Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47885220190701Power Inequalities for the Numerical Radius of Operators in Hilbert Spaces15298420410.22034/kjm.2019.84204ENMohammad H.M.RashidDepartment of Mathematics and Statistics, Faculty of Science P.O.Box(7),
Mu’tah University, Alkarak-Jordan.Journal Article20180802We generalize several inequalities involving powers of the numerical radius for the product of two operators acting on a Hilbert space. Moreover, we give a Jensen operator inequality for strongly convex functions. As a corollary, we improve the operator <span>Hölder-McCarthy</span> inequality under suitable conditions. In particular, we prove that if $f:Jrightarrow mathbb{R}$ is strongly convex with modulus $c$ and differentiable on ${rm int}(J)$ whose derivative is continuous on ${rm int}(J)$ and if $T$ is a self-adjoint operator on the Hilbert space $cal{H}$ with $sigma(T)subset {rm int}(J)$, then<br /> $$langle T^2x,xrangle-langle Tx,xrangle^2leq dfrac{1}{2c}(langle f'(T)Tx,xrangle -langle Tx,xrangle langle f'(T)x,xrangle)$$<br /> for each $xincal{H}$, with $|x|=1$.http://www.kjm-math.org/article_84204_a321253ff5f81d65f8472735f8eb5f80.pdf