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The main focus of this paper is the study of the Selberg operator. It aims to establish appropriate bounds for the norm and numerical radius of the product of three bounded operators, with one of them being a Selberg operator. Moreover, it offers several bounds involving the summation of operators, notably the Selberg operator. Through the examination of these properties and relationships, this study contributes to a better understanding of the Selberg operator and its influence on operator compositions. The paper also highlights the significance of symmetry in mathematics and its potential implications across various mathematical domains. [ABSTRACT FROM AUTHOR]

This paper introduces several refinements of the classical Selberg inequality, which is considered a significant result in the study of the spectral theory of symmetric spaces, a central topic in the field of symmetry studies. By utilizing the contraction property of the Selberg operator, we derive improved versions of the classical Selberg inequality. Additionally, we demonstrate the interdependence among well-known inequalities such as Cauchy–Schwarz, Bessel, and the Selberg inequality, revealing that these inequalities can be deduced from one another. This study showcases the enhancements made to the classical Selberg inequality and establishes the interconnectedness of various mathematical inequalities. [ABSTRACT FROM AUTHOR]