Some berezin radius inequalities related to the Dragomir-Furuta inequality

In this master thesis some basic theories about the numerical radius, the Berezin symbol and the Berezin radius are introduced. Also, the fundamental inequalities for the Berezin radius and the generalized Berezin radius involving a functional Hilbert space operator and the fundamental inequalities involving bounds are studied. This master's thesis is planned to consist of 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, which is the abstracts section, the historical development of the problems studied in this master's thesis and recent studies are examined in detail. In the next section, known auxiliary theorems, notation, definitions and related inequalities, related numerical radius inequalities, Berezin transformation, Berezin radius and generalized cases are given. In addition, some basic auxiliary theorems are presented which we will directly use in our results. The next section, called Research Findings and Discussion, is divided into three subsections. In the first subsection of the fourth section, we use the extension versions of the Dragomir-Furuta inequality and give other related Berezin radius inequalities. We obtain some important auxiliary theorems related to the Berezin symbol, which play a main role in the proof of our main results, and whose proofs are similar to the proofs in \cite{A2019}. In the special case, the inequality \[ f\left( \mathrm{ber}\left( \mathfrak{A}_{4}\mathfrak{A}_{3}\mathfrak{A}% _{2}\mathfrak{A}_{1}\right) \right) \leq\frac{1}{2}\left\Vert f\left( \mathfrak{A}_{1}^{\ast}\left\vert \mathfrak{A}_{2}\right\vert ^{2}% \mathfrak{A}_{1}\right) +f\left( \mathfrak{A}_{4}\left\vert \mathfrak{A}% _{3}^{\ast}\right\vert ^{2}\mathfrak{A}_{4}^{\ast}\right) \right\Vert _{\mathrm{ber}}-\inf_{\Theta\in\Xi}\eta\left( \Theta\right) \] for $\mathfrak{A}_{1},\mathfrak{A}_{2},\mathfrak{A}_{3},\mathfrak{A}_{4}% \in\mathfrak{B}\left( \mathfrak{H}\left( \Xi\right) \right) $ with $\eta\left( \Theta\right) :=\frac{1}{8}\nu\left[ \left( \widetilde {\mathfrak{A}_{1}^{\ast}\left\vert \mathfrak{A}_{2}\right\vert ^{2}% \mathfrak{A}_{1}-\mathfrak{A}_{4}\left\vert \mathfrak{A}_{3}^{\ast}\right\vert ^{2}\mathfrak{A}_{4}^{\ast}}\right) \left( \Theta\right) \right] ^{2}$ is obtained. In the second subsection, some new inequalities for the generalized Berezin radius of operators acting on a functional Hilbert space are given. More related results are also obtained. In the special case, the inequality \[ \mathrm{ber}_{\mathfrak{G}}^{n}\left( \mathfrak{A}\right) \leq\frac{1}{2^{n-1}% }\mathrm{ber}_{\mathfrak{G}}\left( \mathfrak{A}^{n}\right) +\left\Vert \mathfrak{A}% \right\Vert _{\mathfrak{G}-\mathrm{Ber}}\sum_{s=1}^{n-1}\frac{1}{2^{n}}\mathrm{ber}% _{\mathfrak{G}}^{n-s-1}\left( \mathfrak{A}\right) \left\Vert \mathfrak{A}% ^{s}\right\Vert _{\mathfrak{G}-\mathrm{Ber}}, \] for $n=2,3,...,$ is shown. In the last subsection of this section, some generalized inequalities for the generalized Davis-Wielandt-Berezin type radius of operators on Hilbert space with generating kernel are given. Moreover, some new bounds for the generalized Davis-Wielandt-Berezin type radius are studied which improve the existing bounds in the literature. Namely, some lower and upper bounds for the generalized Davis-Wieandt-Berezin radius of operators on a functional Hilbert space are studied. The last chapter of this thesis is the Conclusion and Recommendations 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.