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

Abstract

Let $W_{\beta }(\alpha ,\gamma )$, $\beta <1$, 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 $\varphi \in \mathbb{R}$ with $\alpha \ge 0$, $\gamma \ge 0$ and $\beta <1$. Let $M(\xi )$, $0\le \xi \le 1$, denote the Pascu class of $\xi$-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 $\lambda (t)$ 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 $W_{\beta }(\alpha ,\gamma )$ into $M(\xi )$. As applications, for specific values of $\lambda (t)$, it is found that several known integral operators carry functions from $W_{\beta }(\alpha ,\gamma )$ into $M(\xi )$. Results for a more generalized operator related to $V_{\lambda }(f)(z)$ are also given.

17 pages

References in corpus (1)

• Coefficient Inequalities for Starlikeness and Convexity