Inclusion properties of Generalized Integral Transform using Duality Techniques

Let ${\mathcal{W}}_{\beta }^{\delta }(\alpha ,\gamma )$ be the class of normalized analytic functions $f$ defined in the region $\mathrm{\mid }z\mathrm{\mid }<1$ and 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 $\varphi \in \mathbb{R}$. For a non-negative and real-valued integrable function $\lambda (t)$ with ${\int }_0^1\lambda (t)dt=1$, the generalized non-linear integral transform is defined as \begin{align*} V_{\lambda}^\delta(f)(z)= \left(\int_0^1 \lambda(t) \left({f(tz)}/{t}\right)^\delta dt\right)^{1/\delta}. \end{align*} The main aim of the present work is to find conditions on the related parameters such that $V_{\lambda }^{\delta }(f)(z)\in {\mathcal{W}}_{{\beta }_1}^{{\delta }_1}({\alpha }_1,{\gamma }_1)$, whenever $f\in {\mathcal{W}}_{{\beta }_2}^{{\delta }_2}({\alpha }_2,{\gamma }_2)$. Further, several interesting applications for specific choices of $\lambda (t)$ are discussed.