Hölder-Type Inequalities for Power Series of Operators in Hilbert Spaces

Consider the power series with complex coefficients h(z)=∑k=0∞akzk and its modified version ha(z)=∑k=0∞|ak|zk. In this article, we explore the application of certain Hölder-type inequalities for deriving various inequalities for operators acting on the aforementioned power series. We establish these inequalities under the assumption of the convergence of h(z) on the open disk D(0,ρ), where ρ denotes the radius of convergence. Additionally, we investigate the norm and numerical radius inequalities associated with these concepts.