Starlike And Convex Functions Associated with A Nephroid domain having Cusps On The Real Axis

Abstract

In this paper, we show that the Carathéodory function ${\phi }_{Ne}(z)=1+z-z^3/3$ maps the open unit disk $\mathbb{D}$ onto the interior of the nephroid, a $2$-cusped kidney-shaped curve, \begin{align*} \left((u-1)^2+v^2-\frac{4}{9}\right)^3-\frac{4 v^2}{3}=0, \end{align*} and introduce new Ma-Minda type function classes ${\mathcal{S}}_{Ne}^{\ast }$ and ${\mathcal{C}}_{Ne}$ associated with it. Apart from studying the characteristic properties of the region bounded by this nephroid, the structural formulas, extremal functions, growth and distortion results, inclusion results, coefficient bounds and Fekete-Szegö problems are discussed for the classes ${\mathcal{S}}_{Ne}^{\ast }$ and ${\mathcal{C}}_{Ne}$. Moreover, for $\beta \in \mathbb{R}$ and some analytic function $p(z)$ satisfying $p(0)=1$, we prove certain subordination implications of the first order differential subordination $1+\beta \frac{z{p}^{\prime }(z)}{p^j(z)}\prec {\phi }_{Ne}(z),j=0,1,2,$ and obtain sufficient conditions for some geometrically defined function classes available in the literature.

23 pages, 4 figures