In this research article, new upper bounds are established for the Berezin number of bounded linear operators within the context of functional Hilbert space using the convex function. These bounds are achieved by exploiting an extension of Furuta's inequality and an extension of Kato's inequality. These Berezin number bounds are refinements of already existing bounds found in the literature.

This article presents an investigation to improve the numerical radius bounds for bounded linear operators on a complex Hilbert space through the application of Young’s inequality. This study yields novel numerical radius bounds for operators, improves existing bounds, and achieves new refinements for the numerical radius bounds related to the sum and product of operators.