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Some basic properties of subquadratic functions, i.e. functions fulfilling the inequality phi (x + y) + phi(x - y) is less than or equal to 2 phi(x) + 2 phi(y) are proved. In this note X be always a real linear space and R be denotes the set of all reals. Every function phi : X approaches R satisfying the following inequality (1) phi(x + y) + phi(x - y) is less than or equal 2phi(x) + 2phi(y), x, y is an element of X, is called subquadratic. If the sign "is less than or equal to" is replaced by "is more than or equal to" then phi is called superquadratic and if we have "=" instead of " is less than or equal to" in (1) then we say that phi is quadratic function. There are plenty papers devoted to quadratic functions [1], [2], [3] (and references there). In this note some properties of the solutions of (1) will be proved, particularly we will investigate nonpositive solutions of (1). Also interesting question of finding sucient conditions on subquadratic function to be quadratic one will be considered.
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751--758
Opis fizyczny
Bibliogr. 6 poz.
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Bibliografia
• [1] J. Aczél and J. Dhombres, Functional Equations in Several Variables, Encyclopedia of Mathematics and its Applications, 31, Cambridge University Press, Cambridge, 1989.
• [2] Di -LianYang, The quadratic functional equation on groups, Publ. Math. Debrecen 66/3-4 (2005), no 1, 327–348.
• [3] Pl. Kannappan, On quadratic functional equation, Int. J. Math. Stat. Sci. 9 (2000), no 1, 35–60.
• [4] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities.Cauchy's Equation and Jensen's Inequality, Scientific Publications of the University of Silesia, 489, Warszawa-Kraków-Katowice, 1985, pp. 523, ISBN: 83-01-05508-1.
• [5] W. Smajdor, Subadditive and subquadratic set-valued functions, Scientific Publications of the University of Silesia, 889, Katowice, 1987, 75 pp. ISBN: 83-226-0162-x.
• [6] S. Ulam, A Collection of Mathematical Problems, Interscience Publ., New York, 1960, Problems in Modern Mathematics, Wiley, New York, 1964.
Typ dokumentu
Bibliografia