Some inequality and Berezin number type inequalities

‎In this paper‎, ‎we establish upper bounds that both extend and enhance known‎ ‎inequalities for the Berezin number of bounded linear operators and product of‎ ‎operators‎. ‎In particular‎, ‎this paper aims to establish new upper bounds on the Berezin number for operators on functional Hilbert spaces by introducing important improvements to the Buzano inequality‎. ‎Lastly‎, ‎a few associated‎ ‎upper bounds are also given‎. ‎