In this paper, we show that the Carathéodory function \(\varphi _{\mathrm{Ne}}(z)=1+z-z^3/3\) maps the open unit disk \(\mathbb {D}\) onto the interior of the nephroid, a 2-cusped kidney-shaped curve,

and introduce new Ma–Minda-type function classes \(\mathcal {S}^*_{\mathrm{Ne}}\) and \(\mathcal {C}_{\mathrm{Ne}}\) 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 \(\mathcal {S}^*_{\mathrm{Ne}}\) and \(\mathcal {C}_{\mathrm{Ne}}\). 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 \(1+\beta {zp'(z)}/{p^j(z)}\prec \varphi _{\mathrm{Ne}}(z),j=0,1,2\), and obtain sufficient conditions for some geometrically defined function classes available in the literature.

在本文中，我们证明了Carathéodory函数\（\ varphi _ {\ mathrm {Ne}}（z）= 1 + zz ^ 3/3 \）将打开的单位磁盘\（\ mathbb {D} \）映射到肾上腺的内部，呈2尖的肾形曲线，

和引入新的麻敏达型函数类\（\ mathcal {S} ^ * _ {\ mathrm {氖}} \）和\（\ mathcal {C} _ {\ mathrm {氖}} \）与之相关联的。除了研究该类星状体边界区域的特征以外，还针对\（\ mathcal {S} ^ *类讨论了结构式，极值函数，增长和变形结果，包含结果，系数界和Fekete-Szegö问题。 _ {\ mathrm {Ne}} \）和\（\ mathcal {C} _ {\ mathrm {Ne}} \\）。此外，对于\（\ beta \ in \ mathbb {R} \）和一些满足\（p（0）= 1 \）的解析函数p（z ），我们证明了一阶微分从属\（1+ \ beta {zp'（z）} / {p ^ j（z）} \ prec \ varphi _ {\ mathrm {Ne}}（z）的某些从属含义，j = 0,1,2 \），并为文献中提供的一些几何定义函数类获得了充分的条件。