Bulletin of the Belgian Mathematical Society - Simon Stevin

Application of duality techniques to starlikeness of weighted integral transforms

R. Aghalary, A. Ebadian, and S. Shams

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Abstract

Let $\mathcal{A}$ be the class of normalized analytic functions in the unit disc and let $P_{\gamma}(\alpha, \beta)$ be the class of all functions $f \in \mathcal{A}$ satisfying the condition \[ \exists \ \eta \in \mathbb{R}, \quad \Re \left \{ e^{i \eta}\left[(1-\gamma)\left(\frac{f(z)}{z}\right)^{\alpha} + \gamma \frac{zf'(z)}{f(z)}\left(\frac{f(z)}{z}\right)^{\alpha} - \beta \right] \right \} 0 .\] We consider the integral transform \[ V_{\lambda, \alpha}(f)(z)=\left\{\int_{0}^{1}\lambda(t) \left(\frac{f(tz)}{t} \right)^{\alpha} dt\right\}^{\frac{1}{\alpha}},\] where $\lambda(t)$ is a real-valued nonnegative weight function normalized by\linebreak $\int_{0}^{1}\lambda(t) dt=1$. In this paper we find conditions on the parameters $\alpha, \beta, \gamma, \mu $ such that $V_{\lambda, \alpha}(f)$ maps $P_{\gamma}(\alpha, \beta)$ into the class of starlike functions of order $\mu$. We also provide a number of applications for various choices of $\lambda(t)$. Our results generalize known results on this topic.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 2 (2010), 275-285.

Dates
First available in Project Euclid: 26 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1274896206

Digital Object Identifier
doi:10.36045/bbms/1274896206

Mathematical Reviews number (MathSciNet)
MR2663473

Zentralblatt MATH identifier
1194.30009

Subjects
Primary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Secondary: 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination

Keywords
Convolution Univalent functions Hypergeometric function Duality technique Starlike functions

Citation

Ebadian, A.; Aghalary, R.; Shams, S. Application of duality techniques to starlikeness of weighted integral transforms. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 2, 275--285. doi:10.36045/bbms/1274896206. https://projecteuclid.org/euclid.bbms/1274896206


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