**Title:** Berezin radius and Cauchy-Schwarz inequality

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-21-00068; **Volume 5 / Issue 3 / Year 2023**, Pages 16-22

**Document Type:** Research Paper

**Author(s):** Hamdullah Başaran ^{a} , Verda Gürdal ^{b}

^{a}Süleyman Demirel University, Department of Mathematics, Isparta, Turkey

^{b}Süleyman Demirel University, Department of Mathematics, Isparta, Turkey

Received: 23 November 2021, Accepted: 29 March 2022, Published: 29 April 2022.

**Corresponding Author:** Hamdullah Başaran (Email address: basaranhamdullah@hotmail.com)

**Full Text:** PDF

**Abstract**

In this manuscript, some refinements of the Cauchy-Schwarz inequality for contraction operators on the reproducing kernel Hilbert space are given in terms of the Berezin transform. We show several additional inequalities for the Berezin norm and Berezin radius of operators using these refinements.

**Keywords:** Cauchy-Schwarz inequality, Berezin transform, Berezin radius

**References:**

- N. Aronzajn,
*Theory of reproducing kernels*, Trans. Amer. Math. Soc.**68**, 337–404, 1950. - M. Bakherad and M. T. Garayev,
*Berezin number inequalities for operators*, Concr. Oper.**6 (1)**, 33–43, 2019. - H. Başaran and M. Gürdal,
*Berezin number inequalities via Young inequality*, Honam Math. J.**43 (3)**, 523–537, 2021. - H. Başaran, M. Gürdal and A. N. Güncan,
*Some operator inequalities associated with Kantorovich and Hölder-McCarthy inequalities and their applications*, Turkish J. Math.**43 (1)**, 523–532, 2019. - F. A. Berezin,
*Covariant and contravariant symbols for operators*, Math. USSR-Izv.**6**, 1117–1151, 1972. - J.-C. Bourin and E.-Y. Lee,
*Unitary orbits of Hermitian operators with convex or concave functions*, Bull. Lond. Math. Soc.**44**, 1085–1102, 2012. - S. S. Dragomir,
*Power inequalities for the numerical radius of a product of two operators in Hilbert spaces*, Sarajevo J. Math.**5 (18)**, 269–278, 2009. - M. T. Garayev, M. Gürdal and A. Okudan,
*Hardy-Hilbert’s inequality and power inequalities for Berezin numbers of operators*, Math. Inequal. Appl.**19 (3)**, 883–891, 2016. - M. T. Garayev, H. Guedri, M. Gürdal and G. M. Alsahli,
*On some problems for operators on the reproducing kernel Hilbert space*, Linear Multilinear Algebra**69 (11)**, 2059–2077, 2021. - M. Hajmohamadi, R. Lashkaripour and M. Bakherad,
*Improvements of Berezin number inequalities*, Linear Multilinear Algebra,**68 (6)**, 1218–1229, 2020. - M. B. Huban, H. Başaran and M. Gürdal,
*New upper bounds related to the Berezin number inequalities*, J. Inequal. Spec. Funct.**12 (3)**, 1–12, 2021. - M. B. Huban, H. Başaran and M. Gürdal,
*Berezin number inequalities via convex functions*, Filomat (to appear), 2021. - M. T. Karaev,
*Reproducing kernels and Berezin symbols techniques in various questions of operator theory*, Complex Anal. Oper. Theory**7 (4)**, 983–1018, 2013. - F. Kittaneh,
*Numerical radius inequalities for Hilbert space operators*, Studia Math.**168 (1)**, 73–80, 2005. - M. Sababheh, H. R. Moradi and Z. Heydarbeygi,
*Buzano, Krein and Cauchy-Schwarz inequalities*, arXiv:2010.02464v1 [math.FA] 6 Oct 2020.