Berezin radius inequalities and Selberg operators

This master's thesis focuses on the study and applications of inequalities related to reproducing kernels and the Berezin symbols defined through them. Within the thesis, by employing classical and well-known inequalities such as those of Buzano, Bessel, Bombieri, Jensen, and Young, alongside norm inequalities for convex functions, new inequalities concerning the Berezin norm and Berezin radius of operators on reproducing kernel Hilbert spaces are established. Furthermore, novel upper bounds for Berezin norms and Berezin radii are derived for certain products and sums involving Selberg operators defined on reproducing kernel Hilbert spaces. The thesis is organized into five main chapters. The introduction presents the concepts of the Berezin number and Berezin radius, discussing their properties, applications, and historical development. The second chapter reviews the historical progression and recent advancements in the problems addressed within the thesis, providing a comprehensive literature survey. In the third chapter, referred to as preliminary results, notations, definitions, and relevant inequalities related to Berezin transform, Berezin radius, and Selberg inequalities are introduced; recent studies in these areas are also examined. Additionally, foundational lemmas employed in the subsequent results are provided. The fourth chapter, titled "Research Findings and Discussion," is subdivided into two parts, each containing two subsections. The first subsection, "Upper Bounds for Berezin Radius," proves various norm and Berezin radius inequalities for linear operators in functional Hilbert spaces. The second subsection, "Advancements on Berezin Radius Inequalities for Operators," investigates upper bounds for Berezin radii of bounded linear operators and their products, generalizing existing inequalities. The most significant portion of the thesis, focusing on "Berezin Norm and Berezin Radius Inequalities for Products and Sums Involving Selberg Operators" and "Berezin Number Inequalities for Sums with Selberg Operators," establishes upper estimates for the Berezin norm and radius of the product of three operators including the Selberg operator. Furthermore, new bounds concerning sums involving Selberg operators are demonstrated. Utilizing the fact that the Selberg operator is a contraction, the classical Selberg inequality is improved, allowing for new, enhanced versions applicable to broader cases. By investigating new Berezin norm, radius, and number inequalities for products and sums involving Selberg operators on reproducing kernel Hilbert spaces, this study provides a novel perspective on Berezin number inequalities for operators. The final chapter, "Conclusions and Recommendations," summarizes the main findings, briefly discusses their implications, and proposes future research directions and open problems related to the study.