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Tinaztepe, Gültekin; Yeşilce, İlknur; Adilov, Gabil

A refinement of the Bergström inequality. (English) Zbl 1496.26038

Turk. J. Math. 45, No. 4, 1619-1625 (2021).
Summary: In this paper, the Bergstrom inequality is studied, and a refinement of this inequality is obtained by performing the optimality conditions based on abstract concavity. Some numerical experiments are given to illustrate the efficacy of the refinement.

MSC:

26D15 Inequalities for sums, series and integrals
26B25 Convexity of real functions of several variables, generalizations

Keywords:

abstract concavity; refinement; Bergström inequality; global optimization
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\textit{G. Tinaztepe} et al., Turk. J. Math. 45, No. 4, 1619--1625 (2021; Zbl 1496.26038)
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References:

[1] Adilov GR, Tınaztepe G. The sharpening some inequalities via abstract convexity. Mathematical Inequalities and Applications 2012; 12: 33-51. · Zbl 1178.26012
[2] Budak H, Sarıkaya MZ. On generalized Ostrowski-type inequalities for functions whose first derivatives absolute values are convex. Turkish Journal of Mathematics 2016; 40 (6): 1193-1210. · Zbl 1424.26035
[3] Bullen PS, Mitrinovic DS, Vasic PM. Means and Their Inequalities (Mathematics and Its Applications). Dordrecht, the Netherlands: Springer, 2013.
[4] Crouzeix JP, Legaz JEM, Volle M. Generalized Convexity, Generalized Monotonicity: Recent Results. Dordrecht, the Netherlands: Kluwer, 1998.
[5] Dragomir SS, Dutta J, Rubinov AM. Hermite-Hadamard Type inequalities for increasing convex along rays func-tions. Analysis 2001; 2: 171-181. · Zbl 1064.26017
[6] Jain S, Mehrez K, Baleanu D, Agarwal P. Certain Hermite-Hadamard Inequalities for logarithmically convex functions with applications. Mathematics 2019; 7: 163.
[7] Li Y, Gu XM, Xiao J. A note on the proofs of generalized Radon inequality. Mathematica Moravica 2018; 22 (2): 59-67. · Zbl 1474.26120
[8] Li Y, Gu XM, Zhao J. The weighted arithmetic mean-geometric mean inequality is equivalent to the Hölder inequality, Symmetry 2018; 10 (9): 380. · Zbl 1423.26042
[9] Marghidanu, D. Generalizations and refinements for Bergström and Radon’s inequalities. Journal of Science and Arts 2008; 8 (1): 57-62. · Zbl 1194.26032
[10] Marshall A, Olkin I. Inequalities: theory of majorization and its applications. New York, NY, USA: Springer, 1979. · Zbl 0437.26007
[11] Özdemir ME, Kavurmacı H, Set E. Ostrowski’s type inequalities for ( α , m) -convex functions. Kyungpook Mathematical Journal 2010; 50 (3): 371-378. · Zbl 1204.26040
[12] Pachpatte BG. Analytic Inequalities Recent Advances. Paris, France: Atlantis Press, 2012. · Zbl 1238.26003
[13] Pavic Z, Ardıç MA. The most important inequalities of m-convex functions. Turkish Journal of Mathematics 2017; 41: 625-635. · Zbl 1424.26024
[14] Pop OT. About Bergström Inequality. Journal of Mathematical Inequalities 2009; 3 (2): 237-242. · Zbl 1177.26047
[15] Rubinov AM. Abstract Convexity and Global optimization. Dordrecht, the Netherlands: Kluwer, 2000. · Zbl 0985.90074
[16] Rubinov AM, Wu ZY. Optimality conditions in global optimization and their applications. Mathematical Program-ming 2009; 120 (1): 101-123. · Zbl 1191.90047
[17] Sahir, MJS. Formation of versions of some dynamic inequalities unified on time scale calculus. Ural Mathematical Journal 2018; 4 (2): 88-98. · Zbl 1448.26036
[18] Sarikaya MZ, Set E. Özdemir ME. On new inequalities of Simpson’s type for s -convex functions. Computers and Mathematics with Applications 2010; 60 (8): 2191-2199 · Zbl 1205.65132
[19] Singer I. Abstract Convex Analysis. New York, NY, USA: Wiley, 1997. · Zbl 0898.49001
[20] Tınaztepe G. The sharpening Hölder inequality via abstract convexity. Turkish Journal of Mathematics 2016; 40: 438-444. · Zbl 1424.26065
[21] Tınaztepe G, Kemali S, Sezer S, Eken Z. The sharper form of Brunn-Minkowski type inequality for boxes. Hacettepe Journal of Mathematics and Statistics 2021; 50 (2): 377-386. · Zbl 1499.52016
[22] Tınaztepe G, Tınaztepe R. A sharpened version of Aczel inequality and some remarks, Mathematical Inequalities and Applications 2021; 24 (3): 635-643. · Zbl 1478.26023
[23] Tunç M. On some new inequalities for convex functions. Turkish Journal of Mathematics 2012; 36: 245-251. · Zbl 1251.26009
[24] Yesilce I. Inequalities for B-convex functions via generalized fractional integral. Journal of Inequalities and Appli-cations 2019; 1: 1-15. · Zbl 1499.26205
[25] Yesilce I, Adilov G. Hermite-Hadamard inequalities for B-convex and B −1 -convex functions. The International Journal of Nonlinear Analysis and Applications 2017; 8: 225-233. · Zbl 1387.39028
[26] Yesilce I, Adilov G. Fractional Integral inequalities for B-convex functions. Creative Mathematics and Informatics 2017; 26: 345-351. · Zbl 1424.26018
[27] Yesilce I, Adilov G. Hermite-Hadamard inequalities for L(j) -convex functions and S(j) -convex functions. Malaya Journal of Matematik 2015; 3: 346-359. · Zbl 1371.26040
[28] Yau, SF, Bresler Y. A generalization of Bergstrom’s inequality and some applications. Linear algebra and its applications 1992; 161: 135-151. · Zbl 0746.15011
[29] Zhao CJ, Li XY. On mixed discriminants of positively definite matrix. Ars Mathematica Contemporanea 2015; 9 (2): 261-266. · Zbl 1330.15006
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