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Convexity of the Generalized Integral Transform and Duality Techniques

Let $\mathcal{W}_β^δ(α,γ)$ be the class of normalized analytic functions $f$ defined in the domain $|z|<1$ satisfying \begin{align*} {\rm Re\,} e^{iϕ}\left(\dfrac{}{}(1\!-\!α\!+\!2γ)\!\left({f}/{z}\right)^δ+\left(α\!-\!3γ+γ\left[\dfrac{}{}\left(1-{1}/δ\right)\left({zf'}/{f}\right)+ {1}/δ\left(1+{zf''}/{f'}\right)\right]\right)\right.\\ \left.\dfrac{}{}\left({f}/{z}\right)^δ\!\left({zf'}/{f}\right)-β\right)>0, \end{align*} with the conditions $α\geq 0$, $β<1$, $γ\geq 0$, $δ>0$ and $ϕ\in\mathbb{R}$. Moreover, for $0<δ\leq\frac{1}{(1-ζ)}$, $0\leqζ<1$, the class $\mathcal{C}_δ(ζ)$ be the subclass of normalized analytic functions such that \begin{align*} {\rm Re}{\,}\left(1/δ\left(1+zf''/f'\right)+(1-1/δ)\left({zf'}/{f}\right)\right)>ζ,\quad |z|<1. \end{align*} In the present work, the sufficient conditions on $λ(t)$ are investigated, so that the generalized integral transform \begin{align*} V_λ^δ(f)(z)= \left(\int_0^1 λ(t) \left({f(tz)}/{t}\right)^δdt\right)^{1/δ},\quad |z|<1, \end{align*} carries the functions from $\mathcal{W}_β^δ(α,γ)$ into $\mathcal{C}_δ(ζ)$. Several interesting applications are provided for special choices of $λ(t)$.