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On Superquadratic and Logarithmically Superquadratic Functions

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Abstract

In this paper, we develop a concept of a logarithmically superquadratic function. Such a class of functions is defined via superquadratic functions. We first establish some new properties of superquadratic functions. In particular, we derive the corresponding superadditivity relation and its reverse, as well as the external form of the Jensen inequality and its reverse. Then, as a direct consequence of the established results, we obtain the corresponding properties for logarithmically superquadratic functions. Further, we show that logarithmically superquadratic functions with values greater than or equal to one are convex and logarithmically superadditive. In particular, we also obtain the corresponding refinement of the Jensen inequality in a product form. Finally, we give a variant of the Jensen operator inequality for logarithmically superquadratic functions.

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Acknowledgements

The authors would like to thank the anonymous referee for some valuable comments and useful suggestions.

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Authors and Affiliations

  1. Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, 10000, Zagreb, Croatia

    Mario Krnić

  2. Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran

    Hamid Reza Moradi

  3. Department of Basic Sciences, Princess Sumaya University for Technology, Al Jubaiha, Amman, 11941, Jordan

    Mohammad Sababheh

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  1. Mario Krnić
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  2. Hamid Reza Moradi
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  3. Mohammad Sababheh
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The authors declare that they have contributed equally to this paper. All the authors have read and approved this version.

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Correspondence to Mario Krnić.

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Krnić, M., Moradi, H.R. & Sababheh, M. On Superquadratic and Logarithmically Superquadratic Functions. Mediterr. J. Math. 20, 311 (2023). https://doi.org/10.1007/s00009-023-02514-y

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  • DOI: https://doi.org/10.1007/s00009-023-02514-y

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