--- L(qj)I(Pi) G(Pi)I(qj) Inm(P.Q) Gn(P)Im(Q + In(P)Lm(Q) P e F Q e F. Gk,L k. I(Piq j) WEIGHTED ADDITIVE INFORMATION MEASURES WOLFGANG SANDER

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1 Iterat. J. Math. & Math Sci. VOL. 13 NO. 3 (1990) WEIGHTED ADDITIVE INFORMATION MEASURES WOLFGANG SANDER Istitute for Aalysis Uiversity of Brauschweig Pockelsstr. 14, D 3300 Brauschweig, Geray (Received Jauary 26, 1989 ad i revised for May 25, 1990) ABSTRACT. We deterie all easurable fuctios I,G,L: [O,1] satisfyig the fuctioal equatio I(Piq j) -- - G(Pi)I(qj) + -- i=i 9=I i=1 j=l i=1 9=I for P e F, Q e F L(qj)I(Pi) ad for a fixed pair (,), > 3, > 3, where G(O) L(O) O ad G(1) L(1) I. This fuctioal equatio has iterestig applicatios i iforatio theory. KEY WORDS AND PHRASES. su for type, etropies of degree (e,8). Weighted additive iforatio easures of 1980 AMS SUBJECT CLASSIFICATION CODE. 39B30, 94A17. INTRODUCTION. k Let F k {P (pl,...,pk): Pi > O, Pi k > 2. i=1 We say that a iforatio easure I k F k R, k > 2 is (,)-weighted additive (, e q) if there exist weight fuctios Gk,L k F k R k > 2 such that I(P.Q) G(P)I(Q + I(P)L(Q) P e F Q e F where as usual P-Q (plql,piqj,...,pq) e F. I k, Gk, L k I,G,L [O,1] R, that is have the su property with geeratig fuctios (1.1) If i additio

2 k k k Ik(P) I(Pi Sk(P) S(Pi) Lk(P) i=i i=i i=i L(Pi) the equatio (I.1) goes over ito the fuctioal equatio (I.2) 418 W. SANDER P e F Q e F. the special cases This fuctioal equatio (1.3) is of iterest sice ad play iportat roles i the characterizatio of the etropies of degree a (Losoczi, [I) f a Hk(a,1) (p) (21 a 1)-I (Pi Pi a # a i=1 Ik(P) k Hk(P) Pi log a Pi i=i ad degree (,8) (Shara ad Taeja, [2]) (, k 6 I- 21-6)-I B (Pi Pi 8 (e,b) (p) i=i Ik k a Hk(P) 2a-I Pi log e Pi 8 i=1 respectively. Here we follow the covetios log log2 O.log O O ad oa O a e I. (1.8) G(p) p L(p) p + II(p) I e I (1.4) G(p) p L(p) p e,8 e I (1.5) H k (P) (2 The ai of this paper is to deterie all easurable triples (I,G,L) satisfyig. (1.3) for a fixed pair (,), > 3, > 3 where because of the kow results ad the covetio (I.8) we assue PeF k (1.6) PeF k, (I.7) G(O) L(O) O G(1) L(1) I. (1.9) Thus we deterie ot oly all easurable fuctios I of (I k) (see (I.2)) but also all possible choices for G ad L i (I.3). Therefore the results due to Kaappa [3-5], Losoczi [1], Shara ad Taeja [2,6] are special cases of our ai result. Moreover, if we assue that I is ot costat ad that G ad L are cotiuous the we ca iterpret our result i the for that, without loss of geerality, we ay assue that G ad L i (I.3) are cotiuous, o zero ultiplicative fuctios, that is they are o zero cotiuous solutios of the fuctioal equatio M(p-q) M(p) -M(q) p,q e [O,I] (1.10) 2. MAIN RESULTS. We ake use of the followig well kow result (Kaappa, [5]). LEMMA 1. Let 3 be a fixed iteger ad let F [O,1] be a easurable fuctio satisfyig

3 WEIGHTED ADDITIVE INFORMATION MEASURES 419 F(Pi) O i=i for all P e F The there exists a costat a such that F(p) a(1 p) p e [O,I] Now we are ready to prove our ai result which is a extesio of the results, etioed above. THEOREM 2. Let I,G,L [O, lj R be easurable ad let I be o costat. The I,G,L satisfy (1.9) ad (1.3) for a fixed pair (,), 3, 3 if, ad oly if they are of oe of the followig fors I(p) a(pa pb) G(p) (I b)p A + bp B L(p) bp A + (1 b)p B A B (2.1) I(p) apalog p G(p) pa(i + blog p) L(p) pa(1 blog p) A (2.2) I(p) I(O) + ( )I(O)p + dplog p, G(p) L(p) p, (2.3) I(I) O G(1) L(1) I(p) ap A G(p) (1 b)p A L(p) bp A p e [O,I) (2.4) I(p) a#si(clogp) G(p) pa[cos(clog p) + bsi(clog p)] L(p) pa[cos(clog p) bsi(clog p)] (2.5). Here A,B,a,b,c,d are costats ad we follow the covetios Oa.cos(log O) O Oa. si(log O) O a e PROOF. Obviously, the solutios (I,G,L) give by (2.1) to (2.5) satisfy (1.9) ad (1.3) To prove the coverse let us itroduce the fuctio I [O,I] R defied by I (p) I(p) I(O) (I(1) I(O))p p e [O,1]. (2.6) It is clear that I fulfills I (O) I (1) O. (2.7) We ow show that the triple (I,G,L) also satisfies (1.3). To see this let us put P (1,O,O,...,O) e F ad Q (1,O,O,...,O) e F ito (1.3) Usig (1.9) we arrive at or I(I) + (- 1)I(O) I(I) + (- I)I(O) + I(I) + (- )I(O) I(I) I(O) ( )I(O). (2.8) Thus I ca also be writte i the for I (p) I(p) I(O) ( )I(O)p. (2.9) Substitutig P e F Q (1,O,O,...,O) e F ad P (1,O,O,...,O)EF Q e F separately ito (1 3) we get G(Pi) (I(1) + (- 1)I(O)) (- ) I(O) (2.10) i=1 ad

4 W. SANDER L(qj)(I(I) + ( I)I(O)) ( )I(O) (2.11) j=1 or, usig (2.8) (I G(Pi))( ) I(O) O (2.12) 3=I ad -- (1 L(qj))() I(O) O (2.13) j=l respectively. After these preparatios we ca see iediately that I,G,L satisfy I (piqj) G(Pi)I (qj) + L(qj)I (pi) (2.14) i=i 3=I i=i i=i 3=I for all P e F Q e F Puttig I, give by (2 6) ito (2 14) ad usig (1.3) ad (2.8), we see that (2.14) is equivalet to (1 G(Pi)) ( )I(O) + (1 )I(O) O. (2.15) i=i j=1 L(qj))( But (2.15) is ideed valid because of (2.12) ad (2.13). I a further step we derive a fuctioal equatio for I, G ad L i which o sus will occur. Settig F(p,q) I (p-q) G(p)I (q) L(q)I (p) p,q [O,1] (2.16) we get fro (2.14).= F(Pi qj)= O P E F Q E F - Sice by hypothesis F [O,1] 2 R is easurable i each variable we get fro Lea i Kaappa [3] (This Lea is a applicatio of the above Lea that F ca be represeted i the for F(p,q) F(p,O) (I q) + F(O,q) (I p) F(O,O) (1 q) (1 p) (2.17) Thus (2.16) ad (2.17) iply I (p q) G(p)I (q) + L(q)I (p) p,q e [O,1] (2.18) sice (2.17), (1.9) ad (2.7) yield F(p,O) F(O,q) F(O,O) O. Because of I (O) G(O) L(O) O it is eough to solve (2.18) for all p,q e (O,13. Coplex-valued fuctioal equatios of this type were itesively studied by Vicze [7-93. Fro these results we get the solutios of (2.18) for p,q e (O,1] (Ebaks, [10]) which have oe of the followig fors

5 WEIGHTED ADDITIVE INFORMATION MEASURES 421 I (p) a.m(p).log p, G(p) M(p) (1 + b.log p) L(p) M(p) (I b.log p) (2.19) I (p) a(m (p) M2(p)), G(p) (I b)m1(p) + bm2(p) L(p) bm1(p) + (1 b)m2(p) (2.20) I (p) am(p)si(clog p) G(p) M(p) [cos (clog p) + bsi(clog p)] L(p) M(p)cos(clog p) bsi(clog p)]. (2.21) Here a,b,c are costats ad M, MI, M2: (O,1] R are easurable ultiplicative fuctios. Let us reark that the easurable solutios of (1.10) for p,q e (O,13 are either M O or (2.22) M(p) pa A e or (2.23) M(1) M(p) O for p e (O,1). (2.24) Sice I has the for - I(p) I (p) + I(O) + (- - )I(O)p p e [O,1] (2.25) (see (2.6) (2.8)) we ca derive the solutios (I,G,L) of (1.9) ad (1.3) fro (2.19) to (2.24). Let us first cosider the case that (S(Pi) pi O ad (L(qj) qj) O (2.26) i=i 3=I for all P e F, Q The Lea iplies that F" G(p) rp + s L(p) r p + s p e [O,1] where r,r,s,s are costats. Because of (1.9) we arrive at G(p) L(p) p p [O,1]. This is oly possible if either or if - b O M(p) p i (2.19) b MI(p) M2(p) p i (2.20). (G(Pi) i=i (L(qj) qj) O for soe Q e F 3= I both cases we get the solutio (2.3). Now we assue that pi O for soe P e F or The (2.12) ad (2.13) iply that I(O) O so that i all cases where G(p) p or L(p) # p "

6 22 W SANDER we get that I is equal to I ad thus I is ot depedet upo ad (see (2.25)). Usig G(1) L(1) ad the hypothesis that I is ot costat we obtai fro (2.19) to (2.24) the reaiig solutios (2.1), (2.2), (2.4) ad (2.5) Thus the Theore is prove. It is clear that we ca obtai fro Theore 2 soe ew characterizatio theores for iforatio easures. For istace, we reark that the fuctios G ad L give by (2.4) or (2.5) caot be cotiuous siultaeously. Thus we get the followig extesio of results i Kaappa [3,4], Shara ad Taeja [2,6]. COROLLARY 3. If i additio to the hypotheses of Theore 2, G ad L are cotiuous the the oly solutios (I,G,L) of (1.9) ad (1.3) are give by (2.1), (2.2) ad (2.3). Corollary 3 iplies iediately the followig characterizatio theore Let I k be a (,)-weighted additive iforatio easure where I k, G k, L k - have the su property with cotiuous geeratig fuctios I(O) (0) (0) 0 (1)-- g(1) ad I( the Ik(P) tt,b)(p) or Ik(P) Ilk(p) P e F k. Here,B,C are real costats with A B. Fially we give two iterpretatios of our result. If we put b O ito (2.1), (2.2) ad (2.3) the we get with uchaged I(p) G(p) pa B B pa A L(p) p P pa + I(p) + p B G(p) pa L(p) pa A G(p) p L(p) p, respectively. Thus we ay cosider Corollary 3 as a justificatio for the fact that i the literature oly two special fors of G ad L were cosidered, aely (1.4) ad (1.5). O the other had, the coditio b O i (2.1) ad (2.2) iplies that i Corollary 3 we ay assue without loss of geerality that G ad L are cotiuous, o zero ultiplicative fuctios. This result is aalogous to a result cocerig recursive easures of ultiplicative type (Aczl ad Ng, [113). REFERENCES I. LOSONCZI, L. A characterizatio of etropies of degree, Metrika 28 (1981), SHARMA, B.D. ad TANEJA, I.J. Etropy of type (,8) ad other geeralized easures of iforatio theory, Metrika 2_2 (1975), KANNAPPAN, PIo O soe fuctioal equatios fro additive ad o-additive easures-i, Proc. Edi. Math. Soc. 23 (1980), 45-I 50.

7 WEIGHTED ADDITIVE INFORMATION MEASURES KANNAPPAN, P I. O a geeralizatio of su for fuctioal equatio-iii, Deostratio Math. 13 (1980) KANNAPPAN, PI. O a geeralizatio of su for fuctioal equatio-i, Stochastica 7 (1983), SHARMA, B.D. ad TANEJA, I.J. Three geeralized additive easures of etropy, EI K 13(7/8) (1977), VINCZE, E. Uber die Verallgeeierug der trigooetrische ud verwadte Fuktioalgleichuge, A. Uiv. Sci. Budapest E6tv6s, Sect. Math. 3-4 ( ), Vicze, E. Eie allgeeiere Methode i der Theorie der Fuktioalgleichuge II, Publ. Math. Debrece 9 (1962), Vicze, E. Eie allgeeiere Methode i der Theorie der Fuktioalgleichuge III, Publ. Math. Debrece 10 (1963), Ebaks, B.R. O easures of fuzziess ad their represetatios, J. Math. Aal. AppI. 94 (1983), ACZL, J. ad NG, C.T. Deteriatio of all seisyetric recursive iforatio easures of ultiplicative type o positive discrete probability distributios, Liear Algebra APPI. 52/53 (1983), 1-30.

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