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**Inequalities of convex functions and self-adjoint operators.**
*(English)*
Zbl 1300.47025

Summary: The paper offers generalizations of the Jensen-Mercer inequality for self-adjoint operators and generally convex functions. The obtained results are applied to define the quasi-arithmetic operator means without using operator convexity. The version of the harmonic-geometric-arithmetic operator mean inequality is derived as an example.

### MSC:

47A63 | Linear operator inequalities |

39B62 | Functional inequalities, including subadditivity, convexity, etc. |

### References:

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[2] | M. D. Choi, “A Schwarz inequality for positive linear maps on C*-algebras,” Illinois Journal of Mathematics, vol. 18, pp. 565-574, 1974. · Zbl 0293.46043 |

[3] | F. Hansen, J. Pe, and I. Perić, “Jensen’s operator inequality and its converses,” Mathematica Scandinavica, vol. 100, no. 1, pp. 61-73, 2007. · Zbl 1151.47025 |

[4] | J. Mićić, Z. Pavić, and J. Pe, “Jensen’s inequality for operators without operator convexity,” Linear Algebra and its Applications, vol. 434, no. 5, pp. 1228-1237, 2011. · Zbl 1221.47032 · doi:10.1155/2011/358981 |

[5] | Z. Pavić, “The applications of functional variants of Jensen’s inequality,” Journal of Function Spaces and Applications, vol. 2013, Article ID 194830, 5 pages, 2013. · Zbl 1300.46062 · doi:10.1155/2013/194830 |

[6] | A. M. Mercer, “A variant of Jensen’s inequality,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, article 73, 2 pages, 2003. · Zbl 1048.26016 |

[7] | T. Furuta, J. Mićić Hot, J. Pe, and Y. Seo, Mond-Pe Method in Operator Inequalities, vol. 1, Element, Zagreb, Croatia, 2005. · Zbl 1135.47012 |

[8] | A. Matković, J. Pe, and I. Perić, “Refinements of Jensen’s inequality of Mercer’s type for operator convex functions,” Mathematical Inequalities & Applications, vol. 11, no. 1, pp. 113-126, 2008. · Zbl 1141.47013 · doi:10.7153/mia-11-07 |

[9] | J. E. Pe, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, vol. 187, Academic Press, Boston, Mass, USA, 1992. · Zbl 0749.26004 |

[10] | J. Mićić, Z. Pavić, and J. Pe, “The inequalities for quasiarithmetic means,” Abstract and Applied Analysis, vol. 2012, Article ID 203145, 25 pages, 2012. · Zbl 1246.26019 · doi:10.1155/2012/203145 |

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