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

TL;DR

This paper introduces new function classes associated with a nephroid-shaped domain mapped from the unit disk, analyzing their properties, extremal functions, and differential subordination implications.

Contribution

It defines novel Ma-Minda type classes linked to a nephroid domain and explores their structural, growth, distortion, and coefficient bounds, along with subordination results.

Findings

Characterization of the nephroid domain and its mapping properties.

Derivation of extremal functions and coefficient bounds.

Establishment of subordination conditions for related function classes.

Abstract

In this paper, we show that the Carath\'{e}odory function $\varphi_{\scriptscriptstyle {Ne}}(z)=1+z-z^3/3$ 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 $\mathcal{S}^*_{Ne}$ 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-Szeg\"{o} problems are discussed for the classes $\mathcal{S}^*_{Ne}$ 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 $1+\beta\frac{zp'(z)}{p^j(z)}\prec\varphi_{\scriptscriptstyle {Ne}}(z),\,j=0,1,2,$ and obtain sufficient conditions for some geometrically defined function classes available in the literature.