Some generalized numerical radius inequalities involving Kwong functions

Authors : Mojtaba BAKHERAD

Pages : 951-958

View : 15 | Download : 4

Publication Date : 2019-08-08

Article Type : Research Paper

Abstract :We prove several numerical radius inequalities involving positive semidefinite matrices via the Hadamard product and Kwong functions. Among other inequalities, it is shown that if   $X$ is an arbitrary $n\times n$ matrix and $A,B$ are positive semidefinite, then \[ \omegainsert ignore into journalissuearticles values(H_{f,g}insert ignore into journalissuearticles values(A););\leq k\, \omegainsert ignore into journalissuearticles values(AX+XA);, \]  which is equivalent to \[\omega\biginsert ignore into journalissuearticles values(H_{f,g}insert ignore into journalissuearticles values(A,B);\pm H_{f,g}insert ignore into journalissuearticles values(B,A);\big);\leq k`\,\left\{\omegainsert ignore into journalissuearticles values(insert ignore into journalissuearticles values(A+B);X+Xinsert ignore into journalissuearticles values(A+B););+\omegainsert ignore into journalissuearticles values(insert ignore into journalissuearticles values(A-B);X-Xinsert ignore into journalissuearticles values(A-B););\right\},\]  where  $f$ and $g$ are two continuous functions on $insert ignore into journalissuearticles values(0,\infty);$ such that $hinsert ignore into journalissuearticles values(t);={finsert ignore into journalissuearticles values(t);\over ginsert ignore into journalissuearticles values(t);}$ is Kwong, $k=\max\left\{{finsert ignore into journalissuearticles values(\lambda);ginsert ignore into journalissuearticles values(\lambda);\over \lambda}: {\lambda\in\sigmainsert ignore into journalissuearticles values(A);}\right\}$ and $k`=\max\left\{{finsert ignore into journalissuearticles values(\lambda);ginsert ignore into journalissuearticles values(\lambda);\over \lambda}: {\lambda\in\sigmainsert ignore into journalissuearticles values(A);\cup\sigmainsert ignore into journalissuearticles values(B);}\right\}$.
Keywords : numerical radius, Hadamard product, operator monotone, Kwong function