Convexity of the Generalized Integral Transform and Duality Techniques

Let ${\mathcal{W}}_{\beta }^{\delta }(\alpha ,\gamma )$ be the class of normalized analytic functions $f$ defined in the domain $\mathrm{\mid }z\mathrm{\mid }<1$ satisfying \begin{align*} {\rm Re,} e^{i\phi}\left(\dfrac{}{}(1!-!\alpha!+!2\gamma)!\left({f}/{z}\right)^\delta +\left(\alpha!-!3\gamma+\gamma\left[\dfrac{}{}\left(1-{1}/{\delta}\right)\left({zf’}/{f}\right)+ {1}/{\delta}\left(1+{zf’‘}/{f’}\right)\right]\right)\right.\ \left.\dfrac{}{}\left({f}/{z}\right)^\delta !\left({zf’}/{f}\right)-\beta\right)>0, \end{align*} with the conditions $\alpha \ge 0$, $\beta <1$, $\gamma \ge 0$, $\delta >0$ and $\varphi \in \mathbb{R}$. Moreover, for $0<\delta \le \frac{1}{(1-\zeta )}$, $0\le \zeta <1$, the class ${\mathcal{C}}_{\delta }(\zeta )$ be the subclass of normalized analytic functions such that \begin{align*} {\rm Re}{,}\left(1/\delta\left(1+zf’‘/f’\right)+(1-1/\delta)\left({zf’}/{f}\right)\right)>\zeta,\quad |z|<1. \end{align*} In the present work, the sufficient conditions on $\lambda (t)$ are investigated, so that the generalized integral transform \begin{align*} V_{\lambda}^\delta(f)(z)= \left(\int_0^1 \lambda(t) \left({f(tz)}/{t}\right)^\delta dt\right)^{1/\delta},\quad |z|<1, \end{align*} carries the functions from ${\mathcal{W}}_{\beta }^{\delta }(\alpha ,\gamma )$ into ${\mathcal{C}}_{\delta }(\zeta )$. Several interesting applications are provided for special choices of $\lambda (t)$.