Reverse Jensen Integral Inequalities for Operator Convex Functions in Terms of Fréchet Derivative

Abstract

Let \(f:I\rightarrow {\mathbb {R}}\) be an operator convex function of class \( C^{1}\left( I\right) \). If \((A_{t})_{t\in T}\) is a bounded continuous field of selfadjoint operators in \({\mathcal {B}}\left( H\right) \) with spectra contained in I defined on a locally compact Hausdorff space T with a bounded Radon measure \(\mu \), such that \(\int _{T}{\mathbf {1}}d\mu \left( t\right) =\mathbf {1,}\) then we obtain among others the following reverse of Jensen’s inequality:

$$\begin{aligned} 0&\le \int _{T}f\left( A_{t}\right) d\mu \left( t\right) -f\left( \int _{T}A_{s}d\mu \left( s\right) \right) \\&\le \int _{T}Df(A_{t})\left( A_{t}\right) d\mu \left( t\right) -\int _{T}Df(A_{t})\left( \int _{T}A_{s}d\mu \left( s\right) \right) d\mu \left( t\right) \end{aligned}$$

in terms of the Fréchet derivative \(Df(\cdot )(\cdot ).\) Some applications for the Hermite–Hadamard inequalities are also given.

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Acknowledgements

The author would like to thank the anonymous referees for their valuable comments that have been implemented in the final version of the manuscript.

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Correspondence to S. Silvestru Dragomir.

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Communicated by Hossein Mohebi.

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Dragomir, S.S. Reverse Jensen Integral Inequalities for Operator Convex Functions in Terms of Fréchet Derivative . Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-020-00482-7

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Keywords

  • Unital \(C^{*}\)-algebras
  • Selfadjoint elements
  • Functions of selfadjoint elements
  • Positive linear maps
  • Operator convex functions
  • Jensen’s operator inequality
  • Integral inequalities

Mathematics Subject Classification

  • 47A63
  • 47A99