n(1)n
\bigm| \bigm| \bigm| \bigm| \leq (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda . (2.5)
The left-hand side of (2.5), by (2.3), is less than or equal to
\infty \sum
n=1
(1 + | B| ) (p+ n)
\biggl(
1 + \mu
n
p
\biggr)
(| a| )n(| b| )n
(c)n(1)n
= T1
and so
T1 = p (1 + | B| )
\infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n
+
\biggl(
1 + \mu +
\mu
p
\biggr)
(1 + | B| )
\infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n - 1
+
+
\mu
p
(1 + | B| )
\infty \sum
n=2
(| a| )n(| b| )n
(c)n(1)n - 2
=
= p (1 + | B| )
\biggl[
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
- 1
\biggr]
+
+(1 + | B| )
\biggl(
1 + \mu +
\mu
p
\biggr)
| ab|
c
\Gamma (c+ 1)\Gamma (c - | a| - | b| - 1)
\Gamma (c - | a| )\Gamma (c - | b| )
+
+
\mu (1 + | B| )
p
(| a| )2 (| b| )2
(c)2
\Gamma (c+ 2)\Gamma (c - | a| - | b| - 2)
\Gamma (c - | a| )\Gamma (c - | b| )
=
= (1 + | B| ) \Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl[
p+
\biggl(
1 + \mu +
\mu
p
\biggr)
| ab|
(c - | a| - | b| - 1)
+
+
\mu
p
(| a| )2 (| b| )2
(c - | a| - | b| - 2)2
\biggr]
- p (1 + | B| ) ,
the proof now follows by (2.5).
Theorem 2.2 is proved.
Remark 2.2. Putting p = 1 in Theorem 2.2, we obtain the result of Aouf et al. [3] (Theorem 2.2).
Theorem 2.3. Let a, b \in \BbbC \ast and c > | a| + | b| . If the inequality
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\leq 1 +
1
(1 + | B| )
, (2.6)
is satisfied, then [Ipa,b,c(f)](z) maps the class \scrR \lambda (A,B, p, \alpha ) into itself.
Proof. We need to show that
T2 =
\infty \sum
n=1
(1 + | B| ) (p+ n)
\bigm| \bigm| \bigm| \bigm| (a)n(b)n(c)n(1)n
ap+n
\bigm| \bigm| \bigm| \bigm| \leq (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda . (2.7)
By (2.3) and Lemma 2.2, we have
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1472 M. K. AOUF, A. O. MOSTAFA, H. M. ZAYED
T2 \leq
\infty \sum
n=1
(1 + | B| ) (p+ n)
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
(p+ n)
(| a| )n(| b| )n
(c)n(1)n
=
= (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n
=
= (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
- 1
\biggr]
,
the last expression is bounded above by (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda if (2.6) holds.
Theorem 2.3 is proved.
Theorem 2.4. Let a, b \in \BbbC \ast and c > | a| + | b| + 1. If the inequality
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl[
1 +
\mu
p
| ab|
(c - | a| - | b| - 1)
\biggr]
\leq 1 +
1
(1 + | B| )
, (2.8)
holds, then [Lp,\mu (f)] (z) maps the class \scrR \lambda (A,B, p, \alpha ) into itself.
Proof. We need to prove that
T3 =
\infty \sum
n=1
(1 + | B| ) (p+ n)
\bigm| \bigm| \bigm| \bigm| \biggl( 1 + \mu n
p
\biggr)
(a)n(b)n
(c)n(1)n
ap+n
\bigm| \bigm| \bigm| \bigm| \leq (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda . (2.9)
By (2.3) and Lemma 2.2, we get
T3 \leq
\infty \sum
n=1
(1 + | B| ) (p+ n)
\biggl(
1 +
\mu n
p
\biggr)
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
(p+ n)
(| a| )n(| b| )n
(c)n(1)n
=
= (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\infty \sum
n=1
\biggl(
1 +
\mu n
p
\biggr)
(| a| )n(| b| )n
(c)n(1)n
=
= (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
+
+
\mu
p
| ab|
c
\Gamma (c+ 1)\Gamma (c - | a| - | b| - 1)
\Gamma (c - | a| )\Gamma (c - | b| )
- 1
\biggr]
=
= (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl(
1 +
\mu
p
| ab|
(c - | a| - | b| - 1)
\biggr)
.
It is easy to see that the last expression is bounded above by (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda if (2.8) holds.
Theorem 2.4 is proved.
Theorem 2.5. Assume that a, b \in \BbbC \ast , | a| \not = 1, | b| \not = 1 and c > \mathrm{m}\mathrm{a}\mathrm{x} \{ 0, | a| + | b| - 1\} . If the
inequality
1
(| a| - 1)(| b| - 1)
\Gamma (c)\Gamma (c - | a| - | b| + 1)
\Gamma (c - | a| )\Gamma (c - | b| )
\leq 1 +
(c - 1)
(| a| - 1)(| b| - 1)
+
1
(1 + | B| )
(2.10)
is true, then [\scrM p
a,b,c(f)](z) maps the class \scrR \lambda (A,B, p, \alpha ) into itself.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
MAPPING PROPERTIES FOR CONVOLUTION INVOLVING HYPERGEOMETRIC SERIES 1473
Proof. It is enough to show that
T4 =
\infty \sum
n=1
(1 + | B| ) (p+ n)
\bigm| \bigm| \bigm| \bigm| (a)n(b)n(c)n(1)n
ap+n
(p+ n)
\bigm| \bigm| \bigm| \bigm| \leq (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda . (2.11)
By using (2.3) and Lemma 2.2, we get
T4 \leq (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n+1
=
= (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
(c - 1)
(| a| - 1)(| b| - 1)
\times
\times
\biggl[
\Gamma (c - 1)\Gamma (c - | a| - | b| + 1)
\Gamma (c - | a| )\Gamma (c - | b| )
- 1 - (| a| - 1)(| b| - 1)
(c - 1)
\biggr]
.
Theorem 2.5 is proved.
Theorem 2.6. Assume that a, b \in \BbbC \ast , | a| \not = 1, | b| \not = 1 and c > \mathrm{m}\mathrm{a}\mathrm{x} \{ 0, | a| + | b| - 1\} . If the
inequality
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl[
\mu
p
+
\biggl(
1 - \mu
p
\biggr)
(c - | a| - | b| )
(| a| - 1)(| b| - 1)
\biggr]
\leq
\leq 1 +
1
(1 + | B| )
+
\biggl(
1 - \mu
p
\biggr)
(c - 1)
(| a| - 1)(| b| - 1)
is satisfied, then [\scrN p,\mu (f)] (z) maps the class \scrR \lambda (A,B, p, \alpha ) into itself.
Proof. It suffices to show that
\infty \sum
n=1
(1 + | B| ) (p+ n)
\bigm| \bigm| \bigm| \bigm| \biggl( 1 + \mu n
p
\biggr)
(a)n(b)n
(c)n(1)n
ap+n
(p+ n)
\bigm| \bigm| \bigm| \bigm| \leq (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda . (2.12)
The left-hand side of (2.12), is less than or equal to
\infty \sum
n=1
(1 + | B| )
\biggl(
1 +
\mu n
p
\biggr)
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
(n+ 1)
(| a| )n(| b| )n
(c)n(1)n
= T5,
where
T5 = (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\infty \sum
n=1
\biggl(
1 +
\mu n
p
\biggr)
(| a| )n(| b| )n
(c)n(1)n+1
=
= (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\Biggl[ \biggl(
1 - \mu
p
\biggr) \infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n+1
+
\mu
p
\infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n
\Biggr]
=
= (1 + | B| ) (B - A)(p - \alpha )
\biggl(
1 - \mu
p
\biggr)
(c - 1)
(| a| - 1)(| b| - 1)
\mathrm{c}\mathrm{o}\mathrm{s}\lambda \times
\times
\biggl[
\Gamma (c - 1)\Gamma (c - | a| - | b| + 1)
\Gamma (c - | a| )\Gamma (c - | b| )
- 1 - (| a| - 1)(| b| - 1)
(c - 1)
\biggr]
+
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
1474 M. K. AOUF, A. O. MOSTAFA, H. M. ZAYED
+
\mu
p
(1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
- 1
\biggr]
=
= (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\times
\times
\biggl[ \biggl(
1 - \mu
p
\biggr)
(c - | a| - | b| )
(| a| - 1)(| b| - 1)
+
\mu
p
\biggr]
-
- (1 + | B| ) (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
1 +
\biggl(
1 - \mu
p
\biggr)
(c - 1)
(| a| - 1)(| b| - 1)
\biggr]
.
By a simplification, we see that the last expression is bounded above by (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda if
(2.12) holds.
Theorem 2.6 is proved.
Theorem 2.7. Suppose that a, b \in \BbbC \ast and c be a real number. If the inequality
p | ab|
c(p+ 1)
3F2 (| a| + 1, | b| + 1, p+ 1; c+ 1, p+ 2; 1)+
+\sigma 3F2 (| a| , | b| , p; c, p+ 1; 1) \leq \sigma
\biggl(
1 +
p
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr)
(2.13)
holds, then [Ipa,b,c(f)](z) maps the class \scrR \lambda (A,B, p, \alpha ) to \scrS \ast (p, \sigma ) .
Proof. It is enough to prove that
\infty \sum
n=1
(n+ \sigma )
\bigm| \bigm| \bigm| \bigm| (a)n(b)n(c)n(1)n
ap+n
\bigm| \bigm| \bigm| \bigm| \leq \sigma . (2.14)
The left-hand side of (2.14), is less than or equal to
T6 =
\infty \sum
n=1
(n+ \sigma )
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
(p+ n)
(| a| )n(| b| )n
(c)n(1)n
=
= (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\infty \sum
n=1
\biggl(
n+ \sigma
p+ n
\biggr)
(| a| )n(| b| )n
(c)n(1)n
=
=
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
p
\infty \sum
n=1
(n+ \sigma )
(| a| )n(| b| )n(p)n
(c)n(p+ 1)n(1)n
=
=
(B - A)(p - \alpha ) | ab| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
c(p+ 1)
\infty \sum
n=0
(| a| + 1)n(| b| + 1)n(p+ 1)n
(c+ 1)n(p+ 2)n(1)n
+
+
(B - A)(p - \alpha )\sigma \mathrm{c}\mathrm{o}\mathrm{s}\lambda
p
\Biggl[ \infty \sum
n=0
(| a| )n(| b| )n(p)n
(c)n(p+ 1)n(1)n
- 1
\Biggr]
=
=
(B - A)(p - \alpha ) | ab| \mathrm{c}\mathrm{o}\mathrm{s}\lambda
c(p+ 1)
3F2 (| a| + 1, | b| + 1, p+ 1; c+ 1, p+ 2; 1)+
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
MAPPING PROPERTIES FOR CONVOLUTION INVOLVING HYPERGEOMETRIC SERIES 1475
+
(B - A)(p - \alpha )\sigma \mathrm{c}\mathrm{o}\mathrm{s}\lambda
p
\Bigl[
3F2 (| a| , | b| , p; c, p+ 1; 1) - 1
\Bigr]
,
the last expression is bounded above by \sigma if (2.13) holds.
Theorem 2.7 is proved.
Theorem 2.8. Suppose that a, b \in \BbbC \ast and c > | a| + | b| + 1. If the inequality
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl[
| ab|
(c - | a| - | b| - 1)
+ \sigma
\biggr]
\leq \sigma
\biggl(
1 +
p
(B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggr)
(2.15)
is true, then [Ipa,b,c(f)](z) maps the class \scrR \lambda (A,B, p, \alpha ) to \scrK (p, \sigma ) .
Proof. It suffices to prove that
\infty \sum
n=1
(n+ p)(n+ \sigma )
\bigm| \bigm| \bigm| \bigm| (a)n(b)n(c)n(1)n
ap+n
\bigm| \bigm| \bigm| \bigm| \leq p\sigma . (2.16)
The left-hand side of (2.16), is less than or equal to
T7 = (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\infty \sum
n=1
(n+ \sigma )
(| a| )n(| b| )n
(c)n(1)n
=
= (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\Biggl[ \infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n - 1
+ \sigma
\infty \sum
n=1
(| a| )n(| b| )n
(c)n(1)n
\Biggr]
=
= (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\biggl[
| ab|
c
\Gamma (c+ 1)\Gamma (c - | a| - | b| - 1)
\Gamma (c - | a| )\Gamma (c - | b| )
+ \sigma
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
- \sigma
\biggr]
=
= (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda
\Gamma (c)\Gamma (c - | a| - | b| )
\Gamma (c - | a| )\Gamma (c - | b| )
\biggl[
| ab|
(c - | a| - | b| - 1)
+ \sigma
\biggr]
- \sigma (B - A)(p - \alpha ) \mathrm{c}\mathrm{o}\mathrm{s}\lambda .
We note that the last expression is bounded above by p\sigma if (2.15) holds.
Theorem 2.8 is proved.
References
1. Aouf M. K. On certain subclass of analytic p-valent functions of order alpha // Rend. Mat. – 1988. – 7, № 8. –
P. 89 – 104.
2. Aouf M. K., Mostafa A. O., Zayed H. M. Necessity and sufficiency for hypergeometric functions to be in a subclass
of analytic functions // J. Egypt. Math. Soc. – 2015. – 23. – P. 476 – 481.
3. Aouf M. K., Mostafa A. O., Zayed H. M. Some constraints of hypergeometric functions to belong to certain subclasses
of analytic functions // J. Egypt. Math. Soc. – 2016. – 24. – P. 361 – 366.
4. Duren P. L. Univalent functions. – New York: Springer-Verlag, 1983.
5. Hohlov Yu. E. Operators and operations in the class of univalent functions // Izv. Vyssh. Uchebn. Zaved. Mat. – 1978. –
10. – P. 83 – 89.
6. Kim J. A., Shon K. H. Mapping properties for convolutions involving hypergeometric functions // Int. J. Math. and
Math. Sci. – 2003. – 17. – P. 1083 – 1091.
7. Miller S. S., Mocanu P. T. Differenatial subordinations: theory and applications // Ser. Monographs and Textbooks in
Pure and Appl. Math. – 2000. – № 255.
8. Owa S. On certain classes of p-valent functions with negative coefficients // Simon Stevin. – 1985. – 59. – P. 385 – 402.
9. Owa S. On certain subclass of analytic functions // Math. Japonica. – 1984. – 29. – P. 191 – 198.
10. Patil D. A., Thakare N. K. On convex hulls and extreme points of p-valent starlike and convex classes with
applications // Bull. Math. Soc. Sci. Math. Roumanie (N. S.). – 1983. – 27, № 75. – P. 145 – 160.
11. Robertson M. S. On the theory of univalent functions // Ann. Math. – 1936. – 37. – P. 374 – 408.
Received 12.03.16,
after revision — 25.05.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 11
|
| id | umjimathkievua-article-1650 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:54Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a3/80988d40744db368c3e4004c42865da3.pdf |
| spelling | umjimathkievua-article-16502019-12-05T09:22:19Z Mapping properties for convolution involving hypergeometric series Властивостi вiдображення для згортки, що включає гiпергеометричнi ряди Aouf, M. K. Mostafa, A. O. Zayed, H. M. Ауф, М. К. Мустафа, А. О. Заїд, Г. М. We introduce sufficient conditions of (Gaussian) hypergeometric functions to be in a subclass of analytic functions. In addition, we investigate several mapping properties for convolution and integral convolution involving hypergeometric functions. Введено достатнi умови того, що (гауссовi) гiпергеометричнi функцiї є пiдкласом аналiтичних функцiй. Крiм того, розглянуто деякi властивостi вiдображення для згортки та iнтегральної згортки, що включають гiпергеометричнi функцiї. Institute of Mathematics, NAS of Ukraine 2018-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1650 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 11 (2018); 1466-1475 Український математичний журнал; Том 70 № 11 (2018); 1466-1475 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1650/632 Copyright (c) 2018 Aouf M. K.; Mostafa A. O.; Zayed H. M. |
| spellingShingle | Aouf, M. K. Mostafa, A. O. Zayed, H. M. Ауф, М. К. Мустафа, А. О. Заїд, Г. М. Mapping properties for convolution involving hypergeometric series |
| title | Mapping properties for convolution involving
hypergeometric series |
| title_alt | Властивостi вiдображення для згортки, що включає гiпергеометричнi ряди |
| title_full | Mapping properties for convolution involving
hypergeometric series |
| title_fullStr | Mapping properties for convolution involving
hypergeometric series |
| title_full_unstemmed | Mapping properties for convolution involving
hypergeometric series |
| title_short | Mapping properties for convolution involving
hypergeometric series |
| title_sort | mapping properties for convolution involving
hypergeometric series |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1650 |
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