Original scientific paper

Univalence criteria for general integral operator

Basem Aref Frasin ; Department of Mathematics, Al al--Bayt University, Mafraq, Jordan

Abstract

Let $\mathcal{A}$ be the class of all analytic functions which are analytic
in the open unit disc $\mathcal{U=}\left\{ z:\left\vert z\right\vert
<1\right\} $ and
\[
G_{b}=\left\{ f\in \mathcal{A}:\left\vert \frac{%
1+zf^{\prime \prime }(z)/f^{\prime }(z)}{zf^{\prime }(z)/f(z)}-1\right\vert
\]
In this paper, we derive sufficient
conditions for the integral operator
\[
I_{\gamma }^{\alpha _{i}\ }(f_{1},...,f_{n})(z)=\left\{ \gamma
\int\limits_{0}^{z}t^{\gamma -1}\left( f_{1}^{\prime }(t)\right) ^{\alpha
_{1}}\left( \frac{f_{1}(t)}{t}\right) ^{1-\alpha _{1}}...\left(
f_{n}^{\prime }(t)\right) ^{\alpha _{n}}\left( \frac{f_{n}(t)}{t}\right)
^{1-\alpha _{n}}dt\right\} ^{\frac{1}{\gamma }}
\]
to be analytic and
univalent in the open unit disc$~\mathcal{U}$, when $f_{i}\in G_{b_{i}}$ for
all $i=1,\ldots ,n.$

Keywords

analytic and univalent functions; starlike and convex functions; integral operator

Hrčak ID:

68628

URI

https://hrcak.srce.hr/68628

Publication date:

10.6.2011.

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