Some inequalities for a certain subclass of starlike functions

TL;DR

This paper explores new properties of a specific subclass of starlike functions, including bounds on real parts, starlikeness order, and sharp inequalities, extending previous results by Sokól.

Contribution

It introduces additional properties and sharp inequalities for the class al{SK}(al{alpha}), enhancing understanding of its geometric and coefficient bounds.

Findings

Determined the infimum of al{Re}(f(z)/z) for the class.

Established the order of strongly starlikeness.

Proved sharp inequalities for logarithmic coefficients and Fekete-Szeg
51 inequality.

Abstract

In 2011, Sok\'{o}{\l} (Comput. Math. Appl. 62, 611--619) introduced and studied the class $\mathcal{SK}(\alpha)$ as a certain subclass of starlike functions, consists of all functions $f$ ($f(0)=0=f'(0)-1$) which satisfy in the following subordination relation: \begin{equation*} \frac{zf'(z)}{f(z)}\prec \frac{3}{3+(\alpha-3)z-\alpha z^2} \qquad |z|<1, \end{equation*} where $-3<\alpha\leq1$. Also, he obtained some interesting results for the class $\mathcal{SK}(\alpha)$. In this paper, some another properties of this class, including infimum of $\mathfrak{Re}\frac{f(z)}{z}$, order of strongly starlikeness, the sharp logarithmic coefficients inequality and the sharp Fekete-Szeg\"{o} inequality are investigated.