Order of Starlikeness and Convexity of certain integral transforms using duality techniques

Abstract

For $α\geq 0$, $β<1$ and $γ\geq 0$, the class $\mathcal{W}_β(α,γ)$ satisfies the condition \begin{align*} {\rm Re\,} \left( e^{iϕ}\left((1-α+2γ)f/z+(α-2γ)f'+ γzf''-β\right)\frac{}{}\right)>0, \quad ϕ\in {\mathbb{R}},{\,}z\in {\mathbb{D}}; \end{align*} is taken into consideration. The Pascu class of $ξ$-convex functions of order $σ$ $(M(σ,{\,}ξ))$, having analytic characterization \begin{align*} {\rm Re\,}\frac{ξz(zf'(z))'+(1-ξ)zf'(z)}{ξzf'(z)+(1-ξ)f(z)}>σ,\quad 0\leq σ< 1,\quad z\in {\mathbb{D}}, \end{align*} unifies starlike and convex functions class of order $σ$.The admissible and sufficient conditions on $λ(t)$ are investigated so that the integral transforms \begin{align*} V_λ(f)(z)= \int_0^1 λ(t) \frac{f(tz)}{t} dt, \end{align*} maps the function from $\mathcal{W}_β(α,γ)$ into $M(σ,{\,}ξ)$. Further several interesting applications, for specific choice of $λ(t)$ are discussed which are related to the classical integral transform.