ON THE GENERAL SOLUTION OF A FUNCTIONAL EQUATION CONNECTED TO SUM FORM INFORMATION MEASURES ON OPEN DOMAIN

. In this series, this paper is devoted to the study of a functional equation connected with the characterization of weighted entropy and weighted entropy of degree 8. Here, we find the general solution of the functional equation (2) on an open domain, without using O-probability and 1-probability.

The weigthed entropy of n n + degree B (B R-{I}) of an experiment is defined by Emptoz [2] as n Hn(P'U) (i-21-) -I I u k(pk-pSk )- k=l The measures HI(p,u) satisfy the following functional equation (see Kannappan [3]) Fn, Q Fm, (u I u2,.,Un +, (Vl,V2,.Vm) +, e,B -{0,i}.The measurable solution of (1.2) for = 1 was given by Kannappan in [3].In a recent paper of Kannappan and Sahoo [4], measurable solution of a more general functional equation than (1.2) was given using the result of this paper.In this paper, we deter- We need the following result in this sequel.
Result 1 [5].Let f,g: ]0,i[ -  u for all i 1 2 n and m where u,v Putting v i in (2.3), we get (2.1) and hence its general solutions cna be obtained from Result i.
First we consider the case e 8. Then from Result i, we have A 1 (P,U)u+a(u)up+b(u)up 8 f'(p, u) where a,b: + -]R are real valued functions of u and A 1 is additive in the first variable, with Al(l,u) --0.Letting (2.5) into (2.3),we get m n B.
m n e (a(uv)-a(v)) I Pi I qj+(b(uv)-b(u)) I Pi I q.
.__ From these it is easy to see that for all u,v ]R Now putting (2.7) into (2.5),we get + f(p,u) A l(p,u)u+au(p-p ) with A l(l,u) 0. Again letting (2.8) into (1.2),we get [ A l(qj,v)v + A l(qm,v')v' 0. j=l Since A 1 is additive in the first variable, and Al(l,v) 0, we get Al(qm 'v)v Al('v' )v' (2.11) for all qm ]0, i[, and v,v' +.From equation (2.11) it is clear that A l(x,y)y A(x) (2.12) where A is an additive function with A(1) 0. Now using (2.12) in (2.8), we obtain f(p,u) A(p) + au(pe-pB), p ]0, i[, u m (2.13) + where A is an additive function on with A(1) 0 and a is an arbitrary constant.
Next we consider the case e B.
Thus we have proved the following theorem.ACKNOWLEDGEMENT.This work is partially supported by a NSERC of Canada grant.
, where IR is the set of real numbers.Let n + (,A,B) be a probability space and let us consider an experiment that is a finite measurable partition {AI,A 2 An }' (n > l) of .The weighted entropy of such an experiment is defined by Belis and Guiasu [i] as n HI(p'U)n I ukP klOg Pk k=l where Pk () is the objective probability of the event , piqj,uivj) I PiU--" I f(qj'vj) + I I f(Pi'Ui

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mine the general solution of [1.2) where p and m,n (fixed and) > 3 on an open domain 2. SOLUTION OF (1.2) ON AN OPEN DOMAIN

A 2 )
p)+D(p)pe+c where a,b,c,d are arbitrary constants, A,A' are additive functions on with Now we proceed to determine the general solution of (1.2) on ]0,I[.Let f: ]0,I[ be a real valued function and satisfy the functional equation + (1.2) for an arbitrary but fixed pair of positive integers m,n (> 3), for P F O, I}.Letting u.

Theorem
by  (2.13) when e B and by(2.27)when e B.
b,c are arbitrary constants.Remark.Because of the occurrence of the parameters e,B as powers, f is independent of m and n.