Reverses of the first Hermite-Hadamard type inequality for the square operator modulus in Hilbert spaces
Subject Areas : Operator theory
S. S.
Dragomir
1
(Mathematics‎, ‎College of Engineering \& Science‎, ‎Victoria University‎, ‎PO Box 14428‎, ‎Melbourne City‎, ‎MC 8001‎, ‎Australia|DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences‎, ‎School of Computer Science \& Applied Mathematics‎, ‎University of the Witwatersrand‎, ‎Private Bag 3‎, ‎Johannesburg 2050‎, ‎South Africa)
Keywords: Operator convex functions, Hermite-Hadamard inequality, midpoint inequality, operator power and logarithmic functions,
Abstract :
‎Let $\left( H;\left\langle \cdot‎ ,‎\cdot \right\rangle \right)$ be a complex‎ ‎Hilbert space‎. ‎Denote by $\mathcal{B}\left( H\right)$ the Banach $C^{\ast }$-‎algebra of bounded linear operators on $H$‎. ‎For $A\in \mathcal{B}\left(‎H\right)$ we define the modulus of $A$ by $\left\vert A\right\vert‎ :‎=\left(‎A^{\ast }A\right) ^{1/2}$ and \ $\func{Re}A:=\frac{1}{2}\left( A^{\ast‎‎}+A\right)‎.‎$ In this paper we show among other that‎, ‎if $A,$ $B\in \mathcal{‎‎B}\left( H\right)$ with $0\leq m\leq \left\vert \left( 1-t\right)‎‎A+tB\right\vert ^{2}\leq M$ for all $t\in \left[ 0,1\right]‎,‎$ then \begin{align*}‎ ‎0& \leq \int_{0}^{1}f\left( \left\vert \left( 1-t\right) A+tB\right\vert‎‎^{2}\right) dt-f\left( \frac{\left\vert A\right\vert ^{2}+\func{Re}\left(‎‎B^{\ast }A\right)‎ +‎\left\vert B\right\vert ^{2}}{3}\right) \\‎ ‎& \leq 2\left[ \frac{f\left( m\right)‎ +‎f\left( M\right) }{2}-f\left( \frac{‎m+M}{2}\right) \right] 1_{H}‎ ‎\end{align*} ‎for operator convex functions $f:[0,\infty )\rightarrow \mathbb{R}$‎. ‎Applications for power and logarithmic functions are also provided‎.
[1] R. P. Agarwal, S. S. Dragomir, A survey of Jensen type inequalities for functions of selfadjoint operators in Hilbert spaces, Comput. Math. Appl. 59 (12) (2010), 3785-3812.
[2] N. S. Barnett, P. Cerone, S. S. Dragomir, Some new inequalities for Hermite-Hadamard divergence in information theory, Stochastic Analysis and Applications. 3, Nova Sci. Publ., Hauppauge, NY, 2003.
[3] P. Cerone, S. S. Dragomir, Mathematical Inequalities. A Perspective, CRC Press, Boca Raton, 2011.
[4] S. S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (2002), 2:31.
[5] S. S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math. 3 (2002), 3:35.
[6] S. S. Dragomir, Superadditivity of some functionals associated with Jensen’s inequality for convex functions on linear spaces with applications, Bull. Aust. Math. Soc. 82 (1) (2010), 44-61.
[7] S. S. Dragomir, A note on new refinements and reverses of Young’s inequality, Transylv. J. Math. Mech. 8 (1) (2016), 45-49.
[8] S. S. Dragomir, A note on Young’s inequality, RACSAM. 111 (2) (2017), 349-354.
[9] S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type, Springer Briefs in Mathematics, Springer, New York, 2012.
[10] S. S. Dragomir, Operator Inequalities of the Jensen,ˇCebyˇ sev and Grüss Type, Springer Briefs in Mathematics, Springer, New York, 2012.
[11] S. S. Dragomir, Hermite-Hadamard’s type inequalities for operator convex functions, Appl. Math. Comput. 218 (3) (2011), 766-772.
[12] S. S. Dragomir, Some Hermite-Hadamard type inequalities for operator convex functions and positive maps, Spec. Matrices. 7 (2019), 38-51.
[13] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, 2000.
[14] S. S. Dragomir, Some Hermite-Hadamard type inequalities for the square operator modulus in Hilbert spaces, Preprint RGMIA Res, Rep. Coll. 24 (2021), 1-16.
[15] B. Li, Refinements of Hermite-Hadamard’s type inequalities for operator convex functions, Int. J. Contemp. Math. Sci. 8 (9-12) (2013), 463-467.
[16] J. Pecaric, S. S. Dragomir, A generalization of Hadamard’s inequality for isotonic linear functionals, Radovi Mat.(Sarajevo). 7 (1991), 103-107.
[17] J. E. Pecaric, F. Proschan, Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications (Mathematics in Science and Engineering), 187, Academic Press Inc., 1992.
[18] A. Taghavi, V. Darvish, H. M. Nazari, S. S. Dragomir, Hermite-Hadamard type inequalities for operator geometrically convex functions, Monatsh. Math. 181 (1) (2016), 187-203.
[19] M. Vivas Cortez, E. J. H. Hernández, Refinements for Hermite-Hadamard type inequalities for operator h-convex function, Appl. Math. Inf. Sci. 11 (5) (2017), 1299-1307.
[20] M. Vivas Cortez, E. J. H. Hernández, On some new generalized Hermite-Hadamard-Fejér inequalities for product of two operator h-convex functions, Appl. Math. Inf. Sci. 11 (4) (2017), 983-992.
[21] S.-H. Wang, Hermite-Hadamard type inequalities for operator convex functions on the co-ordinates, J. Nonlinear Sci. Appl. 10 (3) (2017), 1116-1125.
[22] S.-H. Wang, New integral inequalities of Hermite-Hadamard type for operator m-convex and (α,m)-convex functions, J. Comput. Anal. Appl. 22 (4) (2017), 744-753.