%0 Journal Article
%T Power Inequalities for the Numerical Radius of Operators in Hilbert Spaces
%J Khayyam Journal of Mathematics
%I 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)
%Z 2423-4788
%A Rashid, Mohammad H.M.
%D 2019
%\ 07/01/2019
%V 5
%N 2
%P 15-29
%! Power Inequalities for the Numerical Radius of Operators in Hilbert Spaces
%K Numerical Range
%K Numerical radius
%K Aluthge transformation
%K strongly convex
%R 10.22034/kjm.2019.84204
%X We 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 Hölder-McCarthy 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 $$langle T^2x,xrangle-langle Tx,xrangle^2leq dfrac{1}{2c}(langle f'(T)Tx,xrangle -langle Tx,xrangle langle f'(T)x,xrangle)$$ for each $xincal{H}$, with $|x|=1$.
%U http://www.kjm-math.org/article_84204_a321253ff5f81d65f8472735f8eb5f80.pdf