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 $$\Re e^{i\phi}\left((1-\alpha+2\gamma)\frac{f(z)}{z}+(\alpha-2\gamma)f'(z)+\gamma zf''(z)-\beta\right)>0,$$ for all $z\in E=\left\{z : |z|<1\right\}$ and for $\alpha, \gamma\geq0$ and $\beta<1$, $\phi\in \mathbb{R}$ (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 $\displaystyle\int_{0}^{1}\lambda(t)dt=1$. The aim of present paper is to find conditions on $\lambda(t)$ such that $V_{\lambda}(f)$ is convex of order $\delta$ ($0\leq\delta\leq1/2$) whenever $f\in{\mathcal{W}}_{\beta}(\alpha, \gamma)$. As applications, we study various choices of $\lambda(t)$, related to classical integral transforms.