Genelleştirilmiş Mersenne Polinomları ile İlişkili Bi-ünivalent Fonksiyonların Bazı Alt Sınıfları

Abstract

Bi-ünivalent fonksiyonların incelenmesi, hem fonksiyonun hem de tersinin birim disk içinde ünivalent olmasını gerektiren tanımlayıcı özelliği nedeniyle sahip olduğu analitik karmaşıklık sebebiyle önemli ölçüde ilgi görmüştür. Bu çalışmada, genelleştirilmiş Mersenne polinomları ile ilişkili analitik ve bi-ünivalent fonksiyonların yeni bir alt sınıfı tanıtılmaktadır. Söz konusu polinomlar, bu fonksiyon sınıfına ilişkin operatörlerin oluşturulmasında etkili bir araç olarak kullanılmaktadır. Çalışmanın temel amacı, Taylor–Maclaurin açılımının katsayı tahminlerini, özellikle başlangıç katsayıları üzerinde yoğunlaşarak elde etmektir. Subordinasyon teknikleri ve geometrik fonksiyonlar teorisine ait standart yöntemler kullanılarak çeşitli Fekete–Szegö tipli eşitsizlikler türetilmiştir. Ayrıca, daha önce bilinen alt sınıflarla bağlantıların kurulabilmesi amacıyla bazı özel durumlar ve ilgili sonuçlar ele alınmıştır. Elde edilen bulgular, literatürde mevcut çeşitli katsayı eşitsizliklerini genişletmekte ve birleştirmekte olup, Mersenne polinomlarının bi-ünivalent fonksiyon sınıflarındaki katsayı problemlerinde uygulanabilirliğini ortaya koymaktadır.

Keywords

• Bi-ünivalent fonksiyonlar
• Mersenne polinomları
• Fekete-Szegö eşitsizliği
• Katsayı tahminleri
• Analitik fonksiyonlar

Certain Subclasses of Bi-Univalent Functions Associated with Generalized Mersenne Polynomials

Abstract

The study of bi-univalent functions, whose defining property requires both the function and its inverse to be univalent in the unit disk, has received considerable attention due to its intrinsic analytical complexity. In this paper, a novel subclass of analytic and bi-univalent functions is formulated in relation to generalized Mersenne polynomials. These polynomials are employed as an effective tool for constructing operators related to this class of functions. The main focus of the paper is to obtain coefficient estimates for the Taylor–Maclaurin expansion, with particular emphasis on the initial coefficients. Using subordination techniques and standard methods from geometric function theory, several Fekete–Szegö type inequalities are derived. In addition, a number of special cases and related results are discussed to establish connections with previously known subclasses. The results presented here extend and unify various earlier coefficient inequalities in the literature and highlight the applicability of Mersenne polynomials in coefficient problems for bi-univalent function classes.

Keywords

• Bi-univalent functions,
• Mersenne polynomials
• Fekete-Szegö inequality
• Coefficient estimates
• Analytic functions

Ethical Statement

This study does not require ethics committee approval.

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