## Abstract and Applied Analysis

### On an Integral Transform of a Class of Analytic Functions

#### Abstract

For $\alpha ,\gamma \ge 0$ and $\beta <1$, let ${\mathrm{\scr W}}_{\beta }(\alpha ,\gamma )$ denote the class of all normalized analytic functions $f$ in the open unit disc $E=\{z:|z|<1\}$ such that $\Re {e}^{i\varphi }((1-\alpha +2\gamma )\mathrm{(f}(z)/\mathrm{z)}+(\alpha -2\gamma ){f}^{\prime }(z)+$ $\gamma z{f}^{\prime \prime }(z)-\beta )>0$, $z\in E$ for some $\varphi \in \Bbb R$. It is known (Noshiro (1934) and Warschawski (1935)) that functions in ${\mathrm{\scr W}}_{\beta }(1,0)$ are close-to-convex and hence univalent for $0\le \beta <1$. For $f\in {\mathrm{\scr W}}_{\beta }(\alpha ,\gamma )$, we consider the integral transform $F(z)={V}_{\lambda }(f)(z):={\int }_{0}^{1}\lambda (t)(f(tz)/t)dt$, where $\lambda$ is a nonnegative real-valued integrable function satisfying the condition ${\int }_{0}^{1}\lambda (t)dt=1$. The aim of present paper is, for given $\delta <1$, to find sharp values of $\beta$ such that (i) ${V}_{\lambda }(f)\in {\mathrm{\scr W}}_{\delta }(1,0)$ whenever $f\in {\mathrm{\scr W}}_{\beta }(\alpha ,\gamma )$ and (ii) ${V}_{\lambda }(f)\in {\mathrm{\scr W}}_{\delta }(\alpha ,\gamma )$ whenever $f\in {\mathrm{\scr W}}_{\beta }(\alpha ,\gamma )$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 259054, 10 pages.

Dates
First available in Project Euclid: 28 March 2013

https://projecteuclid.org/euclid.aaa/1364475870

Digital Object Identifier
doi:10.1155/2012/259054

Mathematical Reviews number (MathSciNet)
MR2984036

Zentralblatt MATH identifier
1255.30025

#### Citation

Verma, Sarika; Gupta, Sushma; Singh, Sukhjit. On an Integral Transform of a Class of Analytic Functions. Abstr. Appl. Anal. 2012 (2012), Article ID 259054, 10 pages. doi:10.1155/2012/259054. https://projecteuclid.org/euclid.aaa/1364475870

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