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Integral Operator of Analytic Functions with Positive Real Part

  • Frasin, Basem Aref
  • Received : 2010.08.03
  • Accepted : 2010.09.27
  • Published : 2011.03.31

Abstract

In this paper, we introduce the integral operator $I_{\beta}$($p_1$, ${\ldots}$, $p_n$; ${\alpha}_1$, ${\ldots}$, ${\alpha}_n$)(z) analytic functions with positive real part. The radius of convexity of this integral operator when ${\beta}$ = 1 is determined. In particular, we get the radius of starlikeness and convexity of the analytic functions with Re {f(z)/z} > 0 and Re {f'(z)} > 0. Furthermore, we derive sufficient condition for the integral operator $I_{\beta}$($p_1$, ${\ldots}$, $p_n$; ${\alpha}_1$, ${\ldots}$, ${\alpha}_n$)(z) to be analytic and univalent in the open unit disc, which leads to univalency of the operators $\int\limits_0^z(f(t)/t)^{\alpha}$dt and $\int\limits_0^z(f'(t))^{\alpha}dt$.

Keywords

Analytic and univalent functions;Starlike and convex functions;Functions of positive real part;Integral operator

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Cited by

  1. General Integral Operator of Analytic Functions Involving Functions with Positive Real Part vol.2013, 2013, https://doi.org/10.1155/2013/260127
  2. On General Integral Operator of Analytic Functions vol.2013, 2013, https://doi.org/10.1155/2013/621810