# zbMATH — the first resource for mathematics

Some analytical properties of $$\gamma$$-convex functions on the real line. (English) Zbl 0868.26005
Summary: This paper deals with the analytical properties of $$\gamma$$-convex functions, which are defined as those functions satisfying the inequality $$f(x_1')+ f(x_2')\leq f(x_1)+ f(x_2)$$, for $$x_i'\in[x_1,x_2]$$, $$|x_i- x_i'|=\gamma$$, $$i=1,2$$, whenever $$|x_1-x_2|>\gamma$$, for some given positive $$\gamma$$. This class contains all convex functions and all periodic functions with period $$\gamma$$. In general, $$\gamma$$-convex functions do not have ideal properties as convex functions. For instance, there exist $$\gamma$$-convex functions which are totally discontinuous or not locally bounded. But $$\gamma$$-convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length $$\gamma$$. It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of $$\gamma$$-convex functions on the real line. However, $$\gamma$$-convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of $$\gamma$$-convexity are given. Ways for generating and representing $$\gamma$$-convex functions are described.

##### MSC:
 26A51 Convexity of real functions in one variable, generalizations 49J52 Nonsmooth analysis 90C30 Nonlinear programming
Full Text:
##### References:
 [1] Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970. · Zbl 0193.18401 [2] Roberts, A. W. andVarberg, D. E.,Convex Functions, Academic Press, New York, New York, 1973. [3] Beckenbach, E. F.,Generalized Convex Functions, Bulletin of the American Mathematical Society, Vol. 43, pp. 336–371, 1937. · Zbl 0016.35202 · doi:10.1090/S0002-9904-1937-06549-9 [4] Bector, C. R., Suneja, S. K., andLalitha, C. S.,Generalized B-Vex Functions and Generalized B-Vex Programming, Journal of Optimization Theory and Applications, Vol. 76, pp. 561–576, 1993. · Zbl 0802.49027 · doi:10.1007/BF00939383 [5] Behringer, F. A.,Discrete and Nondiscrete Quasiconvexlike Functions and Single-Peakedness (Unimodality), Optimization, Vol. 14, pp. 163–181, 1983. · Zbl 0519.90065 · doi:10.1080/02331938308842844 [6] Ben-Tal, A.,On Generalized Means and Generalized Convex Functions, Journal of Optimization Theory and Applications, Vol. 21, pp. 1–13, 1977. · Zbl 0325.26007 · doi:10.1007/BF00932539 [7] Craven, B. D.,Invex Functions and Constrained Local Minima, Bulletin of the Australian Mathematical Society, Vol. 24, pp. 357–366, 1981. · Zbl 0452.90066 · doi:10.1017/S0004972700004895 [8] Dolecki, S., andKurcyusz, S.,On $$\Phi$$-Convexity in Extremal Problems, SIAM Journal on Control and Optimization, Vol. 16, pp. 227–300, 1978. · Zbl 0397.46013 · doi:10.1137/0316018 [9] Hanson, M. A.,On Sufficiency of Kuhn-Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 545–550, 1981. · Zbl 0463.90080 · doi:10.1016/0022-247X(81)90123-2 [10] Hartwig, H.,On Generalized Convex Functions, Optimization, Vol. 14, pp. 49–60, 1983. · Zbl 0514.26003 · doi:10.1080/02331938308842832 [11] Martos, B.,Nonlinear Programming: Theory and Methods, Akademiai Kiado, Budapest, Hungary, 1975. · Zbl 0357.90027 [12] Komlosi, S., Rapcsak, T., andSchaible, S.,Generalized Convexity, Springer Verlag, Berlin, Germany, 1994. [13] Schaible, S., andZiemba, W. T., Editors,On Generalized Concavity in Optimization and Economics, Academic Press, New York, New York, 1981. [14] Hartwig, H.,Generalized Convexities of Lower Semicontinuous Functions, Optimization, Vol. 16, pp. 663–668, 1985. · Zbl 0585.26008 · doi:10.1080/02331938508843063 [15] Hartwig, H.,Local Boundedness and Continuity of Generalized Convex Functions, Optimization, Vol. 26, pp. 1–13, 1992. · Zbl 0815.26004 · doi:10.1080/02331939208843838 [16] Karamardian, S., andSchaible, S.,Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990. · Zbl 0679.90055 · doi:10.1007/BF00940531 [17] Karamardian, S., Schaible, S., andCrouzeix, J. P.,Characterization of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 76, pp. 399–413, 1993. · Zbl 0792.90070 · doi:10.1007/BF00939374 [18] Hu, T. C., Klee, V., andLarman, D.,Optimization of Globally Convex Functions, SIAM Journal on Control and Optimization, Vol. 27, pp. 1026–1047, 1989. · Zbl 0686.52006 · doi:10.1137/0327055 [19] Söllner, B.,Eigenschaften $$\gamma$$-grobkonvexer Mengen und Funktionen Diplomarbeit, Universität Leipzig, 1991. [20] Phú, H. X.,$$\gamma$$-Subdifferential and $$\gamma$$-Convexity of Functions on the Real Line, Applied Mathematics and Optimization, Vol. 27, pp. 145–160, 1993. · Zbl 0798.49024 · doi:10.1007/BF01195979 [21] Phú, H. X.,$$\gamma$$-Subdifferential and $$\gamma$$-Convexity of Functions on a Normed Space, Journal of Optimization Theory and Applications, Vol. 85, pp. 649–676, 1995. · Zbl 0831.90105 · doi:10.1007/BF02193061 [22] Natanson, I. P.,Theorie der Funktionen einer Reellen Veränderlichen Akademie Verlag, Berlin, Germany, 1975. · Zbl 0301.26007 [23] Hobson, E. W.,The Theory of Functions of a Real Variable and the Theory of Fourier Series University Press, Cambridge, England, 1907. · JFM 38.0414.01
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.