A Third-Order Differential Equation and Starlikeness of a Double Integral Operator

Abstract

Functions$f\left(z\right)=z+{{\displaystyle \sum }}_2^{\infty }{a}_n{z}^n$ that are analytic in the unit disk and satisfy the differential equation $f\text{'}\left(z\right)+\alpha \mathrm{zf}\text{'}\text{'}\left(z\right)+\gamma z^{2}f\text{'}\text{'}\text{'}\left(z\right)=g\left(z\right)$ are considered, where $g$ is subordinated to a normalized convex univalent function $h$. These functions $f$ are given by a double integral operator of the form $f\left(z\right)={{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^1\mathrm{?}G\left(z{t}^{\mu }{s}^{\nu }\right)t^{-\mu }{s}^{-\nu }\text{i}tdsdt$ $f\left(z\right)={{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^1\mathrm{?}G\left(z{t}^{\mu }{s}^{\nu }\right)t^{-\mu }{s}^{-\nu }\text{i}tdsdt$ with $G\text{'}$ subordinated to $h$. The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function $h$.