Application of convolution theory on non-linear integral operators

Abstract

The class ${\mathcal{W}}_{\beta }^{\delta }(\alpha ,\gamma )$ defined in the domain $|z|<1$ satisfying
$$\begin{array}{r}\mathrm{R}\mathrm{e}{e}^{i\varphi }({\displaystyle \frac{}{}}(1-\alpha +2\gamma ){\left(f/z\right)}^{\delta }+(\alpha -3\gamma +\gamma [{\displaystyle \frac{}{}}(1-1/\delta )\left(z{f}^{\prime }/f\right)\\ +1/\delta (1+z{f}^{″}/f^{\prime })]){\displaystyle \frac{}{}}{\left(f/z\right)}^{\delta }\left(z{f}^{\prime }/f\right)-\beta )>0,\end{array}$$
with the conditions $\alpha \ge 0$, $\beta <1$, $\gamma \ge 0$, $\delta >0$ and $\varphi \in \mathbb{R}$ generalizes a particular case of the largest subclass of univalent functions, namely the class of Bazilevi\v c functions. Moreover, for
$0<\delta \le \frac{1}{(1-\zeta )}$, $0\le \zeta <1$, the class ${\mathcal{C}}_{\delta }(\zeta )$ be the subclass of normalized analytic functions such that
$$\begin{array}{r}\mathrm{R}\mathrm{e}(1/\delta (1+z{f}^{″}/f^{\prime })+(1-1/\delta )\left(z{f}^{\prime }/f\right))>\zeta ,|z|<1.\end{array}$$
In the present work, the sufficient conditions on $\lambda (t)$ are investigated, so that the non-linear integral transform
$$\begin{array}{r}{V}_{\lambda }^{\delta }(f)(z)={({\int }_0^1\lambda (t){\left(f(tz)/t\right)}^{\delta }dt)}^{1/\delta },|z|<1,\end{array}$$
carries the functions from ${\mathcal{W}}_{\beta }^{\delta }(\alpha ,\gamma )$ into ${\mathcal{C}}_{\delta }(\zeta )$. Several interesting applications are provided for special choices of $\lambda (t)$. These results are useful in the attempt to generalize the two most important extremal problems in this direction using duality techniques and provide scope for further research.