Some starlikeness and convexity properties for two new $p$−valent integral operators

Year 2016, Volume: 45 Issue: 5, 1367 - 1378, 01.10.2016

Erhan Deniz , Esra Deniz Nizami Mustafa

Abstract

In this paper, we define two new general $p$−valent integral operators in the unit disc $U$ and obtain the properties of $p$−valent starlikeness and $p$−valent convexity of these integral operators of $p$−valent functions on some classes of $\beta$−uniformly $p$−valent starlike and $\beta$−uniformly $p$−valent convex functions of complex order and type α $(0 ≤\leq \alpha < p)$. As special cases, the properties of $p$−valent starlikeness and $p$−valent convexity of the operators $\int_{0}^{z} pt^{p-1} \left( \frac{f(t)}{t^p}\right)^\delta dt$ and $\int_{0}^{z} pt^{p-1} \left( \frac{g'(t)}{pt^{p-1}}\right)^\delta dt$ are given.

Keywords

Analytic functions, Integral operators, $\beta$−uniformly $p$−valent starlike and $\beta$−uniformly $p$−valent convex functions, Complex order

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