## On Korenblum convex functions.(English)Zbl 1416.26022

Summary: We introduce a new class of generalized convex functions called the $$\kappa$$-convex functions, based on Korenblum’s concept of $$\kappa$$-decreasing functions, where $$\kappa$$ is an entropy (distortion) function. We study continuity and differentiability properties of these functions, and we discuss a special subclass which is a counterpart of the class of so-called d.c. functions. We characterize this subclass in terms of the space of functions of bounded second $$\kappa$$-variation, extending a result of F. Riesz. We also present a formal structural decomposition result for the $$\kappa$$-convex functions.

### MSC:

 26A51 Convexity of real functions in one variable, generalizations 39B62 Functional inequalities, including subadditivity, convexity, etc. 26B25 Convexity of real functions of several variables, generalizations 26A48 Monotonic functions, generalizations

### Keywords:

convex function; $$\kappa$$-entropy; generalized convexity
Full Text:

### References:

 [1] J. Appell, J. Banaś, and N. Merentes, Bounded Variation and Around, De Gruyter Series in Nonlinear Analysis and Applications, vol. 17, De Gruyter, Berlin 2014. · Zbl 1282.26001 [2] P. R. Beesack and J. E. Pečarić, On Jessen’s inequality for convex functions, J. Math. Anal. Appl. 110 (1985), no. 2, 536-552,DOI 10.1016/0022-247X(85)90315-4. [3] D. S. Cyphert and J. A. Kelingos, The decomposition of functions of bounded κ-variation into differences of κ-decreasing functions, Studia Math. 81 (1985), no. 2, 185-195. 1 · Zbl 0621.26004 [4] R. Dabrowski, On a natural connection between the entropy spaces and Hardy space Re H, Proc. Amer. Math. Soc. 104 (1988), no. 3, 812-818,DOI 10.2307/2046798. · Zbl 0694.42010 [5] J. Ereú, L. López, and N. Merentes, On second κ-variation, Comment. Math. 56 (2016), no. 2, 209-224, DOI 10.14708/cm.v56i2.1233. 44Jurancy Ereú, Lorena López, and Nelson Merentes [6] R. A. Fefferman, A theory of entropy in Fourier analysis, Adv. in Math. 30 (1978), 171-201, DOI 10.1016/0001-8708(78)90036-1. [7] J. Giménez, L. López, N. Merentes, and J. L. Sánchez, A Burenkov’s type result for functions of bounded κ-variation, Ann. Funct. Anal. 6 (2015), no. 1, 1-11,DOI 10.15352/afa/06-1-1. [8] I. Ginchev and D. Gintcheva, Characterization and recognition of d.c. functions, J. Glob. Optim. 57 (2013), 633-647,DOI 10.1007/s10898-012-9964-6. · Zbl 1284.26014 [9] G. H. Hardy, Weierstrass’s nondifferentiable function, Trans. Amer. Math. Soc. 17 (1916), 301-325, DOI 10.2307/1989005. · JFM 46.0401.03 [10] P. Hartman, On functions representable as a difference of convex functions, Pacific J. Math. 9 (1959), 707-713. · Zbl 0093.06401 [11] R. Horst, P. M. Pardalos, and N. V. Thoai, Introduction to Global Optimization, Nonconvex Optimization and Its Applications, vol. 48, Kluwer, Dordrecht 2000,DOI 10.1007/978-1-4615-0015-5. · Zbl 0966.90073 [12] D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math. Soc. 3 (1952), 821-828, DOI 10.2307/2032186. · Zbl 0047.29505 [13] B. Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), no. 3-4, 187-219, DOI 10.1007/BF02392019. · Zbl 0323.30030 [14] B. Korenblum, A generalization of two classical convergence tests for Fourier series and some new Banach spaces of functions, Bull. Amer. Math. Soc. 9 (1983), no. 2, 15-18,DOI 10.1090/S0273-0979-1983-15160-1. · Zbl 0519.42007 [15] B. Korenblum, On a class of Banach spaces associated with the notion of entropy 290 (1985), 527-553, DOI 10.2307/2000297. · Zbl 0616.46024 [16] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, second ed., Birkhäuser Verlag AG, Basel 2009,DOI 10.1007/978-3-7643-8749-5. · Zbl 1221.39041 [17] W. C. Lang, A growth condition for Fourier coefficients of functions of bounded entropy norm, Proc. Amer. Math. Soc. 112 (1991), no. 2, 433-439,DOI 10.2307/2048737. · Zbl 0718.42007 [18] J. Makó and Z. Páles, On ϕ-convexity, Publ. Math. Debrecen 80 (2012), 107-126, DOI 10.1057/9780230226203.3173. [19] J. E. Pečarić, F. Proschan, and Y. C. Tong, Convex functions, partial orderings and statistical applications, Boston 1992. [20] F. Riesz, Sur certains systems singuliers d’equations integrales, Annales de L’Ecole Norm. Sup. 28 (1911), 33-62. · JFM 42.0374.03 [21] A. W. Roberts and D. E. Varberg, Functions of bounded convexity, Bull. Amer. Math. Soc. 75 (1969), 568-572,DOI 10.1090/S0002-9904-1969-12244-5. · Zbl 0176.01204 [22] A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York 1973. · Zbl 0271.26009 [23] Ch. J. de la Vallée Poussin, Sur la convergence des formules d´interpolation entre ordennées équidistantes, Bull. Accad. Sct. Belg (1908), 314-410.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.