Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

MAKSA, GY ., Functions having quadratic differences in a given class

82 Gyula Maksa for some 77 E R. Since \f"{r])\ = < 1, (8) implies that \A

yAyf(x)\ < y

2 for all x, y E R, that is, A

yA.

yf is bounded for all fixed y E R. Suppose that / has the decomposition (1) for some bounded /*: R —> R, quadratic N and additive A. Then N + A must be bounded on any bounded interval. Thus N(x) + A(x) = ax

2 + ßx, x E R for some a,ß E R. This and (1) imply that (9) f*(x) = x\n(x

2 + 1) + 2arctgx - ax

2 - (2 + ß)x, x £ R. Since /* is bounded, 0 = lim \ ' — —a and thus x—>+oo x 0= Hm r(x

)~

2arctg

x = Mm (ln^

2 + 1) - (2 + ß)) , x—Í- + 00 X X—

^ + OO which is a contradiction. This shows that the set of all bounded functions does not have the quadratic difference property. References [1] AczÉL, Lectures on Functional Equations and Their Applications, Academic Press, New York and London, 1966. [2] J. AczÉL, The general solution of two functional equations by reduction to functions additive in two variables and with the aid of Hamel bases, Glasnik Mat.-Fiz. Astr., 20 (1965), 65-73. [3] N. G. DE BRUIJN, Functions whose differences belong to a given class, Nieuw Arch. Wisk., 23 no. 2 (1951), 194-218. [4] Z. DARÓCZY, 35. Remark, Aequationes Math., 8 (1972), 187-188. [5] M. KUCZMA, An Introduction to the Theory of Functional Equations and Inequalities, Cauchy's Equation and Jensen's Inequality, Panstowowe Wydawnictwo Naukowe, Warszawa • Krakow • Katovice, 1985. GYULA MAKSA LAJOS KOSSUTH UNIVERSITY INSTITUTE OF MATHEMATICS AND INFORMATICS 4010 DEBRECEN P.O. Box 12. HUNGARY E-mail: maksaimath.klte.hu