Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 (2003) A NOTE ON NON-NEGATIVE INFORMATION FUNCTIONS

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1 Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 30 ( A NOTE ON NON-NEGATIVE INFORMATION FUNCTIONS Béla Brindza and Gyula Maksa (Debrecen, Hungary Dedicated to the memory of Professor Péter Kiss Abstract. The purpose of the present paper is to make a first step to prove the conjecture, namely, that not every non-negative information function coincides with the Shannon s one on the algebraic elements of the closed unit interval.. Introduction The characterization of the Shannon entropy, based upon its recursive and symmetric properties is strongly connected with the so-called fundamental equation of information, which is (. f( + ( f ( ( y = f(y + ( yf y where f: [0, ] IR and (. holds for all, y [0, [, + y. The solutions of (. satisfying f(0 = f( and f( 2 = are the information functions. The basic monography Aczél and Daróczy [] contains several results on these functions, like, if f is non-negative and bounded, then f = S, where S( = log 2 ( log 2 (, [0, ], (0 log 2 0 is defined by 0. (See also Daróczy Kátai [2]. A related result is Theorem. (Daróczy Maksa [3]. If f is a non-negative information function, then (.2 f( S(, [0, ] moreover, there eists a non-negative information function different from S. This research has been supported by the Hungarian Research Fund (OTKA Grant T and by the Higher Educational Research and Development Fund (FKFP Grant 025/200.

2 32 B. Brindza and Gy. Maksa The proof of the second part of this theorem is based upon the eistence of a non-identically zero real derivation d: IR IR which is additive, that is d( + y = d( + d(y (, y IR and satisfies the equation d(y = d(y + yd(, (, y IR and different from 0 at some point. (See for eample Kuczma [4]. A computation shows that the function S( + d(2 if ] 0, [ (.3 f( = ( 0 if {0, } is a non-negative information function and different from S if d is a real derivation different from 0. (See Daróczy Maksa [3]. After this result some other natural questions arose, namely, the characterization of the non-negative information functions and (or at least their Shannon kernel { [0, ]: f( = S(} where f is a fied non-negative information function. (See Lawrence Mess Zorzitto [6], Maksa [7] and Lawrence [5]. It is known that the real derivations are vanishing over the field of algebraic numbers (se Kuczma [4], hence (.4 f(α = S(α if f is given by (.3. It is noted that (.4 holds for all non-negative information functions f and for all rational α [0, ]. (See Daróczy Kátai [2]. Our conjecture is that there are non-negative information functions that are different from the Shannon s one at some algebraic element of [0, ]. In the net section we prove a partial result in this direction. 2. Results The base of our investigations is the following theorem. Theorem 2. A function f: [0, ] IR is a non-negative information function, if and only if, there eists an additive function a: IR IR such that a( =, (2. a(log 2 ( a(log 2 ( 0 if ] 0, [,

3 A note on non-negative information functions 33 and a(log 2 ( a(log 2 ( if ] 0, [ (2.2 f( = 0 if {0, }. Furthermore f = S holds, if and only if, there is a real derivation d: IR IR such that (2.3 a( = + 2 d(2 if IR. Proof. The first part of the theorem is an easy consequence of Theorem of Daróczy Maksa [3]. To prove the second part, first suppose that the non-negative information function f coincides with S on [0, ]. Therefore, by the definition of S and by (2.2, we get that (2.4 a(log 2 ( a(log 2 ( = log 2 ( log 2 ( holds for all ] 0, [ where a is an additive function that eists by the first part of the theorem. Define the function ϕ: ] 0, + [ IR by (2.5 ϕ( = a(log 2 + log 2. An easy calculation shows that (2.6 ϕ(y = ϕ(y + yϕ( if > 0, y > 0 and, because of (2.4, This implies that ϕ( + ϕ( = 0 if 0 < <. ( ( y ϕ + ϕ = 0 + y + y for all > 0, y > 0 whence, applying (2.6, we have that ( 0 = ϕ + y = ( + yϕ + ( + y ( ϕ( + yϕ + + y + y ϕ(y + y + (ϕ( + ϕ(y + y = ϕ( (ϕ( + y ϕ( ϕ(y. + y

4 34 B. Brindza and Gy. Maksa Since ϕ( = 0, we dotain that (2.7 ϕ( + y = ϕ( + ϕ(y if > 0, y > 0. If IR define the function d: IR IR by d( = ϕ(u ϕ(v where u > 0, v > 0 and = u v. Equation (2.7 garantees that the definition of d is correct, d is additive, and moreover, by (2.6 and (2.7, d is a real derivation that is an etension of ϕ to IR. Thus, by (2.5, d( = a(log 2 + log 2 if > 0 whence we obtain (2.3 replecing by 2. Finally, if d is an arbitrary real derivation then the function a defined by (2.3 is additive, a( = and the function f given in (2.2 coincides with S on [0, ]. Since every real derivation vanishes at all algebraic points (see, for eample Kuczma [4], in order to prove our conjecture, by (2.3, we have to construct an additive function a for which a( =, a(log 2 β log 2 β for some positive algebraic number β and (2. holds for all ] 0, [. Instead of this we can proof the following weaker result only. Theorem 3. Let Q(α be a real algebraic etension of Q of degree n >. If Q[α] (the ring of algebraic integers in Q(α is a unique factorization domain then there eists an additive a: IR IR with a( = satisfying (2.8 a(log 2 ( a(log 2 ( S( if ] 0, [ Q[α] and (2.9 a(log 2 β log 2 β for some positive algebraic number β. Proof. Let U be the unitgroup of Q[α] generating by a set of fundamental units {ε,..., ε n } and P = {π,..., π s,...} be the set of primes in Q[α]. Since the group of the roots of unity is {, }, only, we may assume that 0 < ε i, i =,...,n ; 0 < π j, j =, 2,.... and every non-zero element of Q[α] can uniquely be written in the form (2.0 = ± ( n i= ε ki i j= π lj j

5 A note on non-negative information functions 35 where the eponents are (rational integers and l j 0, j =, 2,.... The set P is multiplicatively independent, hence the set {log 2 π: π P } is linearly independent (over Q. Therefore there is a Hamel basis H IR for which H and log 2 π H if π P. Let π P be fied. We may assume that π 2. Define the function a 0 π on H by a 0 (log 2 π = log 2 2, a 0 (h = h if h H, h log 2 π, and let a be the additive etension of a 0 to IR. It is obvious that a( = and (2.9 is satisfied by β = π. To prove (2.8 first suppose that the eponent of π is positive in the decomposition (2.0 of ] 0, [ Q[α]. Then the eponent of π in the decomposition of ( is zero. Of course, the same is true also for ( instead of. Therefore (2. a(log 2 ( = log 2 ( or (2.2 a(log 2 = log 2 holds for all ]0, [ Q[α]. Supposing (2. we have that a(log 2 ( a(log 2 ( ( = a log 2 + log 2 π l ( = a π l log 2 π l ( log 2 ( a(log 2 π l ( log 2( = log 2 l a(log π l 2 π ( log 2 ( [ = log 2 ( log 2 ( + l log2 π a(log 2 π ] [ π ] = log 2 ( log 2 ( + l log2 π log 2 2 > log 2 ( log 2 ( = S(. Thus (2.8 holds. In case (2.2 the proof is similar. Finally, if the eponent of π is zero in the decompositions of both and ( then, of course, the equality is valid in (2.8. Remark. According to the classical approimation result of Dirichlet the set D = { ]0, [ Q[α]: l > 0 in (2.0} is dense in [0, ]. Thus the strict inequality holds on the dense set D in (2.8. References [] Aczél, J. and Daróczy, Z., On measures of information and their characterizations, Academic Press, New York, 975.

6 36 B. Brindza and Gy. Maksa [2] Daróczy, Z. and Kátai, I., Additive zahlentheoretische Funktionen und das Mass der Information, Ann. Univ. Sci. Budapest, Eötvös Sect. Math., 3 (970, [3] Daróczy, Z. and Maksa, Gy., Nonnegative information functions. In: Proc. Colloqu. Methods of Comple Anal. in the Theory of Probab. and Statist., Debrecen, 977, Colloquia Mathematica Societatis János Bolyai Vol. 2, North-Holland, Amsterdam, 979, [4] Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities, Państwowe Wydawnictwo Naukowe, Warszawa Kraków Katowice, 985. [5] Lawrence, J., The Shannon kernel of non-negative information function, Aequationes Math., 23 (98, [6] Lawrence, J., Mess, G. and Zorzitto, F., Near-derivations and information functions, Proc. Amer. Math. Soc., 76 (979, [7] Maksa, Gy., On near-derivations, Proc. Amer. Math. Soc., 8, (98, Béla Brindza University of Debrecen Institute of Mathematics 400 Debrecen P.O. Bo 2. Hungary brindza@math.klte.hu Gyula Maksa University of Debrecen Institute of Mathematics 400 Debrecen P.O. Bo 2. Hungary maksa@math.klte.hu