Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Initial Coefficients Bounds for Bi-univalent Functions Related to Gregory Coefficients

Version 1 : Received: 29 May 2023 / Approved: 30 May 2023 / Online: 30 May 2023 (10:38:03 CEST)

A peer-reviewed article of this Preprint also exists.

Murugusundaramoorthy, G.; Vijaya, K.; Bulboacă, T. Initial Coefficient Bounds for Bi-Univalent Functions Related to Gregory Coefficients. Mathematics 2023, 11, 2857. Murugusundaramoorthy, G.; Vijaya, K.; Bulboacă, T. Initial Coefficient Bounds for Bi-Univalent Functions Related to Gregory Coefficients. Mathematics 2023, 11, 2857.

Abstract

In the present paper we introduce three new classes of bi-univalent functions connected with Gregory coefficients. For functions in each of these three bi-univalent function classes we have derived the estimates of the Taylor--Maclaurin coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ and Fekete-Szeg\H{o} functional problems for functions belonging to these new subclasses. We defined three subclasses of the class of the bi-univalent functions $\Sigma$, namely $\mathfrak{HG}_{\Sigma}$, $\mathfrak{GM}_{\Sigma}(\mu)$ and $\mathfrak{G}_{\Sigma}(\lambda)$ by using the subordinations with the function whose coefficients are Gregory's numbers. First, we proved that these classes are not empty, i.e. contains other functions than the identity one. Using the well-known Carath\'eodory Lemma for the functions with real positive parts in the open unit disk, together with an estimation due to P. Zaprawa (see https://doi.org/10.1155/2014/357480) and another one of Libera and Zlotkiewicz, we gave upper bounds for the above mentioned initial coefficients and for the Fekete-Szeg\H{o} functionals. The main results are followed by some particular cases, and the novelty of the definitions and the proofs could involve further studies for such type of similarly defined subclasses.

Keywords

univalent functions; bi-univalent functions; starlike and convex functions of some order; subordination; Fekete-Szeg\H{o} problem

Subject

Computer Science and Mathematics, Analysis

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