- Turkish Journal of Mathematics and Computer Science
- Vol: 14 Issue: 1
- Some New Inequalities via Berezin Numbers
Some New Inequalities via Berezin Numbers
Authors : Mualla Birgül HUBAN, Hamdullah BAŞARAN, Mehmet GÜRDAL
Pages : 129-137
Doi:10.47000/tjmcs.1014841
View : 7 | Download : 3
Publication Date : 2022-06-30
Article Type : Research
Abstract :The Berezin transform $\widetilde{T}$ and the Berezin radius of an operator $T$ on the reproducing kernel Hilbert space $\mathcal{H}\left( Q\right) $ over some set $Q$ with the reproducing kernel $K_{\eta}$ are defined, respectively, by \[ \widetilde{T}(\eta)=\left\langle {T\frac{K_{\eta}}{{\left\Vert K_{\eta }\right\Vert }},\frac{K_{\eta}}{{\left\Vert K_{\eta}\right\Vert }}% }\right\rangle ,\ \eta\in Q\text{ and }\mathrm{ber}(T):=\sup_{\eta\in Q}\left\vert \widetilde{T}{(\eta)}\right\vert . \] We study several sharp inequalities by using this bounded function $\widetilde{T},$ involving powers of the Berezin radius and the Berezin norms of reproducing kernel Hilbert space operators. We also give some inequalities regarding the Berezin transforms of sum of two operators.Keywords : Reproducing kernel Hilbert space, Berezin transform, Berezin radius, Jensen's inequaity, mixed Schwarz inequality