Order of convexity of Integral Transforms and Duality

Abstract

Recently, Ali et al defined the class ${\mathcal{W}}_{\beta }(\alpha ,\gamma )$ consisting of functions $f$ which satisfy $$\Re e^{i\varphi }\left((1-\alpha +2\gamma )\frac{f(z)}{z}+(\alpha -2\gamma )f^{\prime }(z)+\gamma z{f}^{\prime \prime }(z)-\beta \right)>0,$$ for all $z\in E=\left\{z:|z|<1\right\}$ and for $\alpha ,\gamma \ge 0$ and $\beta <1$, $\varphi \in \mathbb{R}$ (the set of reals). For $f\in {\mathcal{W}}_{\beta }(\alpha ,\gamma )$, they discussed the convexity of the integral transform $$V_{\lambda }(f)(z):={\int }_0^1\lambda (t)\frac{f(tz)}{t}dt,$$ where $\lambda$ is a non-negative real-valued integrable function satisfying the condition ${\int }_0^1\lambda (t)dt=1$. The aim of present paper is to find conditions on $\lambda (t)$ such that $V_{\lambda }(f)$ is convex of order $\delta$ ($0\le \delta \le 1/2$) whenever $f\in {\mathcal{W}}_{\beta }(\alpha ,\gamma )$. As applications, we study various choices of $\lambda (t)$, related to classical integral transforms.

15 pages

References in corpus (1)

• Coefficient Inequalities for Starlikeness and Convexity

Cited by in corpus (1)

• Convexity of the Generalized Integral Transform and Duality Techniques