On an Integral Transform of a Class of Analytic Functions

Abstract

For $\alpha ,\gamma \ge 0$ and , let ${𝒲}_{\beta }(\alpha ,\gamma )$ denote the class of all normalized analytic functions $f$ in the open unit disc such that $\Re e^{i\varphi }((1-\alpha +2\gamma )\mathrm{(f}(z)/\mathrm{z)}+(\alpha -2\gamma )f^{\prime }(z)+$ $\gamma z{f}^{\prime \prime }(z)-\beta )>0$, $z\in E$ for some $\varphi \in ℝ$. It is known (Noshiro (1934) and Warschawski (1935)) that functions in ${𝒲}_{\beta }(\mathrm{1,0})$ are close-to-convex and hence univalent for . For $f\in {𝒲}_{\beta }(\alpha ,\gamma )$, we consider the integral transform $F(z)=V_{\lambda }(f)(z):={\int }_0^1\lambda (t)(f(tz)/t)dt$, where $\lambda$ is a nonnegative real-valued integrable function satisfying the condition ${\int }_0^1\lambda (t)dt=1$. The aim of present paper is, for given , to find sharp values of $\beta$ such that (i) $V_{\lambda }(f)\in {𝒲}_{\delta }(\mathrm{1,0})$ whenever $f\in {𝒲}_{\beta }(\alpha ,\gamma )$ and (ii) $V_{\lambda }(f)\in {𝒲}_{\delta }(\alpha ,\gamma )$ whenever $f\in {𝒲}_{\beta }(\alpha ,\gamma )$.