A new subclass of the starlike functions

Abstract

<div>Motivated by the R{\o}nning--starlike class [Proc Amer Math Soc {\bf118}, no. 1, 189--196, 1993], we introduce new class $\mathcal{S}^*_c$ includes of analytic and normalized functions $f$ which satisfy the inequality</div><div>\begin{equation*}</div><div> {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\}\geq\left|\frac{f(z)}{z}-1\right|\quad(|z|<1).</div><div>\end{equation*}</div><div>In this paper, we first give some examples which belong to the class $\mathcal{S}^*_c$. Also, we show that if $f\in\mathcal{S}^*_c$ then ${\rm</div><div>Re} \{f(z)/z\}>1/2$ in $|z|<1$ (Marx--Strohh\"{a}cker problem). Afterwards, upper and lower bounds for $|f(z)|$ are obtained where $f$ belongs to the class $\mathcal{S}^*_c$.</div><div>We also prove that if $f\in\mathcal{S}^*_c$ and $\alpha\in[0,1)$, then $f$ is starlike of order $\alpha$ in the disc $|z|<(1-\alpha)/(2-\alpha)$. At the end, we estimate logarithmic coefficients, the initial coefficients and Fekete--Szeg\"{o} problem for functions $f\in \mathcal{S}^*_c$.</div>