The $n$-valent convexity of Frasin integral operators
UDC 517.5 Let $ f_{i},$ $i\in\lbrace 1,2,\ldots, k\rbrace,$ is an analytic function on the unit disk in the complex plane of the form$f_{i}(z) = z^{n} + a_{i,n+1}z^{n+1} + \ldots, n\in\mathbb{N} = \lbrace 1,2,\ldots\rbrace.$We consider the Frasin integral operator as follows:\begin{gather*}\label{e1...
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| author | Khani , R. Najafzadeh , Sh. Ebadian , A. Nikoufar , I. Khani , R. Najafzadeh, Sh. Ebadian, A. Nikoufar , I. |
| author_facet | Khani , R. Najafzadeh , Sh. Ebadian , A. Nikoufar , I. Khani , R. Najafzadeh, Sh. Ebadian, A. Nikoufar , I. |
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Let $ f_{i},$ $i\in\lbrace 1,2,\ldots, k\rbrace,$ is an analytic function on the unit disk in the complex plane of the form$f_{i}(z) = z^{n} + a_{i,n+1}z^{n+1} + \ldots, n\in\mathbb{N} = \lbrace 1,2,\ldots\rbrace.$We consider the Frasin integral operator as follows:\begin{gather*}\label{e1.3}G_{n}(z)=\int\limits_{0}^{z} n\xi^{(n-1)}\bigg(\dfrac{f'_{1}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{1}}\cdots\bigg(\dfrac{f'_{k}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{k}}d\xi.\end{gather*}In this paper, we obtain a sufficient condition under which this integral operator is $n$-valent convex and get other interesting results.
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DOI: 10.37863/umzh.v73i2.88
UDC 517.5
R. Khani, Sh. Najafzadeh, A. Ebadian, I. Nikoufar (Payame Noor Univ., Tehran, Iran)
THE \bfitn -VALENT CONVEXITY OF FRASIN INTEGRAL OPERATORS
\bfitn -ВАЛЕНТНА ОПУКЛIСТЬ IНТЕГРАЛЬНИХ ОПЕРАТОРIВ ФРАЗIНА
Let fi, i \in \{ 1, 2, . . . , k\} , is an analytic function on the unit disk in the complex plane of the form fi(z) = zn +
+ ai,n+1z
n+1 + . . . , n \in \BbbN = \{ 1, 2, . . .\} . We consider the Frasin integral operator as follows:
Gn(z) =
z\int
0
n\xi (n - 1)
\biggl(
f \prime
1(\xi )
n\xi n - 1
\biggr) \alpha 1
\cdot \cdot \cdot
\biggl(
f \prime
k(\xi )
n\xi n - 1
\biggr) \alpha k
d\xi .
In this paper, we obtain a sufficient condition under which this integral operator is n-valent convex and get other interesting
results.
Нехай fi, i \in \{ 1, 2, . . . , k\} , — аналiтична функцiя на одиничному диску у комплекснiй площинi, яка має вигляд
fi(z) = zn + ai,n+1z
n+1 + . . . , n \in \BbbN = \{ 1, 2, . . .\} . Розглядається iнтегральний оператор Фразiна вигляду
Gn(z) =
z\int
0
n\xi (n - 1)
\biggl(
f \prime
1(\xi )
n\xi n - 1
\biggr) \alpha 1
\cdot \cdot \cdot
\biggl(
f \prime
k(\xi )
n\xi n - 1
\biggr) \alpha k
d\xi .
Отримано достатнi умови, за яких цей iнтегральний оператор є n-валентно опуклим, та iншi цiкавi результати.
1. Introduction. Let \BbbD = \{ z \in \BbbC : | z| < 1\} be the unit disk and let \scrA (n) be the class of all analytic
functions in \BbbD of the form
f(z) = zn + an+1z
n+1 + . . . , n \in \BbbN .
So \scrA := \scrA (1).
For f, g \in \scrA , we say that the function f(z) is subordinate to g(z), written by f(z) \prec g(z), if
exists an analytic function w(z) with w(0) = 0, | w(z)| < 1 for all z \in \BbbD such that f(z) = g
\bigl(
w(z)
\bigr)
.
If g(z) is univalent in \BbbD , then the subordination f(z) \prec g(z) is equivalent to f(0) = g(0) and
f(\BbbD ) \subseteq g(\BbbD ).
A function f \in \scrA (n) is said to be n-valent starlike functions of order \beta in \BbbD , if it satisfies the
inequality
\Re
\biggl(
zf \prime (z)
f(z)
\biggr)
> \beta , z \in \BbbD , 0 \leq \beta < n, n \in \BbbN ,
and we denote this class by S\ast
n(\beta ). If a function f \in \scrA (n) satisfies the following inequality:
\Re
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
> \beta , z \in \BbbD , 0 \leq \beta < n, n \in \BbbN ,
then it is said to be n-valent convex functions of order \beta in \BbbD and we denote this class by Cn(\beta ).
It is known that S\ast
1(\beta ) = S\ast (\beta ) and C1(\beta ) = C(\beta ) (the class of starlike functions of order \beta and
convex functions of order \beta , respectively). These classes are subclasses of the class of univalent
functions and, moreover, C \subseteq S\ast (see [3]), where C = C(0) and S\ast = S\ast (0) (the class of convex
functions and starlike functions, respectively).
c\bigcirc R. KHANI, SH. NAJAFZADEH, A. EBADIAN, I. NIKOUFAR, 2021
278 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
THE n-VALENT CONVEXITY OF FRASIN INTEGRAL OPERATORS 279
For fi \in \scrA and \alpha i > 0, i \in \{ 1, 2, . . . , k\} , Breaz et al. in [1] introduced the following integral
operator:
F\alpha 1,...,\alpha k
(z) =
z\int
0
\bigl(
f \prime
1(\xi )
\bigr) \alpha 1 . . .
\bigl(
f \prime
k(\xi )
\bigr) \alpha k d\xi . (1.1)
The most recent, Frasin [4] introduced the following integral operator, for \alpha i > 0 and fi \in \scrA n,
i \in \{ 1, 2, . . . , k\} :
Gn(z) =
z\int
0
n\xi (n - 1)
\biggl(
f \prime
1(\xi )
n\xi n - 1
\biggr) \alpha 1
. . .
\biggl(
f \prime
k(\xi )
n\xi n - 1
\biggr) \alpha k
d\xi . (1.2)
2. Preliminaries. In order to give our results, we need the following corollary, which is due to
E. Deniz [2].
Corollary 2.1. Let the function f(z) \in \scrA (n) satisfies the inequality
\Re
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
>
2n - 1
2
.
Then
\Re
\biggl(
f \prime (z)
zn - 1
\biggr)
>
n
2
.
3. Main results. In this section, we formulate and prove main results.
Theorem 3.1. Let \alpha i > 0 for all i \in \{ 1, 2, . . . , k\} and fi \in \scrA (n) such that
\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
\geq \beta i, z \in \BbbD ,
where \beta i \in \BbbR and
\sum k
i=1
\beta i\alpha i < n. If
\sum k
i=1
\alpha i \leq 1, then Gn is n-valent convex function of order\sum k
i=1
\beta i\alpha i. Here Gn is the integral operator define as in (1.2).
Proof. From (1.2), we observe that Gn \in \scrA (n) and obtain
G\prime
n(z) = nzn - 1
\biggl(
f \prime
1(z)
nzn - 1
\biggr) \alpha 1
. . .
\biggl(
f \prime
k(z)
nzn - 1
\biggr) \alpha k
.
Differentiating the above expression logarithmically and multiply by z we get
zG\prime \prime
n(z)
G\prime
n(z)
= (n - 1) +
k\sum
i=1
\alpha i
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
- (n - 1)
\biggr]
.
So, we have
zG\prime \prime
n(z)
G\prime
n(z)
+ 1 = n+
k\sum
i=1
\alpha i
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1 - n
\biggr]
=
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
280 R. KHANI, SH. NAJAFZADEH, A. EBADIAN, I. NIKOUFAR
= n+
k\sum
i=1
\alpha i
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
- n
k\sum
i=1
\alpha i. (3.1)
Since
\sum k
i=1
\alpha i \leq 1, then by hypothesis we have
\Re
\biggl[
zG\prime \prime
n(z)
G\prime
n(z)
+ 1
\biggr]
\geq
k\sum
i=1
\alpha i\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
\geq
k\sum
i=1
\beta i\alpha i. (3.2)
Thus, Gn(z) is n-valent convex of order
\sum k
i=1
\beta i\alpha i.
Theorem 3.1 is proved.
Corollary 3.1. Let \alpha i > 0 for all i \in \{ 1, 2, . . . , k\} such that
\sum k
i=1
\alpha i \leq 1. If fi \in Cn(\beta i), then
Gn is n-valent convex function of order
\sum k
i=1
\beta i\alpha i.
Proof. Since fi \in Cn(\beta i), then 0 \leq \beta i < n and so 0 \leq
\sum k
i=1
\beta i\alpha i < n. Therefore by using
the relation (3.2), the proof of this theorem is obvious.
If we put n = 1 in Theorem 3.1, then we get the following corollary.
Corollary 3.2. Let \alpha i > 0 for all i \in \{ 1, 2, . . . , k\} and fi \in \scrA such that
\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
\geq \beta i, z \in \BbbD ,
where \beta i \in \BbbR and
\sum k
i=1
\beta i\alpha i < 1. If
\sum k
i=1
\alpha i \leq 1, then F\alpha 1,...,\alpha k
is the convex function of order\sum k
i=1
\beta i\alpha i. Here F\alpha 1,...,\alpha k
is the integral operator define as in (1.1).
Corollary 3.3. Let \alpha i > 0 for all i \in \{ 1, 2, . . . , k\} such that
\sum k
i=1
\alpha i \leq 1. If fi \in C(\beta i), then
F\alpha 1,...,\alpha k
is convex function of order
\sum k
i=1
\beta i\alpha i.
Theorem 3.2. Let fi be in the class \scrS . If r > 0 satisfies the inequality
r2 - 4r + 1
1 - r2
k\sum
i=1
\alpha i > 0
such that
\sum k
i=1
\alpha i \leq 1, then F\alpha 1,...,\alpha k
is convex univalent function in the disk | z| < r.
Proof. It is known that fi \in \scrS , then for z = rei\theta
\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
>
r2 - 4r + 1
1 - r2
.
Since
\sum k
i=1
\alpha i \leq 1, then we get
1 +
k\sum
i=1
\alpha i\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
\biggr]
\geq
k\sum
i=1
\alpha i\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
. (3.3)
If we put n = 1, in equation (3.1) and use of the hypothesis of this theorem and applying relation
(3.2), then we get that the integral operator F\alpha 1,...,\alpha k
is the convex function.
Theorem 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
THE n-VALENT CONVEXITY OF FRASIN INTEGRAL OPERATORS 281
Theorem 3.3. Let \alpha i > 0 for all i \in \{ 1, 2, . . . , k\} . Also, let fi \in \scrA such that\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
\geq \beta i, z \in \BbbD ,
where \beta i \in \BbbR ,
\sum k
i=1
\beta i\alpha i < 1 and
\sum k
i=1
\alpha i \leq 1. Assume that g(z) = a+ bnz
n + bn+1z
n+1 + . . .
is analytic in \BbbD . If
g(z) +
zg\prime (z)
c
\prec F\alpha 1,...,\alpha k
(z), z \in \BbbD , (3.4)
for \Re (c) \geq 0, c \not = 0, then
g(z) \prec qn(z) \prec F\alpha 1,...,\alpha k
(z), z \in \BbbD , (3.5)
where qn(z) =
c
nzc/n
\int z
0
tc/n - 1F\alpha 1,...,\alpha k
(t)dt. Moreover, the function qn(z) is convex univalent and
is the best dominant of (3.4) in the sense that g \prec qn for all g satisfying (3.4) and if there exists q
such that g \prec q for all g satisfying (3.4), then qn \prec q.
Proof. It is known [5] that the subordination (3.4) with convex univalent right-hand side is
sufficient for (3.5) with the best dominated qn(z). By Theorem 3.1 the function F\alpha 1,...,\alpha k
is convex
univalent in the unit disk and we get the result.
Theorem 3.4. Let fi \in \scrA (n), \alpha i > 0 for all i \in \{ 1, 2, . . . , k\} and
\sum k
i=1
\alpha i \leq 1. If
k\sum
i=1
\alpha i\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
\geq 2n - 1
2
, (3.6)
then
\Re
\Biggl[
k\prod
i=1
\biggl(
f \prime
i(z)
nzn - 1
\biggr) \alpha k
\Biggr]
>
1
2
.
Proof. Since
\sum k
i=1
\alpha i \leq 1, then by relation (3.1) we have
\Re
\biggl[
zG\prime \prime
n(z)
G\prime
n(z)
+ 1
\biggr]
\geq
k\sum
i=1
\alpha i\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
.
We know that the integral operator Gn(z) \in \scrA (n). So, by using Corollary 2.1 and applying equation
(3.6), we get
\Re
\biggl(
G\prime
n(z)
zn - 1
\biggr)
>
n
2
.
Therefore,
\Re
\Biggl[
k\prod
i=1
\biggl(
f \prime
i(z)
nzn - 1
\biggr) \alpha k
\Biggr]
>
1
2
.
Theorem 3.4 is proved.
We put n = 1 in Theorem 3.4, then we get the following corollary.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
282 R. KHANI, SH. NAJAFZADEH, A. EBADIAN, I. NIKOUFAR
Corollary 3.4. Let fi \in \scrA , \alpha i > 0 for all i \in \{ 1, 2, . . . , k\} and
\sum k
i=1
\alpha i \leq 1. If
k\sum
i=1
\alpha i\Re
\biggl[
zf \prime \prime
i (z)
f \prime
i(z)
+ 1
\biggr]
\geq 1
2
,
then
\Re
\Biggl[
k\prod
i=1
\bigl(
f \prime
i(z)
\bigr) \alpha k
\Biggr]
>
1
2
.
If we take k = 1 in Corollary 3.4, then we obtain the following result.
Corollary 3.5. If f \in C(1/2) and 0 < \alpha \leq 1, then \Re (f \prime (z))\alpha >
1
2
.
References
1. D. Breaz, S. Owa, N. Breaz, A new integral univalent operator, Acta Univ. Apulensis. Math. Inform., 16, 11 – 16
(2008).
2. E. Deniz, On p-valent close-to-convex starlike and convex funtions, Hacet. J. Math. and Stat., 41, № 5, 635 – 642
(2012).
3. P. L. Duren, Univalent functions, Springer, New York (1983).
4. B. A. Frasin, New general integral operators of p-valent functions, J. Inequal. Pure and Appl. Math., 10, № 4, Article
109 (2009), 9 p.
5. D. J. Hallenbeck, St. Ruscheweyh, Subordination by convex functions, Proc. Amer. Math. Soc., 52, 191 – 195 (1975).
Received 15.05.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
|
| id | umjimathkievua-article-88 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:14Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b0/5f8c0af77337eab9ba0d437b0c6602b0.pdf |
| spelling | umjimathkievua-article-882025-03-31T08:48:28Z The $n$-valent convexity of Frasin integral operators The $n$-valent convexity of Frasin integral operators Khani , R. Najafzadeh , Sh. Ebadian , A. Nikoufar , I. Khani , R. Najafzadeh, Sh. Ebadian, A. Nikoufar , I. UDC 517.5 Let $ f_{i},$ $i\in\lbrace 1,2,\ldots, k\rbrace,$ is an analytic function on the unit disk in the complex plane of the form$f_{i}(z) = z^{n} + a_{i,n+1}z^{n+1} + \ldots, n\in\mathbb{N} = \lbrace 1,2,\ldots\rbrace.$We consider the Frasin integral operator as follows:\begin{gather*}\label{e1.3}G_{n}(z)=\int\limits_{0}^{z} n\xi^{(n-1)}\bigg(\dfrac{f'_{1}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{1}}\cdots\bigg(\dfrac{f'_{k}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{k}}d\xi.\end{gather*}In this paper, we obtain a sufficient condition under which this integral operator is $n$-valent convex and get other interesting results. &nbsp; УДК 517.5 $n$-валентна опуклість інтегральних операторів Фразіна Нехай $ f_{i},$ $i\in\lbrace 1,2,\ldots, k\rbrace,$, — аналiтична функцiя на одиничному диску у комплекснiй площинi, яка має вигляд$f_{i}(z) = z^{n} + a_{i,n+1}z^{n+1} + \ldots, n\in\mathbb{N} = \lbrace 1,2,\ldots\rbrace.$ Розглядається інтегральний оператор Фразіна вигляду\begin{gather*}\label{e1.3}G_{n}(z)=\int\limits_{0}^{z} n\xi^{(n-1)}\bigg(\dfrac{f'_{1}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{1}}\cdots\bigg(\dfrac{f'_{k}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{k}}d\xi.\end{gather*}Отримано достатні умови, за яких цей інтегральний оператор є $n$-валентно опуклим, та інші цікаві результати. Institute of Mathematics, NAS of Ukraine 2021-02-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/88 10.37863/umzh.v73i2.88 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 2 (2021); 278 - 282 Український математичний журнал; Том 73 № 2 (2021); 278 - 282 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/88/8972 Copyright (c) 2021 shahram Najafzadeh - |
| spellingShingle | Khani , R. Najafzadeh , Sh. Ebadian , A. Nikoufar , I. Khani , R. Najafzadeh, Sh. Ebadian, A. Nikoufar , I. The $n$-valent convexity of Frasin integral operators |
| title | The $n$-valent convexity of Frasin integral operators |
| title_alt | The $n$-valent convexity of Frasin integral operators |
| title_full | The $n$-valent convexity of Frasin integral operators |
| title_fullStr | The $n$-valent convexity of Frasin integral operators |
| title_full_unstemmed | The $n$-valent convexity of Frasin integral operators |
| title_short | The $n$-valent convexity of Frasin integral operators |
| title_sort | $n$-valent convexity of frasin integral operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/88 |
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