Order of convexity of Integral Transforms and Duality

Recently, Ali et al defined the class ${\mathcal{W}}_{\beta }(\alpha ,\gamma )$ consisting of functions $f$ which satisfy $$\mathrm{\Re }{e}^{i\varphi }((1-\alpha +2\gamma )\frac{f(z)}{z}+(\alpha -2\gamma )f^{\mathrm{\prime }}(z)+\gamma z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)-\beta )>0,$$ for all $z\in E=\{z:\mathrm{\mid }z\mathrm{\mid }<1\}$ 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 ${\displaystyle {\int }_0^1\lambda (t)dt=1}$. The aim of present paper is to find conditions on $\lambda (t)$ such that $V_{\lambda }(f)$ is convex of order $\delta$ ($0\le \delta \le 1\mathrm{/}2$) whenever $f\in{$\mathcal{W}$}_{\beta}(\alpha, \gamma)$\mathrm{.}Asapplications,westudyvariouschoicesof$\lambda(t)$, related to classical integral transforms.