Commun. Korean Math. Soc. 2022; 37(4): 1025-1039
Online first article September 21, 2022 Printed October 31, 2022
https://doi.org/10.4134/CKMS.c210322
Copyright © The Korean Mathematical Society.
Somya Malik, Vaithiyanathan Ravichandran
National Institute of Technology; National Institute of Technology
For given non-negative real numbers $\alpha_k$ with $ \sum_{k=1}^{m}\alpha_k =1$ and normalized analytic functions $f_k$, $k=1,\dotsc,m$, defined on the open unit disc, let the functions $F$ and $F_n$ be defined by $ F(z):=\sum_{k=1}^{m}\alpha_k f_k (z)$, and $F_{n}(z):=n^{-1}\sum_{j=0}^{n-1} e^{-2j\pi i/n} F(e^{2j\pi i/n} z)$. This paper studies the functions $f_k$ satisfying the subordination $zf'_{k} (z)/F_{n} (z) \prec h(z)$, where the function $h$ is a convex univalent function with positive real part. We also consider the analogues of the classes of starlike functions with respect to symmetric, conjugate, and symmetric conjugate points. Inclusion and convolution results are proved for these and related classes. Our classes generalize several well-known classes and the connections with the previous works are indicated.
Keywords: Starlike functions, convex functions, symmetric points, conjugate points, convolution
MSC numbers: 30C80, 30C45
Supported by: The first author is supported by the UGC-JRF Scholarship.
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