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Kyungpook Mathematical Journal 2024; 64(1): 57-68

Published online March 31, 2024 https://doi.org/10.5666/KMJ.2024.64.1.57

Copyright © Kyungpook Mathematical Journal.

Polylogarithms and Subordination of Some Cubic Polynomials

Manju Yadav, Sushma Gupta and Sukhjit Singh

Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148106, India
e-mail : manjuyadav1197@gmail.com, sushmagupta1@yahoo.com and sukhjit_d@yahoo.com

Received: January 20, 2023; Accepted: September 15, 2023

Abstract

Let V3(z, f) and σ3(1)(z,f) be the cubic polynomials representing, respectively, the 3rd de la Vallée Poussin mean and the 3rd Cesàro mean of order 1 of a power series f(z). If K denotes the usual class of convex univalent functions in the open unit disk centered at the origin, we show that, in general, V3(z, f) ⊀ σ3(1)(z,f), for all fK. Making use of polylogarithms, we identify a transformation, Λ : KK, such that V3(z, Λ(f)) ≺ σ3(1)(z,(f)) for all fK. Here ‘≺’ stands for subordination between two analytic functions.

Keywords: Convex functions, Cesà,ro means, De la Vallé,e Poussin means, Subordination, Hadamard product