On the properties of strongly hd convex functions

Geanina-Maria Lăchescu, Vasile-Florin Uță

Abstract


We study some optimization properties of  $h_d$ strongly convex functions. More precisely, we discuss the characterization properties/inequalities (existence and uniqueness) of minima  of $h_d$ strongly convex functions. Moreover,  connections with polynomial norms are also presented.

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S. Abramovich, S. Ivelić , and J. E. Pečarić, Improvement of Jensen–Steffensen’s inequality for superquadratic functions, Banach J. Math. Anal. 4 (2010), 159–169.

S. Abramovich, New inequalities related to superquadratic functions, Aequ. Math. 96 (2022), 201–2019.

S. Abramovich, Refinements of Jensen’s inequality by uniformly convex functions, Aequ. Math. 97 (2023), 75–88.

K. Aguilar, Á. Chávez, S. Garcia and J. Volčič, Norms on complex matrices induced by complete homogeneous symmetric polynomials, Bull. London Math. Soc. 54 (2022), 2078–2100.

A. A. Ahmadi, Etienne de Klerk and G. Hall, Polynomial norms, https://arxiv.org/abs/1704.07462.

A. Barvinok, Approximating a norm by a polynomial, Geometric Aspects of Functional Analysis, Springer, 2003, pp. 20–26.

A. Chávez, S. Garcia and J. Hurley, Norms on Complex Matrices Induced by Random Vectors, Can. Math. Bull. (2022), 1–19.

D. B. Hunter, The positive-definiteness of the complete symmetric functions of even order, Math. Proc. Cambridge Philos. Soc. 82 (1977) (2), 255–258.

M. Adil Khan, M. Niezgoda and J. E. Pečarič, On a refinement of the majorisation type in equality, Demonstr. Math. 44 (2011), 49–57.

M. Klaričić Bakula, Jensen-Steffensen inequality for strongly convex functions, J. Inequal. Appl. 306 (2018), 306.

G. M. Lăchescu, M. Mălin and I. Rovența, New Versions of Uniformly Convex Functions via Quadratic Complete Homogeneous Symmetric Polynomials, Mediterranean Journal of Mathe matics 20 279 (2023). 1–20.

G. M. Lăchescu and I. Rovența The Hardy-Littlewood-P´olya inequality of majorization in the context of ω-m-star-convex functions, Aequ. Math. 97 (2023), 523–535.

A.W. Marshall, I. Olkin and B. Arnold, Inequalities: Theory of majorization and its applications, 2nd ed., Springer Series in Statistics, Springer, New York, 2011.

C. P. Niculescu, A new look at the Hardy-Littlewood-Polya inequality of majorization, J. Math. Anal. Appl. 501 (2021), article 125211.

C. P. Niculescu and O. Olteanu, From the Hahn-Banach extension theorem to the isotonicity of convex functions and the majorization theory, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 114 (2020), 1–19.

C. P. Niculescu and F. Popovici, The extension of majorization inequalities within the framework of relative convexity, JIPAM 7 (2006), 6 pp. (Electronic only).

C. P. Niculescu and I. Rovența, An approach of majorization in spaces with a curved geometry, J. Math. Anal. Appl. 411 (2014), 119–128.

C. P. Niculescu and I. Rovența, Relative Schur convexity on global NPC spaces, Math. Inequal. Appl. 18 (2015), 1111–1119.

C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications. A Contemporary Approach, 2nd ed., CMS Books in Mathematics 23, Springer-Verlag, New York, 2018.

C. P. Niculescu and L.-E. Persson, Convex Functions and their Applications. A Contemporary Approach, CMS Books in Mathematics vol. 23, Springer-Verlag, New York, 2006.

M. Niezgoda, Linear maps preserving group majorization, Linear Algebra Appl. 330 (2001), 113–127.

M. Niezgoda, Majorization and refined Jensen Mercer type inequalities for self-adjoint operators, Linear Algebra Appl. 467 (2015), 1–14.

K. Nikodem and Zs. Páles, Characterization of inner product spaces by strongly convex func tions, Banach J. Math. Anal. 5 (2011) (1), 83–87.

I. Rovența, Schur-convexity of a class of symmetric functions, Ann. Univ. Craiova Math. Com put. Sci. Ser. 37 (2010) (1), 12–18.

I. Rovența, A note on Schur-concave functions, J. Inequal. Appl. (2012) 159, 9 pp.

I. Rovența and L. E. Temereancă, A note on the positivity of the even degree complete homo geneous symmetric polynomials, Mediterr. J. Math. 16 (2019), 1–16.

I. Rovența, L. E. Temereancă, and A. M. Tudor, A note on weighted Ingham’s inequality for families of exponentials with no gap, 24th ICSTCC (2020), 43–48.

I. Rovența, L. E. Temereancă and A. M. Tudor, Weighted Ingham’s type inequalities via thepositivity of quadratic polynomials, Aequat. Math. 98 (2024), 865–883.




DOI: https://doi.org/10.52846/ami.v51i2.2026