Norm and Numerical Radius Inequalities for Sums of Power Series of Operators in Hilbert Spaces

The main focus of this paper is on establishing inequalities for the norm and numerical radius of various operators applied to a power series with the complex coefficients h(lambda)= n-ary sumation k=0 infinity ak lambda k and its modified version ha(lambda)= n-ary sumation k=0 infinity|ak|lambda k. The convergence of h(lambda) is assumed on the open disk D(0,R), where R is the radius of convergence. Additionally, we explore some operator inequalities related to these concepts. The findings contribute to our understanding of operator behavior in bounded operator spaces and offer insights into norm and numerical radius inequalities.