Some improvements of numerical radius inequalities via Specht’s ratio

Subject Areas : Functional analysis

Y. Khatib ¹ , M. Hassani ²

1 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran
2 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

Received: 2020-05-31 Accepted : 2020-09-20 Published : 2020-09-01

Keywords: Hermite-Hadamard inequality, Positive operators, numerical radius, Specht's ratio,

Abstract :

We obtain some inequalities related to the powers of numericalradius inequalities of Hilbert space operators. Some results thatemploy the Hermite-Hadamard inequality for vectors in normed linearspaces are also obtained. We improve and generalize someinequalities with respect to Specht's ratio. Among them, we showthat, if $A, B\in \mathcal{B(\mathcal{H})}$ satisfy in someconditions, it follows that \begin{equation*} \omega^2(A^*B)\leq \frac{1}{2S(\sqrt{h})}\Big\||A|^{4}+|B|^{4}\Big\|-\displaystyle{\inf_{\|x\|=1}} \frac{1}{4S(\sqrt{h})}\big(\big\langle \big(A^*A-B^*B\big) x,x\big\rangle\big)^2 \end{equation*} for some $h>0$, where $\|\cdot\|,\,\,\,\omega(\cdot)$ and $S(\cdot)$denote the usual operator norm, numerical radius and the Specht'sratio, respectively.

References:

[1] C. P. Constantin, L-E. Persson, Old and new on the Hermite-Hadamard inequality, Real Anal. Exchange. 29 (2) (2003-04), 663-685.

[2] S. S. Dragomir, Power inequalities for the numerical radius of a product of two operators in Hilbert spaces, Sarajevo J Math. 5 (18) (2009), 269-278.

[3] S. S. Dragomir, Some inequalities generalizing Kato´ s and Furuta´ s results, Filomat. 28 (1) (2014), 179-195.

[4] M. El-Haddad, F. Kittaneh, Numerical radius inequalities for Hilbert space operators, II, Studia Math. 182 (2) (2007), 133-140.

[5] J. I. Fujii, S. Izumino, Y. Seo, Determinant for positive operators and Specht’s theorem, Sci. Math. 1 (3) (1998), 307-310.

[6] S. Furuichi, Refined Young inequalities with Specht’s ratio, J. Egyptian Math. Soc. 20 (1) (2012), 46- 49.

[7] F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math. 158 (1) (2003), 11-17.

[8] F. Kittaneh, Notes on some inequalities for Hilbert space operators, Publ. Res. Inst. Math. Sci. 24 (2) (1988), 283-293.

[9] B. Mond, J. E. Pecaric, Convex inequalities in Hilbert space, Houston J. Math. 19 (3) (1993), 405-420.

[10] H. R. Moradi, S. Furuichi, F. C. Mitroi, R. Naseri, An extension of Jensen’s operator inequality and its application to Young inequality, Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. 113 (2) (2019), 605-614.

[11] J. E. Pecaric, T. Furuta, J. Micic Hot, Y. Seo, Mond-Pecaric, Method in Operator Inequalities, Inequalities for bounded selfadjoint operators on a Hilbert space, Monographs in Inequalities, Zagreb, 2005.

[12] K. Shebrawi, H. Albadawi, Numerical radius and operator norm inequalities, J. Inequal. Appl. 2009, 2009:492154.  

[13] W. Specht, Zur theorie der elementaren mittel, Math. Z. 74 (1960), 91-98.