Integral transforms of functions to be in the Pascu class using duality techniques

Abstract

Let $W_β(α,γ)$, $β<1$, denote the class of all normalized analytic functions $f$ in the unit disc ${\mathbb{D}}=\{z\in {\mathbb{C}}: |z|<1\}$ such that \begin{align*} {\rm Re\,} \left(e^{iϕ}\left((1-α+2γ)\frac{f}{z}+(α-2γ)f'+γzf"-β\right)\frac{}{}\right)>0, \quad z\in {\mathbb{D}}, \end{align*} for some $ϕ\in {\mathbb{R}}$ with $α\geq 0$, $γ\geq 0$ and $β< 1$. Let $M(ξ)$, $0\leq ξ\leq 1$, denote the Pascu class of $ξ$-convex functions given by the analytic condition \begin{align*} {\rm Re\,}\frac{ξz(zf'(z))'+(1-ξ)zf'(z)}{ξzf'(z)+(1-ξ)f(z)}>0 \end{align*} which unifies the class of starlike and convex functions. The aim of this paper is to find conditions on $λ(t)$ so that the integral transforms of the form \begin{align*} V_λ(f)(z)= \int_0^1 λ(t) \frac{f(tz)}{t} dt. \end{align*} carry functions from $W_β(α,γ)$ into $M(ξ)$. As applications, for specific values of $λ(t)$, it is found that several known integral operators carry functions from $W_β(α,γ)$ into $M(ξ)$. Results for a more generalized operator related to $V_λ(f)(z)$ are also given.