# On Applications and Theory of Functional Equations by J. Aczél

By J. Aczél

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In comparison to the functions with values (34) /(*)=** (36) F(x,y)-. the more general forms ( 4 6 ) and ( 4 8 ) of the solutions made these func­ tions more apt for application to the original problems in economics. W. > and ( 4 8 ) as solutions, which conjecture was rejected by the above mentioned result. Other applications of generalized homogeneous functions to problems in economics were made by the Hungarian mathematicians E. VINCZE ( 1 9 6 0 ) and E. VINCZE and M . Hosszu in 1961 and very recently a different generalization with several applications was made by the Polish mathematician S.

But, from ( 4 4 ) we see that k (45) t. H (/,*,) = Here k is a constant, but also z is fixed at zo. For different values of z we always get functions of the form k /(/) == l but k might be different for different z; that is, k which is a constant with respect to / can depend upon z. Hence, we write ( 4 5 ) as (z) (46) H{t,z) = /* . Thus, every solution bounded from below an interval of t-values, of the equation (43) H{tu,z) has to be of the by a positive =H{t,z)H(u,z) H(t,z) (t>0,u>0) = /*<=>. Reciprocally, one sees immediately that every function k of one variable function satisfies = H^9^F(x9y) F(tx,ty) , we have, since \ X that (47) F(tx,ty) = t^FiXty) .

H. B. KEMPERMAN, A General Functional Equation. Trans. Amer. Math. Soc. 86 (1957—1958), 28—56. H. KIESEWETTER, Eine Bemerkung iiber partielle Differentiationen bei N. H. Abel. Publ. Math. Debrecen 5 (1957—1958), 265—268. J . ACZEL, Ober die Differenzierbarkeit der integrierbaren Losungen gewisser Funktionalgleichungen. Annales Univ. Budapest S. Math. 3—4 (1960—1961), 5—8. A. RENYI, On Measures of Entropy and Information. Proc. 4th Berkeley Sympos. Math. Statist, and Prob. 1960, Vol. , Berkeley 1 9 6 1 , 547—561.