A new subclass of starlike functions

Authors: HESAM MAHZOON, RAHIM KARGAR, JANUSZ SOKOL

Abstract: Motivated by the Ronning-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 $$ {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\}\geq\left|\frac{f(z)}{z}-1\right|\quad(|z|<1). $$ 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 ${\rm Re} \{f(z)/z\}>1/2$ in $|z|<1$ (Marx-Strohhacker problem). Afterwards, upper and lower bounds for $|f(z)|$ are obtained where $f$ belongs to the class $\mathcal{S}^*_c$. 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-Szegö problem for functions $f\in \mathcal{S}^*_c$.

Keywords: Starlike, subordination, Marx?Strohhäcker problem, logarithmic coefficients, Fekete?Szegö problem

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