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New Versions of Uniformly Convex Functions via Quadratic Complete Homogeneous Symmetric Polynomials

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Abstract

We introduce new versions of uniformly convex functions, namely \(h_d\) strongly (weaker) convex functions. Based on the positivity of complete homogeneous symmetric polynomials with even degree, recently studied in Rovenţa and Temereanc (Mediterr J Math 16:1–16, 2019), Rovenţa et al. (A note on weighted Ingham’s inequality for families of exponentials with no gap, In: 24th ICSTCC, pp 43–48, 2020; Weighted Ingham’s type inequalities via the positivity of quadratic polynomials, submitted), and Tao (https://terrytao.wordpress.com/2017/08/06/schur-convexity-and-positive-definiteness-of-the-even-degree-co-mplete-homogeneous-symmetric-polynomials/), we introduce stronger and weaker versions of uniformly convexity. In this context, we recover well-known type inequalities such as: Jensen’s, Hardy–Littlewood–Polya’s and Popoviciu’s inequalities. Some final remarks related to Sherman’s and Ingham’s type inequalities are also discussed.

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Acknowledgements

The work of G. M. Lăchescu, M. Malin and I. Rovenţa has been supported by a grant of the Romanian Ministry of Research, Innovation and Digitalization (MCID), project number 22–Nonlinear Differential Systems in Applied Sciences, within PNRR-III-C9-2022-I8.

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Authors and Affiliations

  1. Department of Mathematics, University of Craiova, A.I. Cuza Street 13, 200585, Craiova, Romania

    Geanina Maria Lăchescu, Maria Malin & Ionel Rovenţa

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  1. Geanina Maria Lăchescu
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  2. Maria Malin
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  3. Ionel Rovenţa
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Correspondence to Ionel Rovenţa.

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Lăchescu, G.M., Malin, M. & Rovenţa, I. New Versions of Uniformly Convex Functions via Quadratic Complete Homogeneous Symmetric Polynomials. Mediterr. J. Math. 20, 279 (2023). https://doi.org/10.1007/s00009-023-02484-1

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  • DOI: https://doi.org/10.1007/s00009-023-02484-1

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