Subordination-implication problems concerning the nephroid starlikeness of analytic functions

Abstract

Abstract Let A be the set of all analytic functions f defined on the open unit disk D satisfying f(0) = f'(0) − 1 = 0. Let φNe (z) := 1 + z − z 3 =3 be the recently introduced Carathéodory function which maps the unit circle ∂ D $ \partial \mathbb{D} $ D onto a 2-cusped kidney-shaped curve called nephroid given by ( ( u − 1 ) 2 + v 2 − 4 a ) 3 − 4 v 2 2 = 0. ${{\left( {{(u-1)}^{2}}+{{v}^{2}}-\frac{4}{a} \right)}^{3}}-\frac{4{{v}^{2}}}{2}=0.$ In this paper, we determine the best possible estimate on the real β so that for some analytic p satisfying p(0) = 1 the following subordination-implication holds: 1 + β z p ′ ( z ) p j ( z ) ≺ F ( z ) ⇒ p ( z ) ≺ φ N e ( z ) ,   j = 0 , 1 , 2 , $$1+\beta \frac{z{{p}^{\prime }}(z)}{{{p}^{j}}(z)}\prec \mathcal{F}(z)\Rightarrow p(z)\prec {{\varphi }_{Ne}}(z),\quad j=0,1,2,$$ where F(z) is some Carathéodory function with special geometries like right/left-half of Bernoulliφs lemniscate, cardioid, lune, eight-shaped, etc. As applications, we establish sufficient conditions for the Ma-Minda family of nephroid starlike functions given by S N e * : = { f ∈ A : z f ′ ( z ) f ( z ) ≺ φ N e ( z ) } . $$\mathcal{S}_{Ne}^{*}:=\left\{ f\in \mathcal{A}:\frac{z{{f}^{\prime }}(z)}{f(z)}\prec {{\varphi }_{Ne}}(z) \right\}.$$