A sum form functional equation on a closed domain and its role in information theory
aa r X i v : . [ c s . I T ] A ug A sum form functional equation on a closed domainand its role in information theory
P. Nath and D.K. Singh (India)
Abstract.
This paper is devoted to finding the general solutions of thefunctional equation n X i =1 m X j =1 h ( p i q j ) = n X i =1 h ( p i ) + m X j =1 k j ( q j ) + λ n X i =1 h ( p i ) m X j =1 k j ( q j )valid for all complete probability distributions ( p , . . . , p n ), ( q , . . . , q m ),0 p i
1, 0 q j i = 1 , . . . , n ; j = 1 , . . . , m , n P i =1 p i = 1, m P j =1 q j = 1; n > m > λ ∈ R , λ = 0 and the mappings h : I → R , k j : I → R , j = 1 , . . . , m ; I = [0 , R denoting the set of all real numbers.A special case of the above functional equation was treated earlier by L. Losoncziand Gy. Maksa.2000 Mathematics Subject Classification.
Key words and phrases. sum form functional equation, additive function, multiplicativefunction, entropy of degree α . P. NATH AND D.K. SINGH
1. Introduction
Let Γ n = { ( p , . . . , p n ) : 0 p i , i = 1 , . . . , n ; n P i =1 p i = 1 } , n = 2 , , . . . denote the set of all discrete n -component complete probability distributions withnonnegative elements. Let R denote the set of all real numbers and∆ = { ( x, y ) : 0 x , y , x + y } , the unit triangle ; I = { x ∈ R : 0 x } = [0 ,
1] ; I = { x ∈ R : 0 < x < } . A mapping a : I → R is said to be additive on I if a ( x + y ) = a ( x ) + a ( y )holds for all ( x, y ) ∈ ∆. A mapping A : R → R is said to be additive on R if A ( x + y ) = A ( x ) + A ( y ) (1.1)holds for all x ∈ R , y ∈ R .It is known [2] that every mapping a : I → R , additive on the unit triangle∆, has a unique additive extension A : R → R in the sense that A satisfies theequation (1.1) for all x ∈ R , y ∈ R .A mapping M : I → R is said to be multiplicative on I if M (0) = 0 (1.2) M (1) = 1 (1.3)and M ( pq ) = M ( p ) M ( q ) (1.4)holds for all p ∈ I , q ∈ I .The functional equation (see [1]) n X i =1 m X j =1 f ( p i q j ) = n X i =1 f ( p i ) + m X j =1 f ( q j ) + λ n X i =1 f ( p i ) m X j =1 f ( q j ) (1.5) sum form functional equation . . . 3 where f : I → R , ( p , . . . , p n ) ∈ Γ n , ( q , . . . , q m ) ∈ Γ m , λ = 2 − α − = 0 is usefulin characterizing the entropy of degree α (see [3]) defined as H αn ( p , . . . , p n ) = (1 − − α ) − − n X i =1 p αi ! , (1.6)where H αn : Γ n → R , n = 2 , , . . . and 0 α := 0, α = 1, α ∈ R . For λ ∈ R , λ = 0, the general solutions of (1.5), for fixed integers n > m > p , . . . , p n ) ∈ Γ n , ( q , . . . , q m ) ∈ Γ m have been obtained in [6]. A generalizationof (1.5) is the following functional equation (see [5]) n X i =1 m X j =1 f ij ( p i q j ) = n X i =1 h i ( p i ) + m X j =1 k j ( q j ) + λ n X i =1 h i ( p i ) m X j =1 k j ( q j ) (1.7)with f ij : I → R , h i : I → R , k j : I → R , i = 1 , . . . , n ; j = 1 , . . . , m . Forfixed integers n > m > p , . . . , p n ) ∈ Γ n , ( q , . . . , q m ) ∈ Γ m , themeasurable (in the sense of Lebesgue) solutions of (1.7) have been obtained in(see [5], Theorem 6 on p-69) but it seems that the general solutions of (1.7), forfixed integers n > m > p , . . . , p n ) ∈ Γ n , ( q , . . . , q m ) ∈ Γ m are stillnot known. As mentioned in [5], equations like (1.7) arise while characterizingmeasures of information concerned with two probability distributions. In thispaper, we study the equation n X i =1 m X j =1 h ( p i q j ) = n X i =1 h ( p i ) + m X j =1 k j ( q j ) + λ n X i =1 h ( p i ) m X j =1 k j ( q j ) (1.8)where h : I → R , k j : I → R , j = 1 , . . . , m ; λ ∈ R , λ = 0 and n > m > f : I → R and g j : I → R , j = 1 , . . . , m as (with λ = 0) f ( x ) = x + λ h ( x ) and g j ( x ) = x + λ k j ( x ) (1.9)for all x ∈ I , then (1.8) reduces to the functional equation n X i =1 m X j =1 f ( p i q j ) = n X i =1 f ( p i ) m X j =1 g j ( q j ) . (1.10)Also, (1.9) and (1 .
10) yield (1.8). Thus, if the general solutions of (1.10), for fixedintegers n > m > p , . . . , p n ) ∈ Γ n , ( q , . . . , q m ) ∈ Γ m are known; P. NATH AND D.K. SINGH the corresponding general solutions of (1.8), for fixed integers n > m > p , . . . , p n ) ∈ Γ n , ( q , . . . , q m ) ∈ Γ m can be determined with the aid of (1.9).We would like to mention that, on open domain, namely when f : I → R , h : I → R , g j : I → R , k j : I → R , j = 1 , . . . , m , the general solutionsof (1.8) and (1.10) for fixed integers n > m > p , . . . , p n ) ∈ Γ n ,( q , . . . , q m ) ∈ Γ m have been found in [4]. The object of this paper is to determinethe general solutions of (1.8) and (1.10), on the closed domain, namely when f : I → R , h : I → R , g j : I → R , k j : I → R , j = 1 , . . . , m ; for fixed integers n > m > p , . . . , p n ) ∈ Γ n , ( q , . . . , q m ) ∈ Γ m . While investigatingthese solutions, the functional equation n X i =1 m X j =1 ϕ ( p i q j ) = n X i =1 ϕ ( p i ) m X j =1 ϕ ( q j ) + m ( n − ϕ (0) n X i =1 ϕ ( p i ) (1.11)arises with ϕ : I → R , n > m > p , . . . , p n ) ∈ Γ n ,( q , . . . , q m ) ∈ Γ m .To deal with equations (1.8), (1.10) and (1.11), we need the results andmethods from [5] and [6].
2. Some preliminary results
We require the following two results in sections 3 and 4.
Result 1. [6]. Let k > c be a given constant.Suppose that a mapping ψ : I → R satisfies the functional equation k X i =1 ψ ( p i ) = c (2.1)for all ( p , . . . , p k ) ∈ Γ k . Then there exists an additive mapping B : R → R suchthat ψ ( p ) = B ( p ) − k B (1) + ck (2.2)for all p ∈ I . sum form functional equation . . . 5 Result 2. [5]. If the mappings ψ j : I → R , j = 1 , . . . , m satisfy the functionalequation m X j =1 ψ j ( q j ) = 0 (2.3)for an arbitrary but fixed integer m > q , . . . , q m ) ∈ Γ m , then thereexists an additive mapping A : R → R and the constants c j ( j = 1 , . . . , m ) suchthat ψ j ( p ) = A ( p ) + c j (2.4)for all p ∈ I and j = 1 , . . . , m with A (1) + m X j =1 c j = 0 . (2.5)
3. The functional equation (1.11)
In this section, we prove:
Theorem 1.
Let n > , m > be fixed integers and ϕ : I → R be amapping which satisfies the functional equation (1.11) for all ( p , . . . , p n ) ∈ Γ n and ( q , . . . , q m ) ∈ Γ m . Then ϕ is of the form ϕ ( p ) = a ( p ) + ϕ (0) (3.1) where a : R → R is an additive mapping with (i) a (1) = − nm ϕ (0) if ϕ (1) + ( n − ϕ (0) = 1or(ii) a (1) = 1 − n ϕ (0) if ϕ (1) + ( n − ϕ (0) = 1 (3.2) or ϕ ( p ) = M ( p ) − B ( p ) (3.3) where B : R → R is an additive mapping with B (1) = 0 and M : I → R is multiplicative on I in the sense that it satisfies (1.2) , (1.3) and (1.4) for all p ∈ I , q ∈ I . P. NATH AND D.K. SINGH
Proof.
Let us put p = 1, p = . . . = p n = 0 in (1.11). We obtain[ ϕ (1) + ( n − ϕ (0) − " m X j =1 ϕ ( q j ) + m ( n − ϕ (0) = 0 (3.4)for all ( q , . . . , q m ) ∈ Γ m . We divide our discussion into two cases.Case 1. ϕ (1) + ( n − ϕ (0) − = 0.In this case, (3.4) reduces to m X j =1 ϕ ( q j ) = − m ( n − ϕ (0) (3.5)for all ( q , . . . , q m ) ∈ Γ m . By Result 1, there exists an additive mapping a : R → R such that ϕ ( p ) = a ( p ) − m a (1) − ( n − ϕ (0) (3.6)for all p ∈ I . The substitution p = 0, in (3.6), gives a (1) = − nm ϕ (0) . (3.7)From (3.6) and (3.7), (3.1) follows. Thus, we have obtained the solution (3.1)satisfying (i) in(3.2).Case 2. ϕ (1) + ( n − ϕ (0) − m X j =1 ( n X i =1 ϕ ( p i q j ) − ϕ ( q j ) n X i =1 ϕ ( p i ) − m ( n − ϕ (0) q j n X i =1 ϕ ( p i ) ) = 0 . (3.8)Choose ( p , . . . , p n ) ∈ Γ n and fix it. Define ψ : Γ n × I → R as ψ ( p , . . . , p n ; q ) = n X i =1 ϕ ( p i q ) − ϕ ( q ) n X i =1 ϕ ( p i ) − m ( n − ϕ (0) q n X i =1 ϕ ( p i ) (3.9)for all q ∈ I . By Result 1, there exists a mapping A : Γ n × R → R , additive inthe second variable, such that n X i =1 ϕ ( p i q ) − ϕ ( q ) n X i =1 ϕ ( p i ) − m ( n − ϕ (0) q n X i =1 ϕ ( p i )= A ( p , . . . , p n ; q ) − m A ( p , . . . , p n ; 1) (3.10) sum form functional equation . . . 7 The substitution q = 0, in (3.10), gives A ( p , . . . , p n ; 1) = m ϕ (0) " n X i =1 ϕ ( p i ) − n (3.11)as A ( p , . . . , p n ; 0). From (3.10) and (3.11), we obtain n X i =1 ϕ ( p i q ) − ϕ ( q ) n X i =1 ϕ ( p i ) − m ( n − ϕ (0) q n X i =1 ϕ ( p i )= A ( p , . . . , p n ; q ) − ϕ (0) n X i =1 ϕ ( p i ) + n ϕ (0) . (3.12)Since ( p , . . . , p n ) ∈ Γ n was chosen arbitrarily and then fixed, equation (3.12),indeed, holds for all ( p , . . . , p n ) ∈ Γ n and all q ∈ I .Let x ∈ I and ( r , . . . , r n ) ∈ Γ n . Putting q = xr t , t = 1 , . . . , n in (3.12);adding the resulting n equations and using the additivity of A in the secondvariable, it follows that n X i =1 n X t =1 ϕ ( xp i r t ) − n X t =1 ϕ ( xr t ) n X i =1 ϕ ( p i ) − m ( n − ϕ (0) x n X i =1 ϕ ( p i )= A ( p , . . . , p n ; x ) − n ϕ (0) n X i =1 ϕ ( p i ) + n ϕ (0) . (3.13)Also, if we put q = x and p i = r i , i = 1 , . . . , n in (3.12), we obtain n X t =1 ϕ ( xr t ) = ϕ ( x ) n X t =1 ϕ ( r t ) + m ( n − ϕ (0) x n X t =1 ϕ ( r t )+ A ( r , . . . , r n ; x ) − ϕ (0) n X t =1 ϕ ( r t ) + n ϕ (0) . (3.14)From (3.13) and (3.14), we can obtain the equation n X i =1 n X t =1 ϕ ( xp i r t ) − [ ϕ ( x ) + m ( n − ϕ (0) x − ϕ (0)] × n X i =1 ϕ ( p i ) n X t =1 ϕ ( r t ) − n ϕ (0)= A ( p , . . . , p n ; x ) + m ( n − ϕ (0) x n X i =1 ϕ ( p i )+ A ( r , . . . , r n ; x ) n X i =1 ϕ ( p i ) . (3.15) P. NATH AND D.K. SINGH
The symmetry of the left hand side of (3.15), in p i and r t , i = 1 , . . . , n ; t = 1 , . . . , n gives rise to the equation A ( p , . . . , p n ; x ) + m ( n − ϕ (0) x n X i =1 ϕ ( p i ) + A ( r , . . . , r n ; x ) n X i =1 ϕ ( p i )= A ( r , . . . , r n ; x ) + m ( n − ϕ (0) x n X t =1 ϕ ( r t )+ A ( p , . . . , p n ; x ) n X t =1 ϕ ( r t )which can be written in the form[ A ( p , . . . , p n ; x ) + m ( n − ϕ (0) x ] " n X t =1 ϕ ( r t ) − = [ A ( r , . . . , r n ; x ) + m ( n − ϕ (0) x ] " n X i =1 ϕ ( p i ) − . (3.16)Equation (3.16) holds for all ( r , . . . , r n ) ∈ Γ n , ( p , . . . , p n ) ∈ Γ n and all x ∈ I .Subcase 2.1. n P t =1 ϕ ( r t ) − n .In this case, n X t =1 ϕ ( r t ) = 1 (3.17)holds for all ( r , . . . , r n ) ∈ Γ n . By Result 1, there exists an additive map a : R → R such that ϕ ( p ) = a ( p ) − n a (1) + 1 n (3.18)for all p ∈ I . The substitution p = 0, in (3.18), yields a (1) = 1 − n ϕ (0) . (3.19)From (3.18) and (3.19), (3.1) follows again. Thus, we have obtained the solution(3.1) satisfying (ii) in (3.2).Subcase 2.2. n P t =1 ϕ ( r t ) − n .Then, there exists a probability distribution ( r ∗ , . . . , r ∗ n ) ∈ Γ n such that n X t =1 ϕ ( r ∗ t ) − = 0 . (3.20) sum form functional equation . . . 9 Setting r = r ∗ , . . . , r n = r ∗ n in (3.16), we obtain[ A ( p , . . . , p n ; x ) + m ( n − ϕ (0) x ] " n X t =1 ϕ ( r ∗ t ) − = [ A ( r ∗ , . . . , r ∗ n ; x ) + m ( n − ϕ (0) x ] " n X i =1 ϕ ( p i ) − which gives, for all x ∈ I , A ( p , . . . , p n ; x ) = A ( x ) " n X i =1 ϕ ( p i ) − − m ( n − ϕ (0) x (3.21)where A : R → R is defined as A ( y ) = " n X t =1 ϕ ( r ∗ t ) − − [ A ( r ∗ , . . . , r ∗ n ; y ) + m ( n − ϕ (0) y ] (3.22)for all y ∈ R . From (3.22), it is easy to verify that A : R → R is additive. Also,from (3.11) (with p i = r ∗ i , i = 1 , . . . , n ) and (3.22), it is easy to derive A (1) = m ϕ (0) . (3.23)From (3.12) and (3.21), it follows that n X i =1 ϕ ( p i q ) − ϕ ( q ) n X i =1 ϕ ( p i ) − m ( n − ϕ (0) q n X i =1 ϕ ( p i )= A ( q ) n X i =1 ϕ ( p i ) − A ( q ) − m ( n − ϕ (0) q − ϕ (0) n X i =1 ϕ ( p i ) + n ϕ (0)which, upon using (3.23), gives n X i =1 [ ϕ ( p i q ) + A ( p i q ) + m ( n − ϕ (0) p i q − ϕ (0)] − [ ϕ ( q ) + A ( q ) + m ( n − ϕ (0) q − ϕ (0)] × n X i =1 [ ϕ ( p i ) + A ( p i ) + m ( n − ϕ (0) p i − ϕ (0)]+ [ ϕ ( q ) + A ( q ) + m ( n − ϕ (0) q − ϕ (0)] n ( m − ϕ (0) = 0 . (3.24)Define a mapping B : R → R as B ( x ) = A ( x ) + m ( n − ϕ (0) x (3.25) for all x ∈ R . Then, B : R → R is additive. Moreover, from (3.23) and (3.25), itfollows that B (1) = mn ϕ (0) . (3.26)With the help of (3.25), equation (3.24) can be written in the form n X i =1 [ ϕ ( p i q ) + B ( p i q ) − ϕ (0)] − [ ϕ ( q ) + B ( q ) − ϕ (0)] × n X i =1 [ ϕ ( p i ) + B ( p i ) − ϕ (0)] + n ( m − ϕ (0) [ ϕ ( q ) + B ( q ) − ϕ (0)]= 0 . (3.27)Define a mapping M : I → R as M ( x ) = ϕ ( x ) + B ( x ) − ϕ (0) (3.28)for all x ∈ I . Notice that though B : R → R but, in (3.28), we are restricting itsuse only for all x ∈ I .From (3.28), it is easy to see that (1.2) follows as B (0) = 0. Also, from (3.26),(3.28) and the fact that ϕ (1) + ( n − ϕ (0) = 1, it follows that M (1) = 1 + n ( m − ϕ (0) . (3.29)Moreover, from (3.27) and (3.28), we get (for all q ∈ I ) n X i =1 M ( p i q ) − M ( q ) n X i =1 M ( p i ) + n ( m − ϕ (0) M ( q ) = 0 (3.30)which can be written in the form n X i =1 [ M ( p i q ) − M ( q ) M ( p i ) + n ( m − ϕ (0) M ( q ) p i ] = 0 . (3.31)By Result 1, there exists a mapping E : R × I → R , additive in the first variable,such that M ( pq ) − M ( p ) M ( q ) + n ( m − ϕ (0) M ( q ) p = E ( p, q ) − n E (1 , q ) (3.31a) sum form functional equation . . . 11 for all p ∈ I , q ∈ I . The substitution p = 0 in (3.31a) and the use of (1.2) gives E (1 , q ) = 0 for all q ∈ I . Consequently, (3.31a) reduces to the equation M ( pq ) − M ( p ) M ( q ) + n ( m − ϕ (0) M ( q ) p = E ( p, q ) (3.32)for all p ∈ I , q ∈ I .Now we prove that n ( m − ϕ (0) = 0 is not possible.If possible, suppose n ( m − ϕ (0) = 0. Then, (3.29) gives M (1) = 1. Putting q = 1 in (3.30), using (3.29) and the fact that M (1) − = 0, we get n X i =1 M ( p i ) = M (1)for all ( p , . . . , p n ) ∈ Γ n . By Result 1, there exists an additive mapping A : R → R such that M ( p ) = A ( p ) − n A (1) + 1 n M (1) (3.33)for all p ∈ I . The substitution p = 0, in (3.33), gives A (1) = M (1) as A (0) = 0and M (0) = 0. Hence M ( p ) = A ( p )for all p ∈ I . Thus M is additive on I . Now, from (3.20), (3.26), (3.28), (3.29)and the additivity of M on I , we have1 = n X t =1 ϕ ( r ∗ t ) = M (1) − B (1) + n ϕ (0)= 1 + n ( m − ϕ (0) − nm ϕ (0) + n ϕ (0) = 1a contradiction.So, the only possibility is that n ( m − ϕ (0) = 0. Since n > m > ϕ (0) = 0 and hence ϕ (1) = 1. From this and (3.29),(1.3) follows. Since ϕ (0) = 0, equation (3.32) reduces to the equation M ( pq ) − M ( p ) M ( q ) = E ( p, q ) (3.34) for all p ∈ I , q ∈ I . The left hand side of (3.34) is symmetric in p and q . Hence E ( p, q ) = E ( q, p ) for all p ∈ I , q ∈ I . Consequently, E is also additive on I in thesecond variable. We may assume that E ( p, · ) has been extended additively to thewhole of R .Let p ∈ I , q ∈ I , r ∈ I . From (3.34), we have E ( pq, r ) + M ( r ) E ( p, q ) = M ( pqr ) − M ( p ) M ( q ) M ( r )= E ( qr, p ) + M ( p ) E ( q, r ) . (3.35)Now we prove that E ( p, q ) = 0 for all p ∈ I , q ∈ I . If possible, suppose thereexists a p ∗ ∈ I and a q ∗ ∈ I such that E ( p ∗ , q ∗ ) = 0. Then, from (3.35) M ( r ) = [ E ( p ∗ , q ∗ )] − { E ( q ∗ r, p ∗ ) + M ( p ∗ ) E ( q ∗ , r ) − E ( p ∗ q ∗ , r ) } from which it follows that M is additive on I . Now, making use of (3.20), (3.26),(3.28), (1.3), the additivity of M and the fact that ϕ (0) = 0, we obtain1 = n X t =1 ϕ ( r ∗ t ) = M (1) − B (1) + n ϕ (0) = 1 − mn ϕ (0) + n ϕ (0) = 1a contradiction. Hence E ( p, q ) = 0 for all p ∈ I , q ∈ I . Now, (3.34) reduces tothe equation M ( pq ) = M ( p ) M ( q ) (3.36)for all p ∈ I , q ∈ I . From (3.36), (1.4) follows immediately for all p ∈ I , q ∈ I .Also, since ϕ (0) = 0, (3.28) reduces to (3.3) and (3.26) gives B (1) = 0. Thiscompletes the proof of Theorem 1. (cid:3)
4. The functional equation (1.10)
In this section, we prove:
Theorem 2.
Let n > , m > be fixed integers and f : I → R , g j : I → R , j = 1 , . . . , m be mappings which satisfy the functional equation (1.10) for all sum form functional equation . . . 13 ( p , . . . , p n ) ∈ Γ n and ( q , . . . , q m ) ∈ Γ m . Then, any general solution of (1.10) isof the form f ( p ) = b ( p ) , g j any arbitrary real-valued mapping (4.1) where b : R → R is an additive mapping with b (1) = 0 or f ( p ) = [ f (1) + ( n − f (0)] a ( p ) + f (0) g j ( p ) = a ( p ) + A ∗ ( p ) + g j (0) ) (4.2) for all j = 1 , . . . , m ; with a : R → R , A ∗ : R → R being additive maps and a (1) = 1 − n f (0) f (1) + ( n − f (0) , f (1) + ( n − f (0) = 0 A ∗ (1) = − m P j =1 g j (0) + nm f (0) f (1) + ( n − f (0) , f (1) + ( n − f (0) = 0 (4.3) or f ( p ) = f (1)[ M ( p ) − B ( p )] , f (1) = 0 g j ( p ) = M ( p ) − B ( p ) + A ∗ ( p ) + g j (0) ) (4.4) for all j = 1 , . . . , m ; with B : R → R , A ∗ : R → R being additive maps, B (1) = 0 , A ∗ (1) = − m P j =1 g j (0) and M : I → R a multiplicative function in the sense that itsatisfies (1.2) , (1.3) and (1.4) for all p ∈ I , q ∈ I . Proof.
Put p = 1, p = . . . = p n = 0 in (1.10). We obtain m X j =1 [ f ( q j ) + ( n − f (0)] = [ f (1) + ( n − f (0)] m X j =1 g j ( q j ) (4.5)for all ( q , . . . , q m ) ∈ Γ m .Case 1. f (1) + ( n − f (0) = 0 . (4.6)Then, (4.5) reduces to the equation m X j =1 f ( q j ) = − m ( n − f (0) (4.7) valid for all ( q , . . . , q m ) ∈ Γ m . By Result 1, there exists an additive mapping b : R → R such that f ( p ) = b ( p ) − m b (1) − ( n − f (0) (4.8)for all p ∈ I . The substitution p = 0, in (4.8), gives b (1) = − nm f (0) . (4.9)From (4.8) and (4.9), it follows that f ( p ) = b ( p ) + f (0) (4.10)for all p ∈ I . From (4.6), (4.9) and (4.10), using the fact that n > m > f (0) = 0 . (4.11)From (4.9) and (4.11), it follows that b (1) = 0 . (4.12)Also, (4.10) and (4.11) give f ( p ) = b ( p ) (4.13)for all p ∈ I . Also, from (1.10), (4.12), (4.13) and the additivity of b : R → R , itfollows that g j can be any arbitrary real-valued mapping. Thus, we have obtainedthe solution (4.1) in which b satisfies (4.12).Case 2. f (1) + ( n − f (0) = 0.In this case, (4.5) gives m X j =1 g j ( q j ) = [ f (1) + ( n − f (0)] − m X j =1 [ f ( q j ) + ( n − f (0)] (4.14) sum form functional equation . . . 15 which can be written in the form m X j =1 (cid:8) g j ( q j ) − [ f (1) + ( n − f (0)] − [ f ( q j ) + ( n − f (0)] (cid:9) = 0 . (4.15)This holds for all ( q , . . . , q m ) ∈ Γ m . By Result 2, there exists an additive mapping A ∗ : R → R and constants c j ( j = 1 , . . . , m ) such that g j ( p ) − [ f (1) + ( n − f (0)] − [ f ( p ) + ( n − f (0)] = A ∗ ( p ) + c j (4.16)with A ∗ (1) + m X j =1 c j = 0 . (4.17)The substitution p = 0, in (4.16), gives c j = g j (0) − [ f (1) + ( n − f (0)] − nf (0) (4.18)for j = 1 , . . . , m . From (4.17) and (4.18), we get A ∗ (1) as mentioned in (4.3).Also, from (4.16) and (4.18), g j ( p ) = [ f (1) + ( n − f (0)] − [ f ( p ) − f (0)] + A ∗ ( p ) + g j (0) (4.19)for j = 1 , . . . , m . Equation (4.19) tells us that if f is known, then thecorresponding form of g j ( p ), j = 1 , . . . , m , can be determined. To determine f ,we eliminate m P j =1 g j ( q j ) from equations (1.10) and (4.14). We obtain the equation n X i =1 m X j =1 f ( p i q j ) = [ f (1) + ( n − f (0)] − n X i =1 f ( p i ) m X j =1 f ( q j )+ [ f (1) + ( n − f (0)] − m ( n − f (0) n X i =1 f ( p i ) (4.20)valid for all ( p , . . . , p n ) ∈ Γ n and ( q , . . . , q m ) ∈ Γ m .Define a mapping ϕ : I → R as ϕ ( x ) = [ f (1) + ( n − f (0)] − f ( x ) (4.21) for all x ∈ I . Then (4.20) reduces to the functional (1.11) which also holds for all( p , . . . , p n ) ∈ Γ n and ( q , . . . , q m ) ∈ Γ m . Moreover, ϕ satisfies the condition ϕ (1) + ( n − ϕ (0) = 1 . (4.22)Also, from (4.21), f ( p ) = [ f (1) + ( n − f (0)] ϕ ( p ) (4.23)for all p ∈ I with f (1) + ( n − f (0) = 0 and ϕ (0) = f (0) f (1) + ( n − f (0) . (4.24)From, (4.19), (4.23), (4.24), (3.1) and (ii) in (3.2), the forms of f ( p ), g j ( p ) and a (1), as mentioned in (4.2) and (4.3), follow. Thus, we have obtained the solution(4.2), of (1.10), subject to a (1) and A ∗ (1) as mentioned in (4.3).The form of ϕ , given by (3.3), with B (1) = 0, is also acceptable as in thiscase, ϕ (0) = 0, ϕ (1) = 1 and hence ϕ (1) + ( n − ϕ (0) = 1. Now, from (4.24), f (0) = 0. The solution (4.4), of (1.10), follows from (4.23), (4.19), (3.3), (1.2),(1.3), (1.4) and the fact that f (0) = 0, B (1) = 0, A ∗ (1) = − m P j =1 g j (0). Thiscompletes the proof of Theorem 2. (cid:3)
5. The functional equation (1.8)
In this section, we prove:
Theorem 3.
Let n > , m > be fixed integers and h : I → R , k j : I → R , j = 1 , . . . , m be mappings which satisfy the functional equation (1.8) for all ( p , . . . , p n ) ∈ Γ n and ( q , . . . , q m ) ∈ Γ m and λ = 0 . Then, any general solution of (1.8) is of the form h ( p ) = 1 λ [ b ( p ) − p ] , k j any arbitrary real-valued mapping (5.1) sum form functional equation . . . 17 where b : R → R is an additive mapping with b (1) = 0 or h ( p ) = 1 λ (cid:8) [ λ ( h (1) + ( n − h (0)) + 1] a ( p ) + λ h (0) − p (cid:9) k j ( p ) = 1 λ (cid:8) a ( p ) + A ∗ ( p ) + λ k j (0) − p (cid:9) (5.2) for all j = 1 , . . . , m ; with a : R → R , A ∗ : R → R being additive maps and a (1) = 1 − nλ h (0) λ ( h (1) + ( n − h (0)) + 1 ,λ ( h (1) + ( n − h (0)) + 1 = 0 A ∗ (1) = − λ m P j =1 k j (0) + nmλ h (0) λ ( h (1) + ( n − h (0)) + 1 ,λ ( h (1) + ( n − h (0)) + 1 = 0 . (5.3) or h ( p ) = 1 λ (cid:8) [ λ h (1) + 1][ M ( p ) − B ( p )] − p (cid:9) , [ λ h (1) + 1] = 0 k j ( p ) = 1 λ (cid:8) M ( p ) − B ( p ) + A ∗ ( p ) + λ k j (0) − p (cid:9) (5.4) with B : R → R , A ∗ : R → R being additive maps such that B (1) = 0 , A ∗ (1) = − λ m X j =1 k j (0) (5.5) and M : I → R a multiplicative function in the sense that it satisfies (1.2) , (1.3) and (1.4) for all p ∈ I , q ∈ I . Proof.
Let us write (1.8) in the form n X i =1 m X j =1 [ λ h ( p i q j ) + p i q j ] = n X i =1 [ λ h ( p i ) + p i ] m X j =1 [ λ k j ( q j ) + q j ] . (5.6)Define the mappings f : I → R and g j : I → R , j = 1 , . . . , m (with λ = 0), asin (1.9), for all x ∈ I . Then, (5.6) reduces to the functional equation (1.10) whoserespective solutions are given by (4.1); (4.2) subject to the condition (4.3); and(4.4) subject to B (1) = 0, A ∗ (1) = − m P j =1 g j (0); in which b : R → R , a : R → R , A ∗ : R → R , B : R → R are all additive functions and M : [0 , → R is a multiplicative function. Now, making use of (1.9) along with (4.1); (4.2) subject to(4.3); and (4.4) subject to B (1) = 0 and A ∗ (1) = − m P j =1 g j (0); the required solutions(5.1); (5.2) subject to (5.3); and (5.4) subject to (5.5); follow respectively. (cid:3) REFERENCES [1]
M. Behara and P. Nath.
Additive and non-additive entropies of finite measurablepartitions,
Probability and Information Theory II , Lecture Notes in Math., 296, Berlin-Heidelberg-New York, 1973, 102–138.[2]
Z. Dar´oczy and L. Losonczi. ¨Uber die Erweiterung der auf einer Punktmenge additivenFunktionen,
Publ. Math. (Debrecen), 14 (1967), 239–245.[3]
J. Havrda and F. Charvat.
Quantification method of classification process, concept ofstructural α -entropy, Kybernetika (Prague), 3 (1967), 30–35.[4]
PL. Kannappan and P.K. Sahoo.
On the general solution of a functional equationconnected to sum from information measures on open domain-VI,
Radovi Matematicki , 8(1992), 231–239.[5]
L. Losonczi.
Functional equations of sum form,
Publ. Math. (Debrecen), 32 (1985), 57–71.[6]
L. Losonczi and Gy. Maksa.
On some functional equations of the information theory,
Acta Math. Acad. Sci. Hung. , 39 (1982), 73–82.
Prem Nath
Department of MathematicsUniversity of DelhiDelhi 110 007India
E-mail: [email protected]
Dhiraj Kumar Singh