@article {
author = {Rashid, Mohammad},
title = {Power Inequalities for the Numerical Radius of Operators in Hilbert Spaces},
journal = {Khayyam Journal of Mathematics},
volume = {5},
number = {2},
pages = {15-29},
year = {2019},
publisher = {Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)},
issn = {2423-4788},
eissn = {2423-4788},
doi = {10.22034/kjm.2019.84204},
abstract = {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 $f:J\rightarrow \mathbb{R}$ 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 $$\langle T^2x,x\rangle-\langle Tx,x\rangle^2\leq \dfrac{1}{2c}(\langle f'(T)Tx,x\rangle -\langle Tx,x\rangle \langle f'(T)x,x\rangle)$$ for each $x\in\cal{H}$, with $\|x\|=1$.},
keywords = {Numerical Range,Numerical radius,Aluthge transformation,strongly convex},
url = {http://www.kjm-math.org/article_84204.html},
eprint = {http://www.kjm-math.org/article_84204_a321253ff5f81d65f8472735f8eb5f80.pdf}
}