A new subclass of analytic functions

Year 2020, Volume: 49 Issue: 4, 1346 - 1354, 06.08.2020

Hesam Mahzon Janusz Sokol

https://doi.org/10.15672/hujms.549313

Abstract

In the present paper, we introduce a class $\mathcal{B}_\theta(\alpha,\beta)$ of functions, analytic in $|z|<1$, such that $f(0)=0$, $f'(0)=1$ and
\[\alpha< Re\left(f'(z)+\frac{1+e^{i\theta}}{2}zf''(z)\right)<\beta\quad (|z|<1),\]
where $\theta\in(-\pi,\pi]$, $0\leq\alpha<1$ and $\beta>1$. Integral representation, differential subordination results and coefficient estimates are considered. Also Fekete-Szegö coefficient functional associated with the $k$--th root transform $[f(z^k)]^{1/k}$ for functions in the class $\mathcal{B}_\theta(\alpha,\beta)$ is investigated.

Keywords

analytic functions, univalent functions, Carathéodory functions, differential subordination, Fekete-Szegö inequality

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