Starlikeness associated with crescent-shaped region

Abstract

In this paper, we investigate the sufficient conditions for an analytic function f defined in the open unit disk ${\mathbb {D}}\subseteq {\mathbb {C}}$ satisfying $f(0)=f'(0)-1= 0$, such that the function f satisfies

$$\begin{aligned}\left| \left( \frac{zf'(z)}{f(z)}\right) ^2-1\right| <2\left| \frac{zf'(z)}{f(z)}\right| ,\quad (z\in {\mathbb {D}}).\end{aligned}$$

Let ${\mathcal {H}}({\mathbb {D}})$ denote the class of analytic functions defined in ${\mathbb {D}}$ and let ${\mathcal {H}}[1,n]:=\{p_n\in {\mathcal {H}}({\mathbb {D}}): p_n(z)=1+a_nz^n+a_{n+1}z^{n+1}+...\}$, $n\in {\mathbb {N}}$. We derive the admissibility conditions for the function $q(z):= z+\sqrt{1+z^2}$, which maps ${\mathbb {D}}$ to the crescent-shaped region and use it to establish the result if the function $p_n \in {\mathcal {H}}[1,n]$ with ${\text {Re}}p_n(z)>\sqrt{2}-1$ satisfies the second ordered differential condition $\vert z^2p_n^{\prime \prime }(z)/p_n(z)\vert < (n^2(1+\sqrt{2})-2n)/2\sqrt{2}$, $(z\in {\mathbb {D}})$ then $p_n\prec q$. We also find several first order differential sufficient conditions for the function $p_n$ to obey the subordination $p_n\prec q$. Moreover, we provide examples to demonstrate the existence of the functions $p_n\in {\mathcal {H}}[1,n]$ to satisfy these sufficient conditions.