Reverses of the first Hermite-Hadamard type inequality for the square operator modulus in Hilbert spaces

Subject Areas : Operator theory

S. S. Dragomir ¹

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

Received: 2021-11-30 Accepted : 2022-01-16 Published : 2022-03-01

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.

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