On the sum form functional equation related to diversity index

1 Introduction

The term Information generally presents a descriptive statement whose interpretation relies on the specific context in which it is considered. Although commonly used, the necessity to quantify it was initially addressed by Hartley [1]. Building upon his work, Shannon [2] emphasized scientifically conceptualizing information. In his conception, he envisioned the communication system as one where the channel was characterized statistically by giving a suitable probability distribution over the set of all possible outcomes that can be attained corresponding to each permissible input. Within this framework, information emerged as a physical parameter widely recognized as entropy. Over the years, this phenomenon has been studied comprehensively, resulting in the emergence of its multiple facets, different approaches, and applications. Entropies emerging from the information theory first entered the corpus of sum form functional equations with the seminal article of Chaundy and McLeod [3]. These equations are the core of various branches of pure and applied mathematics. They have evolved with a multidisciplinary approach over the years since they have been used to characterize various entropies, which are further appearing in branches of sociology [4,5]; ecology [6]; economics [7]; geometry [8]; biology [9], and many more. Following a similar integrative aspect of functional equations, we have discussed the general solution and stability of the functional equation containing four unknown mappings in this article. Further, we analyze its solutions given information theory and diversity index.

Let N denote the set of natural numbers; R denote the set of real numbers; I denote the closed unit interval [ 0 , 1 ] , i.e., I = [ 0 , 1 ] = { x ∈ R : 0 ≤ x ≤ 1 } . For n ∈ N , let

denote the set of all n -component discrete probability distributions.

A mapping a : I → R is said to be additive on I or on the unit triangle △ = { ( x , y ) : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , 0 ≤ x + y ≤ 1 } if it satisfies a ( x + y ) = a ( x ) + a ( y ) for all ( x , y ) ∈ △ . Analogously, a mapping A : R → R is said to be additive on R if it satisfies A ( x + y ) = A ( x ) + A ( y ) for all x ∈ R , y ∈ R . Daróczy and Losonczi [10] established an interesting relation between these aforementioned additive mappings and proved that there exists a unique additive extension of the additive mapping a : I → R to the set of real numbers.

A mapping ℓ : I → R is said to be logarithmic on I if it satisfies ℓ ( 0 ) = 0 and ℓ ( x y ) = ℓ ( x ) + ℓ ( y ) for all x ∈ ] 0 , 1 ] , y ∈ ] 0 , 1 ] .

A mapping m : I → R is said to be multiplicative on I if it satisfies m ( 0 ) = 0 , m ( 1 ) = 1 and m ( x y ) = m ( x ) m ( y ) for all x ∈ ] 0 , 1 [ , y ∈ ] 0 , 1 [ .

For a given probability distribution ( p 1 , … , p n ) ∈ Γ n , consider a real valued discrete random variable Z n taking n distinct values z 1 , … , z n with respective probabilities p 1 , … , p n where

such that α is a fixed positive real power with α ≠ 1 , 0 α ≔ 0 , 1 α ≔ 1 and 0 α − 1 log 2 0 ≔ 0 . Then

For r ∈ N , define the mapping ϕ ( α , r ) : I → R as follows:

for all p ∈ I . Furthermore for r ∈ N , we obtain

Clearly from equations (1.1) and (1.2), we obtain

This implies the mean of random variable Z n admits a sum representation. Moreover, the mapping ϕ ( α , 1 ) is called its generating function [11].

Now from equation (1.2), it is easy to verify that mapping ϕ ( α , 1 ) satisfies the functional equation

with ϕ ( α , 1 ) ( 0 ) = 0 and ϕ ( α , 1 ) ( 1 ) = 0 . Consequently for all ( p 1 , … , p n ) ∈ Γ n , ( q 1 , … , q m ) ∈ Γ m , the functional equation

holds for any pair of positive integers ( n , m ) , where ϕ ( α , 1 ) ( 0 ) = 0 and ϕ ( α , 1 ) ( 1 ) = 0 . Interestingly, we observe that (1.5) is a particular case of a famous functional equation (with f in place of ϕ ( α , 1 ) ) given by Behara and Nath [12], that is,

where f : I → R ; ( p 1 , … , p n ) ∈ Γ n , ( q 1 , … , q m ) ∈ Γ m ; α and β are fixed positive real powers satisfying

Behara and Nath [12] were first to study (1.6), which was useful in characterizing entropies of type ( α , β ) . For n ∈ N , ( p 1 , … , p n ) ∈ Γ n , the entropies H n ( α , β ) : Γ n → R of type ( α , β ) are defined as follows:

where 0 β log 2 0 ≔ 0 ; α and β are fixed positive real powers satisfying (1.7). Clearly, the functional equation (1.5) is useful in characterizing entropies of type ( α , β ) and also related to statistics (follows from (1.4)).

This methodology poses a significant question that:

Do we obtain another functional equation which is useful to characterize entropies of type ( α , β ) for different values of r in ϕ ( α . r ) , and also is it related to the random variable Z n ?

So considering r = 1 , 2 in (1.2), we see that mappings ϕ ( α , 1 ) and ϕ ( α , 2 ) satisfies the functional equation

with

Thus, for all ( p 1 , … , p n ) ∈ Γ n , ( q 1 , … , q m ) ∈ Γ m and for any pair of positive integers ( n , m ) , the functional equation

holds with (1.9) (which follows from (1.3)). Clearly, the aforementioned sum form of the functional equation is useful in characterizing entropies of type ( α , β ) (for α = β ) and is also related to the random variable Z n (as last term on the right-hand side is related to Z n ). Thus, it seems interesting to consider its Pexiderized form

where f : I → R , g : I → R , h : I → R , k ¯ : I → R are unknown mappings; ( p 1 , … , p n ) ∈ Γ n , ( q 1 , … , q m ) ∈ Γ m ; α and β are fixed positive real powers satisfying (1.7); 0 ≠ d is a real constant and boundary conditions (following from (1.9))

holds.

Since d ≠ 0 , we may define a mapping k : I → R as k ( x ) = d k ¯ ( x ) for all x ∈ I in equation (1.10) and obtain

where f : I → R , g : I → R , h : I → R , k : I → R are unknown mappings satisfying boundary conditions (1.11); ( p 1 , … , p n ) ∈ Γ n , ( q 1 , … , q m ) ∈ Γ m ; α and β are fixed positive real powers fulfilling (1.7).

Consequently, we have arrived at a new functional equation (A) related to statistics and useful in characterizing entropies of type ( α , β ) but seem to have missed the attention so far. Thus, this functional equation connects two branches, i.e., mathematical statistics and information theory, providing us with an important reason to address it.

Our motive in this article is to obtain the general solutions of the functional equation (A). Further, once the general solution of any functional equation is obtained, its stability is recommended to be examined. We would prefer references [13–15] for the readers to be familiar with the stability problem for the functional equations and the sum form functional equations.

To discuss the stability of functional equation (A) for the fixed integers n ≥ 3 , m ≥ 3 , we replace it by an inequality and consider its perturbation as follows:

for all ( p 1 , … , p n ) ∈ Γ n , ( q 1 , … , q m ) ∈ Γ m ; n ≥ 3 , m ≥ 3 be fixed integers and ε ∈ R is a fixed real constant.

This article is structured as follows:

In Section 1, we have briefly mentioned the notations and definitions and then explicated the relation between functional equations characterizing entropies and statistics, which helped us to arrive at functional equation (A). Section 2 presents a few previous results that will be used in subsequent sections of the article. In Section 3, we obtain general solutions of the functional equation (A) for n ≥ 3 , m ≥ 3 being fixed integers. In Section 4, we examine the stability of (A) for n ≥ 3 , m ≥ 3 being fixed integers. Finally, in Section 5, we discuss the significance of the general solutions of (A) obtained in Section 3, with reference to the information theory and the diversity index.

2 Auxiliary results

In this section, we state some known previous results, which will be used in the upcoming sections.

3 The general solution of functional equation (A)

The functional equation (A) has eight boundary conditions given by (1.11), eliminating these one by one as many as we could; here, we present the main result of this section as follows: