Starlikeness of the generalized integral transform using duality techniques

Abstract

For $\alpha \ge 0$, $\delta >0$, $\beta <1$ and $\gamma \ge 0$, the class ${\mathcal{W}}_{\beta }^{\delta }(\alpha ,\gamma )$ consist of analytic and normalized functions $f$ along with the condition \begin{align*} {\rm Re\,} e^{iϕ}(\dfrac{}{}(1\!-\!α\!+\!2γ)\!({f}/{z})^δ+(α\!-\!3γ\!+\!γ[\dfrac{}{}(1-{1}/δ)({zf'}/{f})+ {1}/δ(1+{zf''}/{f'})]).\\ .\dfrac{}{}({f}/{z})^δ\!({zf'}/{f})-β)>0, \end{align*} where $\varphi \in \mathbb{R}$ and $|z|<1$, is taken into consideration. The class ${\mathcal{S}}_s^{\ast }(\zeta )$ be the subclass of the univalent functions, defined by the analytic characterization $\mathrm{Re}(z{f}^{\prime }/f)>\zeta$, for $0\le \zeta <1$, $0<\delta \le \frac{1}{(1-\zeta )}$ and $|z|<1$. The admissible and sufficient conditions on $\lambda (t)$ are examined, so that the generalized and non-linear integral transforms \begin{align*} V_λ^δ(f)(z)= (\int_0^1 λ(t) ({f(tz)}/{t})^δdt)^{1/δ}, \end{align*} maps the function from ${\mathcal{W}}_{\beta }^{\delta }(\alpha ,\gamma )$ into ${\mathcal{S}}_s^{\ast }(\zeta )$. Moreover, several interesting applications for specific choices of $\lambda (t)$ are discussed, that are related to some well-known integral operators.

24 pages

Cited by in corpus (1)

• Convexity of the Generalized Integral Transform and Duality Techniques