Interpolation inequalities and berezin radius

In this master thesis, some basic theories about the numerical radius, the Berezin symbol, and the Berezin radius are introduced. Also, the basic inequalities for the Berezin radius involving a functional Hilbert space operator are analyzed. We plan to structure this master thesis into five main chapters. Firstly, in the introduction part, the usage areas, properties, and historical process of the concepts of Berezin number and numerical radius are mentioned. In the second chapter, the historical development of the problems studied in the thesis and recent studies are analyzed in detail. In the next section, known auxiliary theorems, notation, definitions, and related inequalities, the Berezin transformation, the Berezin radius and generalized cases are given. In addition, some basic auxiliary theorems that we will use in our results are presented. The next section, called Research Results and Discussion, is divided into two subsections. In the first subsection of the fourth section, several refinements of the Berezin radius inequalities are established. Also, as the main aim of this chapter, new interpolation inequalities of some known inequalities for the numerical radius using the properties of operator convex functions are presented. A new inequality is obtained in the special case by using another improvement of the basic inequality related to the Berezin radius and the operator norm. In the special case, a new inequality is shown for a nonnegative increasing operator convex function defined on a functional Hilbert space, a hyponormal operator, and a set of positive real numbers containing zero. The last chapter of this thesis is the Conclusion and Suggestions section. In this section, which is the fifth chapter, the main results given in the previous chapter are briefly discussed, and some suggestions and problems related to the continuation of these results or related results are given.