Abstract

A considerable amount of literature on the theory of inequality is devoted to studying Jensen's and Young's inequality. This article presents a number of new inequalities involving the log-convex functions and the geometrically convex functions. As their consequences, we derive the refinements for Young's and Jensen's inequality. In addition, the operator Jensen's type inequality is also developed for conditioned two functions. Utilizing these new inequalities, we investigate the operator inequalities related to the relative operator entropy.

Author information

Contact details are reproduced from the original publication and may be historical.

Supriyo Dutta

Dept. of Mathematics, National Institute of Technology Agartala, Tripura, India

dosupriyo@gmail.com

Keywords

  • Convexity
  • log-convex
  • geometrically convex
  • Jensen's inequality
  • relative operator entropy
  • operator inequality

Mathematics Subject Classification

26D15, 26A51, 39B62, 47A63, 47A64, 94A17

References

  1. Y. J. Cho, M. Matic, J. Pečarić: Two mappings in connection to Jensen’s inequality, Panamerican Math. J. 12 (2002) 43–50.Google Scholar
  2. S. Dutta, S. Furuichi: On log-sum inequalities, Linear Multilinear Algebra 72/5 (2024) 812–827.https://doi.org/10.1080/03081087.2023.2165611Google Scholar
  3. S. Dutta, S. Furuichi, P. Guha: A two-parameter entropy and its fundamental properties, Rev. Math. Phys. 33 (2021), art. no. 2130003.https://doi.org/10.1142/S0129055X2130003XGoogle Scholar
  4. J. I. Fujii, E. Kamei: Relative operator entropy in noncommutative information theory, Math. Japon. 34 (1989) 341–348.Google Scholar
  5. S. Furuichi, H. R. Moradi: Advances in Mathematical Inequalities, De Gruyter, Berlin (2020).https://doi.org/10.1515/9783110643473Google Scholar
  6. T. Furuta: Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators, Linear Algebra Appl. 381 (2004) 219–235.https://doi.org/10.1016/j.laa.2003.11.017Google Scholar
  7. H. Isa, M. Ito, E. Kamei, H. Tohyama, M. Watanabe: Relative operator entropy, operator divergence, and Shannon inequality, Sci. Math. Japon. Online e-2012(30) (2012) 353–362.Google Scholar
  8. B. Mond: On Jensen’s inequality for operator convex functions, Houston J. Math. 21 (1995) 739–754.Google Scholar
  9. C. P. Niculescu, L. E. Persson: Convex Functions and their Applications, 2nd ed., Springer, Berlin (2018).https://doi.org/10.1007/978-3-319-78337-6Google Scholar
  10. J. Pečarić, T. Furuta, J. Mićić Hot, Y. Seo: Mond-Pečarić Method in Operator Inequalities, Element, Zagreb (2005).Google Scholar
  11. K. Yanagi, K. Kuriyama, S. Furuichi: Generalized Shannon inequalities based on Tsallis relative operator entropy, Linear Algebra Appl. 394 (2005) 109–118.https://doi.org/10.1016/j.laa.2004.06.025Google Scholar
  12. L. Zou: Operator inequalities associated with Tsallis relative operator entropy, Math. Inequal. Appl. 18/2 (2015) 401–406.https://doi.org/10.7153/mia-18-29Google Scholar

Suggested citation

S. Furuichi, H. R. Moradi, S. Dutta. “On Some Relative Operator Entropies by Convex Inequalities.” Journal of Convex Analysis 31 (2024), No. 3, 983–998. https://doi.org/10.68381/jca31047

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.