Some functional inequalities and inclusion relationships associated with certain families of integral operators
type: Item
instance of: scholarly article
title: Some functional inequalities and inclusion relationships associated with certain families of integral operators
zbMATH Open document ID: 1085.30014
author: Chun-Yi Gao
author: Shao-Mou Yuan
author: H. M. Srivastava
publication date: 2006-02-02T00:00:00Z
review text: Several subclasses of starlike, convex, close-to-convex and quasi-convex functions are considered. Let \(A\) be the class of functions \(f\) normalized by \[ f(z)=z+\sum^\infty_{n=2}a_nz^n, \] which are analytic in the open unit disk \(U:=\{z\in\mathbb{C}\mid|z|<1\}\). Let \(\alpha,\beta\) be real numbers with \(0\leq \alpha\), \(\beta<1\). Let \[ \begin{aligned} {\mathcal S}^* (\alpha): & =\left\{f\in A\mid \text{Re}\left(\frac{zf'(z)}{f(z)}\right)> \alpha,\quad\forall z\in U\right\},\\ \mathbb{K} (\alpha): & =\left\{f\in A\mid \text{Re}\left(1+\frac {zf''(z)}{f'(z)} \right)> \alpha,\quad\forall z\in U \right\},\\ {\mathcal C}(\beta,\alpha): & =\left\{ f\in A\mid g \in {\mathcal S}^*(\alpha)\text{ and }\text{Re}\left(\frac {zf'(z)}{g(z)}\right)> \beta, \quad \forall z\in U\right\},\\ {\mathcal C}^*(\beta,\alpha): & =\left\{f\in A \mid g\in\mathbb{K} (\alpha)\text{ and }\text{Re}\left(\frac{zf''(z)+f'(z)} {g'(z)} \right)> \beta,\quad\forall z\in U\right\},\end{aligned} \] and let \[ Q_\lambda^\mu f(z):=z+ \frac{\Gamma(\lambda+\mu+1)} {\Gamma(\lambda+1)} \sum^\infty_{n=2}\frac {\Gamma (\lambda+n)}{\Gamma(\lambda+ \mu+n)}a_nz^n \] for \(\lambda>-1\), \(\mu>0\), \(f\in A\), where \(\Gamma\) denotes the Gamma function. Let \[ \begin{aligned} {\mathcal S}^*_{\lambda, \mu} (\alpha): & =\bigl\{f\in A\mid Q_\lambda^\mu \mid Q^\mu_\lambda f\in{\mathcal S}^* (\alpha)\bigr\},\\ \mathbb{K}_{\lambda, \mu}(\alpha): & =\bigl\{f\in A\mid Q_\lambda^\mu f\in \mathbb{K}(\alpha) \bigr\},\\ {\mathcal C}_{\lambda,\mu}(\beta,\alpha): & =\bigl\{ f\in A\mid Q_\lambda^\mu f\in{\mathcal C}(\beta,\alpha)\bigr\} \end{aligned} \] and \[ {\mathcal C}^*_{\lambda,\mu}(\beta,\alpha):=\bigl\{f\in A\mid Q_\lambda^\mu f \in{\mathcal C}^*(\beta,\alpha)\bigr\}. \] Using the Miller-Mocanu lemma [\textit{S. S. Miller} and \textit{P. T. Mocanu}, J. Math. Anal. Appl. 65, 289--305 (1978; Zbl 0367.34005)], the authors prove that the inclusions \({\mathcal S}^*_{\lambda,\mu} (\alpha)\subset{\mathcal S}^*_{\lamb…
reviewed by: Choonkil Park
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 30C50
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 5003100
zbMATH Keywords: analytic functions
zbMATH Keywords: univalent functions
zbMATH Keywords: starlike functions
zbMATH Keywords: convex functions
zbMATH Keywords: close-to-convex functions
zbMATH Keywords: quasi-convex functions
zbMATH Keywords: Jung-Kim-Srivastava operators
zbMATH Keywords: Bernardi operator
zbMATH Keywords: Miller-Mocanu Lemma
zbMATH Keywords: multiplier transformation
zbMATH Keywords: fractional integral
zbMATH Keywords: fractional derivative
MaRDI profile type: Publication
full work available at URL: j.camwa.2004.10.041
OpenAlex ID: W2082833212
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DOI: 10.1016/J.CAMWA.2004.10.041
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 5003100