New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality

Abstract

In this paper, we introduce a new subclass of harmonic functions $f=s+\overline{t}$ in the open unit disk $U=\left\{z\in C:\left|z\right|<1\right\}$ satisfying $\text{Re}\left[\gamma s^{\prime }(z)+\delta z{s}^{\prime \prime }(z)+\left(\frac{\delta -\gamma }{2}\right)z^2{s}^{\prime \prime \prime }\left(z\right)-\lambda \right]>\left|\gamma t^{\prime }(z)+\delta z{t}^{\prime \prime }(z)+\left(\frac{\delta -\gamma }{2}\right)z^2{t}^{\prime \prime \prime }\left(z\right)\right|,$ where $0\le \lambda <\gamma \le \delta ,z\in U.$ We determine several properties of this class such as close-to-convexity, coefficient bounds, and growth estimates. We also prove that this class is closed under convex combination and convolution of its members. Furthermore, we investigate the properties of fully starlikeness and fully convexity of the class.