×

Characterizations and decomposition of strongly Wright-convex functions of higher order. (English) Zbl 1333.26011

Summary: Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 2, 661–665 (2011; Zbl 1204.26017)] we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states that a function \(f\) is strongly Wright-convex of order \(n\) if and only if it is of the form \(f(x)=g(x)+p(x)+c x^{n+1}\), where \(g\) is a (continuous) \(n\)-convex function and \(p\) is a polynomial function of degree \(n\). This is a counterpart of Ng’s decomposition theorem for Wright-convex functions. We also characterize higher order strongly Wright-convex functions via generalized derivatives.

MSC:

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

Citations:

Zbl 1204.26017
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Azócar, J. Giménez, K. Nikodem, J. L. Sánchez, On strongly midconvex functions, Opuscula Math. 31 (2011), 15–26. · Zbl 1234.26035
[2] A. Dinghas, Zur Theorie der gewöhnlichen Differentialgleichungen, Ann. Acad. Sci. Fennicae, Ser. A I 375 (1966). · Zbl 0143.30602
[3] G. Friedel, Zur Theorie der Intervallableitung reller Funktionen, Diss., Freie Univ. Berlin, 1968.
[4] R. Ger, K. Nikodem, Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661–665. · Zbl 1204.26017
[5] A. Gilányi, Zs. Páles, On Dinghas-type derivatives and convex functions of higher order, Real Anal. Exchange 27 (2001/2002), 485–493.
[6] A. Gilányi, Zs. Páles, On convex functions of higher order, Math. Inequal. Appl. 11 (2008), 271–282.
[7] E. Hopf, Über die Zusammenhänge zwischen gewissen höheren Differenzenquotienten reeller Funktionen einer reellen Variablen und deren Differenzierbarkeitseigenschaften, Diss., Friedrich Wilhelms Univ., Berlin, 1926.
[8] M.V. Jovanovič, A note on strongly convex and strongly quasiconvex functions, Math. Notes 60 (1996), 778–779.
[9] Z. Kominek, On additive and convex functionals, Radovi Mat. 3 (1987), 267–279. · Zbl 0643.39006
[10] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Państwowe Wydawnictwo Naukowe – Uniwersytet Śląski, Warszawa-Kraków-Katowice, 1985.
[11] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, 2nd ed., Birkhäuser Verlag, 2009. · Zbl 1221.39041
[12] Gy. Maksa, Zs. Páles, Decomposition of higher order Wright-convex functions, J. Math. Anal. Appl. 359 (2009), 439–443.
[13] N. Merentes, K. Nikodem, Remarks on strongly convex functions, Aequationes Math. 80 (2010), 193–199. · Zbl 1214.26007
[14] N. Merentes, K. Nikodem, S. Rivas, Remarks on strongly Wright-convex functions, Ann. Polon. Math. 102 (2011), 271–278. · Zbl 1229.26021
[15] L. Montrucchio, Lipschitz continuous policy functions for strongly concave optimization problems, J. Math. Economy 16 (1987), 259–273. · Zbl 0636.90077
[16] C.T. Ng, Functions generating Schur-convex sums, [in:] W. Walter (ed.), General Inequalities 5, Oberwolfach, 1986, International Series of Numerical Mathematics, vol. 80, Birkhäuser Verlag, Basel, Boston, 1987, 433–438.
[17] K. Nikodem, Zs. Páles, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal. 5 (2011), 83–87. · Zbl 1215.46016
[18] K. Nikodem, T. Rajba, Sz. Wąsowicz, Functions generating strongly Schur-convex sums, in C. Bandle, A. Gilányi, L. Losonczi, M. Plum (eds.), Inequalities and Applications 2010, International Series of Numerical Mathematics, vol. 161, Birkhäuser Verlag, Basel, Boston, Berlin, 2012, 175–182. · Zbl 1253.26022
[19] B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl. 7 (1966), 72–75.
[20] T. Popoviciu, Sur quelques propriétés des fonctions d’une ou de deux variables réelles, Mathematica (Cluj) 8 (1934), 1–85. · Zbl 0009.05901
[21] T. Popoviciu, Les fonctions convexes, Hermann et Cie, Paris, 1944.
[22] T. Rajba, Sz. Wąsowicz, Probabilistic characterization of strong convexity, Opuscula Math. 31 (2011), 97–103. · Zbl 1234.26031
[23] A.W. Roberts, D.E. Varberg, Convex Functions, Academic Press, New York–London, 1973. · Zbl 0271.26009
[24] J. P. Vial, Strong convexity of sets and functions, J. Math. Economy 9 (1982), 187–205. · Zbl 0479.52005
[25] P. Volkmann, Die Äquivalenz zweier Ableitungsbegriffe, Diss., Freie Univ. Berlin, 1971.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.