Ratio Mathematica Vol. 32, 2017, pp. 21–35 ISSN: 1592-7415 eISSN: 2282-8214 On a Functional Equation Related to Information Theory P. Nath1, D.K. Singh2∗ 1Department of Mathematics, University of Delhi, Delhi - 110007, India pnathmaths@gmail.com 2Department of Mathematics, Zakir Husain Delhi College (University of Delhi) Jawaharlal Nehru Marg, Delhi - 110002, India dhiraj426@rediffmail.com, dksingh@zh.du.ac.in Received on: 28-05-2017. Accepted on: 15-06-2017. Published on: 30-06-2017 doi: 10.23755/rm.v32i0.332 c©P. Nath and D.K. Singh Abstract The main aim of this paper is to obtain the general solutions of the functional equation (1.3) without imposing any regularity condition on the mappings appearing in it. To do so, the general solutions of the functional equation (1.5), without imposing any regularity condition on the mappings appearing in it are needed. To meet this need, the general solutions of the functional equation (1.6) without imposing any regularity condition on a mapping appearing have to be investigated. One solution of (1.3) is useful in information theory. Thus, indeed, is the reason to consider (1.3). Keywords: Functional equation; additive mapping; multiplicative mapping. 2010 AMS subject classifications: 39B22, 39B52, 94A15, 94A17. ∗Corresponding author 21 P. Nath and D.K. Singh 1 Introduction For n = 1, 2, . . ., let Γn = {(p1, . . . ,pn) : 0 6 pi 6 1, i = 1, . . . ,n; n∑ i=1 pi = 1}, denote the set of all discrete n-component complete probability distributions with non-negative elements. Let I = {x ∈ R : 0 6 x 6 1} = [0, 1], R denoting the set of all real numbers. The axiomatic characterization of the non-additive entropy of degree α (see [2]) defined as Hαn (p1, . . . ,pn) = (2 1−α − 1)−1 ( n∑ i=1 pαi − 1 ) , α 6= 1 leads to the study of the functional equation n∑ i=1 m∑ j=1 f(piqj) = n∑ i=1 f(pi) + m∑ j=1 f(qj) + λ n∑ i=1 f(pi) m∑ j=1 f(qj) (1.1) in which f : I → R is an unknown mapping, (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm, λ 6= 0, λ ∈ R and n, m being positive integers. By a general solution of a functional equation, we mean a solution obtained without imposing any condition such as differentiability, continuity, continuity at a point, measurability, boundedness, monotonicity etc on a(the) mapping(s) appearing in the functional equation under consideration. The general solutions of (1.1), for fixed integers n > 3, m > 3 and (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm have been obtained in [5]. Losonczi [4] considered the functional equation n∑ i=1 m∑ j=1 fij(piqj) = n∑ i=1 hi(pi) + m∑ j=1 kj(qj) + λ n∑ i=1 hi(pi) m∑ j=1 kj(qj) (1.2) with (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm, λ 6= 0, λ ∈ R, fij : I → R, hi : I → R, kj : I → R, i = 1, . . . ,n; j = 1, . . . ,m, as unknown mappings. He found the measurable (in the sense of Lebesgue) solutions of (1.2) for all (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm by taking n > 3, m > 3 as fixed integers, in Theorem 6 on p-69 in [4]. For the last more than two decades, the general solutions of (1.2) for all (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm, n > 3, m > 3 being fixed integers, are still not known so far. The main aim of this paper is to obtain the general solutions of the functional equation n∑ i=1 m∑ j=1 h(piqj) = n∑ i=1 h(pi) + m∑ j=1 kj(qj) + λ n∑ i=1 h(pi) m∑ j=1 kj(qj) (1.3) 22 On a Functional Equation Related to Information Theory which contains m + 1 unknown real-valued mappings h and kj (j = 1, . . . ,m), each defined on I = [0, 1]; λ ∈ R, λ 6= 0 and n > 3, m > 3 being fixed integers. These general solutions have been obtained without making use of the difference operator Dri suggested on p-58 by Losonczi [4]. This paper is an improved version of the manuscript [9]. Nath and Singh [8] have also obtained the general solutions of n∑ i=1 m∑ j=1 F(piqj) = n∑ i=1 G(pi) + m∑ j=1 Hj(qj) + λ n∑ i=1 G(pi) m∑ j=1 Hj(qj) with F : I → R, G : I → R, Hj : I → R, j = 1, . . . ,m; λ 6= 0, (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm, n > 3, m > 3 being fixed integers. The functional equation (1.3) is a special case of (1.2). A particular case of (1.3) is the following n∑ i=1 m∑ j=1 h(piqj) = n∑ i=1 h(pi) + m∑ j=1 k(qj) + λ n∑ i=1 h(pi) m∑ j=1 k(qj) in which h : I → R, k : I → R and (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm. Nath and Singh [7] have obtained its general solution(s) for fixed integers n > 3, m > 3. Let us define the mappings f : I → R and gj : I → R, j = 1, . . . ,m as f(x) = x + λh(x); gj(x) = x + λkj(x) (1.4) for all x ∈ I. Then (1.3) reduces to the Pexider type functional equation n∑ i=1 m∑ j=1 f(piqj) = n∑ i=1 f(pi) m∑ j=1 gj(qj) . (1.5) We would like to mention that Kannappan and Sahoo [3] have obtained the general solutions of (1.3) and (1.5) on an open domain. In our case, the process of finding the general solutions of (1.5), for fixed integers n > 3, m > 3, needs determining the general solutions of the functional equation n∑ i=1 m∑ j=1 ϕ(piqj) = n∑ i=1 ϕ(pi) m∑ j=1 ϕ(qj) + m(n− 1) ϕ(0) n∑ i=1 ϕ(pi) (1.6) where ϕ : I → R and n > 3, m > 3 are fixed integers. This task has been accomplished in section 3. The corresponding general solutions of (1.5) and (1.3) have been investigated in sections 4 and 5 respectively. At the end of section 5, we have analysed the importance of the solutions of functional equation (1.3) from information-theoretic point of view. Section 2 contains some known definitions and results needed for the subsequent development of this paper. 23 P. Nath and D.K. Singh 2 Some preliminary results In this section, we mention some known definitions and results. A mapping a : I → R is said to be additive on I or on the unit triangle ∆ = {(x,y) : 0 6 x 6 1, 0 6 y 6 1, 0 6 x + y 6 1} if it satisfies the equation a(x + y) = a(x) + a(y) for all (x,y) ∈ ∆. A mapping A : R → R is said to be additive on R if it satisfies the equation A(x + y) = A(x) + A(y) for all x ∈ R, y ∈ R. It is known [1] that if a mapping a : I → R is additive on I, then it has a unique additive extension A : R → R in the sense that A is additive on R and A(x) = a(x) for all x ∈ I. A mapping M : I → R is said to be multiplicative if M(pq) = M(p) M(q) holds for all p ∈ I, q ∈ I. Result 2.1. [5] Let n > 3 be a fixed integer and c be a given constant. Suppose that a mapping ψ : I → R satisfies the functional equation n∑ i=1 ψ(pi) = c for all (p1, . . . ,pn) ∈ Γn. Then there exists an additive mapping b : R → R such that ψ(p) = b(p) − 1 n b(1) + c n for all p ∈ I. Result 2.2. [4] Let d be a given real constant and ψj : I → R, j = 1, . . . ,m, be mappings which satisfy the functional equation m∑ j=1 ψj(qj) = d for all (q1, . . . ,qm) ∈ Γm, m > 3 being a fixed integer. Then there exists an additive mapping a : R → R and real constants cj (j = 1, . . . ,m) such that ψj(p) = a(p) + cj for all p ∈ I with a(1) + m∑ j=1 cj = d. 3 The functional equation (1.6) In this section, we prove: Theorem 3.1. Let n > 3, m > 3 be fixed integers and ϕ : I → R be a map- ping which satisfies the functional equation (1.6) for all (p1, . . . ,pn) ∈ Γn and (q1, . . . ,qm) ∈ Γm. Then ϕ is of the form ϕ(p) = a(p) + ϕ(0) (3.1) 24 On a Functional Equation Related to Information Theory where a : R → R is an additive mapping with   (i) a(1) = −nmϕ(0) if ϕ(1) + (n− 1) ϕ(0) 6= 1 or (ii) a(1) = 1 −nϕ(0) if ϕ(1) + (n− 1) 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 a multiplicative mapping which is not additive and M(0) = 0, M(1) = 1. Proof. Let us put p1 = 1, p2 = . . . = pn = 0 in (1.6). We obtain [ϕ(1) + (n− 1) ϕ(0) − 1] [ m∑ j=1 ϕ(qj) + m(n− 1) ϕ(0) ] = 0 (3.4) for all (q1, . . . ,qm) ∈ Γm. We divide our discussion into two cases. Case 1. ϕ(1) + (n− 1) ϕ(0) 6= 1. In this case, (3.4) reduces to m∑ j=1 ϕ(qj) = −m(n−1) ϕ(0) for all (q1, . . . ,qm) ∈ Γm. By Result 2.1, there exists an additive mapping a : R → R such that ϕ is of the form (3.1) with a(1) as in (3.2)(i). Thus, we have obtained the solution (3.1) satisfying (i) in (3.2). Case 2. ϕ(1) + (n− 1) ϕ(0) − 1 = 0. Let us write (1.6) in the form m∑ j=1 { n∑ i=1 ϕ(piqj) −ϕ(qj) n∑ i=1 ϕ(pi) −m(n− 1)ϕ(0)qj n∑ i=1 ϕ(pi) } = 0 . 25 P. Nath and D.K. Singh By Result 2.1, there exists a mapping A1 : Γn × R → R, additive in the second variable, such that n∑ i=1 ϕ(piq) −ϕ(q) n∑ i=1 ϕ(pi) −m(n− 1) ϕ(0) q n∑ i=1 ϕ(pi) (3.5) = A1(p1, . . . ,pn; q) −ϕ(0) n∑ i=1 ϕ(pi) + nϕ(0) valid for all (p1, . . . ,pn) ∈ Γn and q ∈ I with A1(p1, . . . ,pn; 1) = mϕ(0) [ n∑ i=1 ϕ(pi) −n ] . (3.6) Let x ∈ I and (r1, . . . ,rn) ∈ Γn. Putting successively q = xrt, t = 1, . . . ,n in (3.5), adding the resulting n equations so obtained and then substituting the value of n∑ t=1 ϕ(xrt) calculated from (3.5), we get the equation n∑ i=1 n∑ t=1 ϕ(xpirt) − [ϕ(x) + m(n− 1) ϕ(0) x−ϕ(0)] (3.7) × n∑ i=1 ϕ(pi) n∑ t=1 ϕ(rt) −n2 ϕ(0) = A1(p1, . . . ,pn; x) + m(n− 1) ϕ(0) x n∑ i=1 ϕ(pi) + A1(r1, . . . ,rn; x) n∑ i=1 ϕ(pi) . The symmetry of the left hand side of (3.7), in pi and rt, i = 1, . . . ,n; t = 1, . . . ,n gives rise to the equation [A1(p1, . . . ,pn; x) + m(n− 1) ϕ(0) x] [ n∑ t=1 ϕ(rt) − 1 ] (3.8) = [A1(r1, . . . ,rn; x) + m(n− 1) ϕ(0) x] [ n∑ i=1 ϕ(pi) − 1 ] . Case 2.1. n∑ t=1 ϕ(rt) − 1 vanishes identically on Γn. 26 On a Functional Equation Related to Information Theory In this case, by Result 2.1, there exists an additive mapping a : R → R such that ϕ is of the form (3.1) but now a(1) is as in (3.2)(ii). Case 2.2. n∑ t=1 ϕ(rt) − 1 does not vanish identically on Γn. Then, there exists a probability distribution (r∗1, . . . ,r ∗ n) ∈ Γn such that[ n∑ t=1 ϕ(r∗t ) − 1 ] 6= 0 . (3.9) Setting rt = r∗t , t = 1, . . . ,n in (3.8) and using (3.9), we obtain the equation A1(p1, . . . ,pn; x) = B(x) [ n∑ i=1 ϕ(pi) − 1 ] −m(n− 1) ϕ(0) x (3.10) where B : R → R is defined as B(x) = [ n∑ t=1 ϕ(r∗t ) − 1 ]−1 [A1(r ∗ 1, . . . ,r ∗ n; x) + m(n − 1) ϕ(0) x] for all x ∈ R. It can be easily verified that B : R → R is an additive mapping with B(1) = mϕ(0). From (3.5), (3.10), B(1) = mϕ(0) and the additivity of B : R → R, it follows that n∑ i=1 [M(piq) −M(q)M(pi) + n(m− 1) ϕ(0) M(q) pi] = 0 (3.11) where M : I → R is defined as M(x) = ϕ(x) + B(x) + m(n− 1)ϕ(0)x−ϕ(0) (3.12) for all x ∈ I. From (3.12), it is easy to see that M(0) = 0 as B(0) = 0. Applying Result 2.1 on (3.11), there exists a mapping E : R × I → R, additive in the first variable such that M(pq) −M(p)M(q) + n(m− 1) ϕ(0) M(q) p = E(p,q) − 1 n E(1,q) (3.13) for all p ∈ I, q ∈ I. The substitution p = 0 in (3.13) and the use of M(0) = 0 gives E(1,q) = 0 for all q ∈ I. Consequently, M(pq) −M(p)M(q) + n(m− 1) ϕ(0) M(q) p = E(p,q) (3.14) 27 P. Nath and D.K. Singh for all p ∈ I, q ∈ I. Since E(1,q) = 0, therefore E(1, 1) = 0. Now, putting p = q = 1 in equation (3.14), we obtain M(1)[1 −M(1) + n(m− 1)ϕ(0)] = 0 . (3.14a) We prove that M(1) 6= 0. To the contrary, suppose that M(1) = 0. Putting q = 1 in (3.14) and using M(1) = 0, we get M(p) = E(p, 1) for all p ∈ I. So, M is additive on I. Also, if we put x = 1 in (3.12), use M(1) = 0 and ϕ(1) + (n− 1)ϕ(0) = 1, we obtain n(m− 1)ϕ(0) = −1. Now from (3.9), (3.12) and the additivity of M on I, we have 1 6= n∑ t=1 ϕ(r∗t ) = 1 a contradiction. Hence M(1) 6= 0. Now, from (3.14a), it follows that M(1) − 1 = n(m− 1)ϕ(0) . (3.15) Our next task is to prove that M : I → R, defined by (3.12), is not additive. To the contrary, suppose that M is additive. Now from (3.9), (3.12), the additivity of M on I and (3.15), we have 1 6= n∑ t=1 ϕ(r∗t ) = M(1) −n(m− 1)ϕ(0) = 1 a contradiction. Hence M : I → R is not additive. Now we prove that, indeed, M(1)−1 = 0. If possible, suppose [M(1)−1] 6= 0. Putting q = 1 in (3.14) and using (3.15), we obtain [M(1)p−M(p)] = [M(1) − 1]−1E(p, 1) for all p ∈ I. Since p 7−→ E(p, 1) is additive on I, it follows that p 7−→ M(1)p−M(p) must also be additive on I. But p 7−→ M(1)p is additive on I. Hence M is additive on I contradicting the fact that M is not additive. Hence M(1) − 1 = 0, that is, M(1) = 1. 28 On a Functional Equation Related to Information Theory Now, from (3.15), it follows that ϕ(0) = 0. Consequently, equation (3.14) reduces to the equation M(pq) −M(p) M(q) = E(p,q) (3.16) for all p ∈ I, q ∈ I and (3.12) reduces to (3.3) for all p ∈ I with B(1) = 0. The left hand side of (3.16) 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 the second variable. We may assume that E(p, ·) has been extended additively to the whole of R. Let p ∈ I, q ∈ I, r ∈ I. From (3.16), we have E(pq,r) + M(r) E(p,q) = M(pqr) −M(p) M(q) M(r) (3.17) = E(qr,p) + M(p) E(q,r) . We prove that E(p,q) = 0 for all p ∈ I, q ∈ I. If possible, suppose there exists a p∗ ∈ I and a q∗ ∈ I such that E(p∗,q∗) 6= 0. Then, (3.17) gives M(r) = [E(p∗,q∗)]−1{E(q∗r,p∗) + M(p∗)E(q∗,r) −E(p∗q∗,r)} from which it follows that M is additive on I contradicting the fact that M is not additive. Hence E(p,q) = 0 for all p ∈ I, q ∈ I. Now, from (3.16), it follows that M(pq) = M(p) M(q) for all p ∈ I, q ∈ I. Thus, M : I → R is a multiplicative mapping which is not additive and M(0) = 0, M(1) = 1. 4 The functional equation (1.5) In this section, we prove: Theorem 4.1. Let n > 3, m > 3 be fixed integers and f : I → R, gj : I → R, j = 1, . . . ,m be mappings which satisfy the functional equation (1.5) for all 29 P. Nath and D.K. Singh (p1, . . . ,pn) ∈ Γn and (q1, . . . ,qm) ∈ Γm. Then, any general solution of (1.5), for all p ∈ I, is of the form{ f(p) = b(p) gj any arbitrary real-valued mapping (4.1) or   f(p) = [f(1) + (n− 1) f(0)] a(p) + f(0), [f(1) + (n− 1) f(0)] 6= 0 gj(p) = a(p) + A ∗(p) + gj(0) (4.2) or { f(p) = f(1)[M(p) −B(p)] , f(1) 6= 0 gj(p) = M(p) −B(p) + A∗(p) + gj(0) (4.3) where b : R → R, a : R → R, A∗ : R → R, B : R → R are additive mappings with   (i) b(1) = 0 (ii) B(1) = 0 (iii) a(1) = 1 −nf(0)[f(1) + (n− 1)f(0)]−1 (iv) A∗(1) = − m∑ j=1 gj(0) + nmf(0)[f(1) + (n− 1)f(0)]−1 (4.4) and M : I → R is a multiplicative mapping which is not additive and M(0) = 0, M(1) = 1. Proof. Put p1 = 1, p2 = . . . = pn = 0 in (1.5). We obtain m∑ j=1 [f(qj) + (n− 1)f(0)] = [f(1) + (n− 1) f(0)] m∑ j=1 gj(qj) (4.5) for all (q1, . . . ,qm) ∈ Γm. Case 1. f(1) + (n− 1) f(0) = 0 . Then, (4.5) reduces to the equation m∑ j=1 f(qj) = −m(n−1) f(0). Put q1 = 1, q2 = . . . = qm = 0 in this equation and using the fact f(1) + (n−1) f(0) = 0, we have 30 On a Functional Equation Related to Information Theory f(0) = 0 = f(1). Hence m∑ j=1 f(qj) = 0. By Result 2.1, there exists an additive mapping b : R → R such that f(p) = b(p) with b(1) = 0. Consequently, for all (p1, . . . ,pn) ∈ Γn, (q1, . . . ,qm) ∈ Γm, it is easy to verify that n∑ i=1 m∑ j=1 f(piqj) = n∑ i=1 f(pi) = b(1) = 0. Now, from (1.5), it follows that gj (j = 1, . . . ,m) are, indeed, arbitrary real-valued mappings. Thus, we have obtained the solution (4.1) of (1.5) where b(1) is given by (4.4)(i). Case 2. f(1) + (n− 1) f(0) 6= 0. In this case, (4.5) can be written in the form m∑ j=1 { gj(qj) − [f(1) + (n− 1) f(0)]−1[f(qj) + (n− 1) f(0)] } = 0 . (4.6) By Result 2.2, there exists an additive mapping A∗ : R → R such that gj(p) = [f(1) + (n− 1) f(0)]−1[f(p) −f(0)] + A∗(p) + gj(0) (4.7) for j = 1, . . . ,m with A∗(1) given by (iv) in (4.4). The elimination of m∑ j=1 gj(qj) from equations (1.5) and (4.6) gives the equation n∑ i=1 m∑ j=1 f(piqj) = [f(1) + (n− 1) f(0)]−1 n∑ i=1 f(pi) m∑ j=1 f(qj) (4.8) + [f(1) + (n− 1) f(0)]−1m(n− 1) f(0) n∑ i=1 f(pi) valid for all (p1, . . . ,pn) ∈ Γn and (q1, . . . ,qm) ∈ Γm. Define a mapping ϕ : I → R as ϕ(x) = [f(1) + (n− 1) f(0)]−1 f(x) (4.9) for all x ∈ I. Then (4.8) reduces to the functional equation (1.6) with ϕ(1) + (n− 1) ϕ(0) = 1. So, we need to consider only those solutions of (1.6) which satisfy the requirement ϕ(1) + (n− 1) ϕ(0) = 1. 31 P. Nath and D.K. Singh The solutions (3.1) (with (3.2)(ii)) and (3.3) of (1.6) satisfy the condition ϕ(1) + (n − 1) ϕ(0) = 1. Making use of (4.9), (4.7), (3.1) (with (3.2)(ii)) and (3.3), the solutions (4.2) and (4.3) can be obtained in which B(1), a(1) and A∗(1) are given respectively by (ii), (iii) and (iv) in (4.4). 5 The functional equation (1.3) In this section, we prove: Theorem 5.1. Let n > 3, m > 3 be fixed integers and h : I → R, kj : I → R, j = 1, . . . ,m be mappings which satisfy the functional equation (1.3) for all (p1, . . . ,pn) ∈ Γn and (q1, . . . ,qm) ∈ Γm and λ 6= 0. Then, any general solution of (1.3), for all p ∈ I, is of the form   h(p) = 1 λ [b(p) −p] kj any arbitrary real-valued mapping (5.1) or   h(p) = 1 λ { [λ(h(1) + (n− 1) h(0)) + 1] a(p) −p } + h(0), [λ(h(1) + (n− 1) h(0)) + 1] 6= 0 kj(p) = 1 λ { a(p) + A∗(p) −p } + kj(0) (5.2) or   h(p) = 1 λ { [λh(1) + 1][M(p) −B(p)] −p } , [λh(1) + 1] 6= 0 kj(p) = 1 λ { M(p) −B(p) + A∗(p) −p } + kj(0) (5.3) where b : R → R, a : R → R, A∗ : R → R, B : R → R are additive mappings 32 On a Functional Equation Related to Information Theory with  (i) b(1) = 0 (ii) B(1) = 0 (iii) a(1) = 1 −λnh(0)[λ(h(1) + (n− 1)h(0)) + 1]−1 (iv) A∗(1) = −λ m∑ j=1 kj(0)+λnmh(0)[λ(h(1) +(n− 1)h(0))+ 1]−1 (5.4) and M : I → R is a multiplicative mapping which is not additive and M(0) = 0, M(1) = 1. Proof. From (1.4) and the solutions of the functional equation (1.5) i.e., (4.1), (4.2), (4.3) with (4.4); we obtain respectively the solutions (5.1), (5.2), (5.3) with (5.4); of the functional equation (1.3). The details are omitted. Remarks. The object of this remark is to point out the importance of various solutions of Theorem 5.1 from information-theoretic point of view. 1. The summand n∑ i=1 h(pi) of the mapping h appearing in (5.1) is independent of the probabilities p1, . . . ,pn. The solution (5.1) may be of some importance in information theory provided kj is chosen as a suitable mapping of probability p, p ∈ I. 2. In solution (5.2), the summands n∑ i=1 h(pi) and m∑ j=1 kj(qj) are independent of the probabilities p1, . . . ,pn and q1, . . . ,qm respectively. So, this solution does not seem to be of any relevance in information theory. 3. In solution (5.3) n∑ i=1 h(pi) = 1 λ { β1 n∑ i=1 M(pi) − 1 } 33 P. Nath and D.K. Singh and m∑ j=1 kj(qj) = 1 λ { m∑ j=1 M(qj) − 1 } + β2 where β1 = λh(1) + 1 β2 = nmh(0)[λ(h(1) + (n− 1)h(0)) + 1]−1 . If β1 = 1 and β2 = 0, then n∑ i=1 h(pi) = L λ n(p1, . . . ,pn) and m∑ j=1 kj(qj) = L λ m(q1, . . . ,qm) where (see Nath and Singh [6]) Lλt (x1, . . . ,xt) = 1 λ [ t∑ i=1 M(xi) − 1 ] . (5.5) The non-additive measure of entropy Hαt (x1, . . . ,xt) = (2 1−α−1)−1( t∑ i=1 xαi −1), α 6= 1, is a particular case of (5.5) when λ = 21−α − 1, α > 0, α 6= 1 and M : I → R is of the form M(p) = pα, p ∈ I, α 6= 1, α > 0, 0α := 0, 1α := 1. References [1] Z. Daróczy and L. Losonczi, Über die Erweiterung der auf einer Punkt- menge additiven Funktionen, Publ. Math. (Debrecen) 14 (1967), 239–245. [2] J. Havrda and F. Charvát, Quantification method of classification process, concept of structural α-entropy, Kybernetika (Prague) 3 (1967), 30–35. [3] PL. Kannappan and P.K. Sahoo, On the general solution of a functional equation connected to sum form information measures on open domain-VI, Radovi Matematicki 8 (1992), 231–239. [4] L. Losonczi, Functional equations of sum form, Publ. Math. (Debrecen) 32 (1985), 57–71. 34 On a Functional Equation Related to Information Theory [5] L. Losonczi and Gy. Maksa, On some functional equations of the information theory, Acta Math. Acad. Sci. Hung. 39 (1982), 73–82. [6] P. Nath and D.K. Singh, On a multiplicative type sum form functional equa- tion and its role in information theory, Applications of Mathematics 51(5) (2006), 495–516. [7] P. Nath and D.K. Singh, On a sum form functional equation and its role in information theory, in Proc.: 8th National Conf. ISITA on “Information Technology: Setting Trends in Modern Era”, (2008), 88–94. [8] P. Nath and D.K. Singh, On a functional equation containing an indexed family of unknown mappings, Functional Equations in Mathematical Analy- sis (Edited by Themistocles Rassias and Janusz Brzdek), Springer 52 (2011), 671–687. [9] P. Nath and D.K. Singh, A sum form functional equation on a closed domain and its role in information theory, arXiv:1508.05910v1 (2015), 1–18. 35