A1 Refereed original research article in a scientific journal
A new subclass of the starlike functions
Authors: Hesam Mahzoon, Rahim Kargar, Janusz Sokol
Publisher: Scientific and Technical Research Council of Turkey
Publication year: 2019
Journal: Turkish Journal of Mathematics
Volume: 43
Issue: 5
First page : 2354
Last page: 2365
Number of pages: 12
ISSN: 1300-0098
eISSN: 1303-6149
DOI: https://doi.org/10.3906/mat-1906-64
Web address : http://journals.tubitak.gov.tr/math/issues/mat-19-43-5/mat-43-5-23-1906-64.pdf
Self-archived copy’s web address: https://research.utu.fi/converis/portal/Publication/42411498
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\begin{equation*} {\rm Re}\left\{\frac{zf'(z)}{f(z)}\right\}\geq\left|\frac{f(z)}{z}-1\right|\quad(|z|<1).\end{equation*}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 ${\rmRe} \{f(z)/z\}>1/2$ in $|z|<1$ (Marx--Strohh\"{a}cker problem). Afterwards, upper and lower bounds for $|f(z)|$ are obtained where $f$ belongs to the class $\mathcal{S}^*_c$.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 $f\in \mathcal{S}^*_c$.
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