COMPLEMENTARY RESULTS TO HEUVERS S CHARACTERIZATION OF LOGARITHMIC FUNCTIONS. Martin Himmel. 1. Introduction

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1 Annales Mathematicae Silesianae 3 207), DOI: 0.55/amsil COMPLEMENTARY RESULTS TO HEUVERS S CHARACTERIZATION OF LOGARITHMIC FUNCTIONS Martin Himmel Abstract. Based on a characterization of logarithmic functions due to Heuvers we develop analogous results for multiplicative, exponential and additive functions, respectivel.. Introduction Heuvers [] see also Kannappan [3, p. 3]), using a coccle equation, gave the following characterization of logarithmic functions: a function f : 0, ) R satisfies the functional equation if and onl if fx + ) fx) f) = f x + ), x, > 0, fx) = fx) + f), x, > 0. On this basis, we present analogous characterizations for the remaining three Cauch equations, i.e. for multiplicative, exponential and additive functions. In the case of exponential and additive functions the classical log function appears. Received: Accepted: Published online: ) Mathematics Subject Classification: 26E60, 39B22, 97I70. Ke words and phrases: means, functional equation.

2 00 Martin Himmel Let us observe that Heuvers result can be formulated also in terms of the Cauch difference C f : 0, ) 2 R, C f x, ) := fx + ) fx) f), x, > 0, of a generator f : 0, ) R, and the harmonic mean H : 0, ) 2 0, ), Hx, ) = 2x, x, > 0. x + Remark [2]). The generator f : 0, ) R of the Cauch difference satisfies the functional equation.) C f x, ) = f 2 Hx, ) if and onl if f is a logarithmic function, i.e. ), x, > 0 fx) = fx) + f), x, > Characterization of multiplicative functions The following result characterizes multiplicative functions. Theorem. The function g : 0, ) 0, ) satisfies the functional equation 2.) gx + ) gx)g) = g x + ), x, > 0, if and onl if g is a multiplicative function. Proof. Assume that g : 0, ) 0, ) satisfies 2.). Taking the logarithm of both sides, we obtain log ) gx + ) = log g gx)g) x + )), x, > 0,

3 Complementar results to Heuvers s characterization of logarithmic functions 0 and thus, using properties of the function log : 0, ) R, we have log gx + )) log gx)) log g)) = log g x + )), x, > 0. Due to Heuvers result, the function f := log g is logarithmic, i.e. whence log gx)) = log gx)) + log g)), x, > 0, log gx)) = log gx)g)), x, > 0, and the injectivit of the function log implies that gx) = gx)g), x, > 0. To prove the converse implication, assume that g : 0, ) 0, ) is multiplicative. Since g is positive, the multiplicativit implies that g) = and g ) = x gx), x > 0. Hence, we have for all x, > 0, g x + ) = g x + ) ) ) = gx + )g x x = gx + ) gx) = gx + ) gx)g), which completes the proof. Motivated b the relationship between the Euler Gamma function and the Beta function, let us introduce the following

4 02 Martin Himmel Definition [2]). For arbitrar g : 0, ) 0, ) we call B g : 0, ) 2 0, ) defined b the beta-tpe function of generator g. B g x, ) = gx)g), x, > 0, gx + ) Note that for an function g : 0, ) 0, ) we have B g = B g. This fact, Theorem and the definition of beta-tpe functions allow us to formulate the following Remark 2. The generator g : 0, ) 0, ) of the beta-tpe function B : 0, )2 0, ) satisfies the functional equation g ) x, ) = g 2, x, > 0, g Hx, ) B if and onl if g is a multiplicative function, i.e. gx) = gx)g), x, > 0. This shows that the harmonic mean is indirectl related to the Beta function cf. [2]). 3. Characterization of exponential functions From Theorem we get the following. Corollar. A function h: R 0, ) is exponential, i.e. if and onl if 3.) hs + t) = hs)ht), s, t R, h logx + ) h logx))h log)) = h log x + ), x, > 0.

5 Complementar results to Heuvers s characterization of logarithmic functions 03 Proof. Assume that h is exponential. Hence, using properties of the function log: 0, ) R, we have, for all x, > 0, h log x + )) h log x)h log ) = = h log h log = h log = h log x + x + x + x + = h logx + ), )) h log x + log )) )) h logx)) ) ) + logx) ) )) x = hlog + x)) thus 3.) holds. Conversel, assume that h satisfies 3.), and put g := h log. Then h = g exp and, b 3.), g exp logx + ) = g exp log g exp logx))g exp log)) x + ), x, > 0. Hence, using properties of the functions exp and log, we have gx + ) gx)g) = g x + ), x, > 0. B Theorem, the function g is multiplicative, i.e. gx) = gx)g), x, > 0. For all x, > 0 there exist s, t R such that Thus, b the multiplicativit of g, x = e s, = e t. g e s+t) = g e s ) g e t), s, t R,

6 04 Martin Himmel which means that the function h = g exp is exponential. This finishes the proof. In the spirit of the preceeding two remarks, we have the following. Remark 3. A function h: R 0, ) is exponential if and onl if ) x, ) = h log 2, x, > 0. h log Hx, ) B 4. Characterization of additive functions The characterization of additive functions with the aid of Heuvers result is given in the following. Corollar 2. A function α: R R is additive, i.e. if and onl if αs + t) = αs) + αt), s, t R, 4.) α logx + ) α logx) α log) = α log x + ), x, > 0. Proof. Assume that α is additive. Using this and properties of the function log, we have, for all x, > 0, α logx + ) α logx) α log) α log which shows 4.). x + = α logx + ) log x log log x + )) = α log x + ) x = αlog ) = α0) = 0, x + ) )

7 Complementar results to Heuvers s characterization of logarithmic functions 05 Vice versa, assume that α satisfies 4.). Put h := α log. Hence, α = h exp and, b 4.), using properties of the functions exp and log, we have, for all x, > 0, α logx + ) α logx) α log) = h exp logx + ) h exp logx) h exp log) = h exp log x + ) = α log x + ) ; thus hx + ) hx) h) = h x + ), x, > 0. Appling the function exp to both sides and using its properties we obtain e hx+) e hx) e h) = eh x + ), x, > 0. Thus, b the definition of h, the function g := exp α log satisfies gx + ) gx)g) = g x + ), x, > 0, and Theorem implies that g is multiplicative, i.e. gx) = gx)g), x, > 0. Appling here the function log to both sides and writing each x, > 0 in the form x = e s, = e t, for some s, t R gives us, using properties of exponential functions, log ge s+t ) = log ge s ) + log ge t ), s, t R, which means that function α = log g exp is additive. This finishes the proof.

8 06 Martin Himmel We reformulate this result in terms of the harmonic mean and the Cauch difference in the following. Remark 4. A function α: R R is additive if and onl if C α log x, ) = α log 2 Hx, ) ), x, > 0. Remark, Remark 2, Remark 3 and Remark 4 lead to the question on the relation between different tpes of Cauch differences or beta-tpe functions and means, which will be treated in our next paper. References [] Heuvers K.J., Another logarithmic functional equation, Aeq. Math ), [2] Himmel M., Matkowski J., Homogeneous beta-tpe functions, J. Math. Inequal. To appear. [3] Kannappan Pl., Functional equations and inequalities with applications, Springer Monographs in Mathematics, Springer, New York, Facult of Mathematics Computer Science and Econometrics Universit of Zielona Góra Szafrana 4A Zielona Góra Poland himmel@mathematik.uni-mainz.de