Inclusion properties of Generalized Integral Transform using Duality Techniques

Let $\mathcal{W}_β^δ(α,γ)$ be the class of normalized analytic functions $f$ defined in the region $|z|<1$ and 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 $ϕ\in\mathbb{R}$. For a non-negative and real-valued integrable function $λ(t)$ with $\int_0^1λ(t) dt=1$, the generalized non-linear integral transform is defined as \begin{align*} V_λ^δ(f)(z)= \left(\int_0^1 λ(t) \left({f(tz)}/{t}\right)^δdt\right)^{1/δ}. \end{align*} The main aim of the present work is to find conditions on the related parameters such that $V_λ^δ(f)(z)\in\mathcal{W}_{β_1}^{δ_1}(α_1,γ_1)$, whenever $f\in\mathcal{W}_{β_2}^{δ_2}(α_2,γ_2)$. Further, several interesting applications for specific choices of $λ(t)$ are discussed.

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