# New Estimates for Csiszár Divergence and Zipf–Mandelbrot Entropy via Jensen–Mercer’s Inequality

Authors: Muhammad Adil Khan; Zakir Husain; Yu-Ming Chu
Journal: Complexity
 (2020)

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## Abstract

Jensen’s inequality is one of the fundamental inequalities which has several applications in almost every field of science. In 2003, Mercer gave a variant of Jensen’s inequality which is known as Jensen–Mercer’s inequality. The purpose of this article is to propose new bounds for Csiszár and related divergences by means of Jensen–Mercer’s inequality. Also, we investigate several new bounds for Zipf–Mandelbrot entropy. The idea of this article may further stimulate research on information theory with the help of Jensen–Mercer’s inequality.


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## Body

## 1. Introduction

In theory of inequalities, convex functions play an important role. The definition of convex function [1] is as follows.Letϕ:δ,ɛ⊂ℝ⟶ℝ be a function, then ϕ is said to be convex if ∀w,z∈δ,ɛ and 0≤t≤1, the inequality(1)ϕtw+1−tz≤tϕw+1−tϕz,holds, and ϕ is said to be strictly convex if ∀w≠z and t∈0,1, (1) holds strictly. If inequality (1) holds in reversed directions, then ϕ is said to be concave, and ϕ is said to be strictly concave if the inequality (1) holds strictly in reversed direction ∀w≠z and t∈0,1.There are several important inequalities which have been established with the help of convex functions. In 2003, Mercer [2] proved the following variant of Jensen’s inequality, which is known as Jensen–Mercer’s inequality:(2)ϕδ+ɛ−1Θn∑λ=1nθλκλ≤ϕδ+ϕɛ−1Θn∑λ=1nθλϕκλ,where ϕ:δ,ɛ⟶ℝ is a convex function, δ,ɛ⊂ℝ with κλ∈δ,ɛ, and θλ∈ℝ+∀λ∈1,2,…,n with ∑λ=1nθλ=Θn.This inequality has been refined, extended, and generalized in different direction. Recently, Niezgoda [3] generalized this inequality by using the concept of majorization and doubly stochastic matrices. The author gave applications for separable sequences. For some other recent results related to Jensen’s and Jensen–Mercer’s inequality, we recommend [4–14].In the remaining part of this paper, we present some basic terms in information theory which we deal in this paper. We start with divergence. Divergence measure is basically the measure of distance between two probability distributions. The concept of divergence measure is working efficiently to resolve different problems related to probability theory. A few divergence measures are reasonable relying upon the idea of the issue. Recently, Dragomir [15, 16], Jain [17, 18], and Taneja [19] have made many contributions in this field; they introduced different divergence measures, obtained their bounds, presented relations with other divergences, and discussed their properties.Divergence measures have gigantic applications in assortment of fields, such as approximation of probability distributions [20, 21], biology [22], economics and political science [23, 24], analysis of contingency tables [25], signal processing [26, 27], color image segmentation [28], pattern recognition [29, 30], and magnetic resonance image analysis [31].Some specific operational facts about divergence measure have been introduced, characterized, and applied in different fields, such as Bregman’sϕ-divergence [32], Csiszár’s ϕ-divergence [33], Burbea–Rao’s ϕ-divergence [34], and Renyi’s like ϕ-divergence [35]. By suitable choice of the function ϕ, many divergence measures can be obtained from these generalized divergence. Because of its compact nature, Csiszár ϕ-divergence is one of the most important divergences, which is given as follows:(3)Iϕθ,γ=∑λ=1nγλϕθλγλ,where θ=θ1,θ2,…,θn and γ=γ1,γ2,…,γn are positive real n-tuples and ϕ:ℝ+⟶ℝ is a convex function. Here, convexity of the function ϕ ensures that Iϕθ,γ≥0.By choosing appropriate convex function instead of functionϕ, many well-known divergences or distance functions can be acquired from (3), such as Hellinger, Renyi, Bhattacharyya, Chi-square, Kullback–Leibler, triangular discrimination, and Jefferys divergences. A concise presentation of these divergences is given as follows.The observed data are approximated by the probability distribution in probability and statistics. This approximation leads to information loss. The primary purpose is to assess how much information is contained in the data. Approximating a distribution byγκ for which the real distribution is θκ results in loss of data. Kullback–Leibler divergence is the deficiency of encoding the information regarding the distribution γ instead of genuine distribution θ. Kullback–Leibler divergence [36] can be acquired by choosing ϕκ=κlogκ, κ>0, in (3):(4)Kθ,γ=∑λ=1nθλlogθλγλ.Kullback–Leibler divergence is nonnegative and is zero if and only ifθλ=γλ. It satisfies the two properties of metric, but Kθ,γ≠Kγ,θ, and does not obey the triangle inequality. Kullback–Leibler divergence is also called relative entropy. We can construct Shannon entropy [37] from Kullback–Leibler divergence, which is given as follows:(5)Hθ=∑λ=1nθλlog1θλ,where θλ are positive real numbers with ∑λ=1nθλ=1. Shannon entropy has been used widely in physics, particularly in many quantum soluble systems, for example, see [38–41].The extension of Kullback–Leibler divergence is Jefferys divergence [42]. It is the summation of Kullback–Leibler divergence in both directions. Jeffery's divergence can be obtained by selecting ϕκ=κ−1logκ, κ>0, in (3):(6)Jθ,γ=∑λ=1nθλ−γλlogθλγλ,Jθ,γ≥0, and Jθ,γ=Jγ,θ, but does not obey the triangle inequality; therefore, it is not a metric. The uses of Jefferys divergence are similar to Kullback–Leibler divergence.Bhattacharyya divergence [43] can be obtained by choosing ϕκ=κ, κ>0, in (3):(7)Bθ,γ=∑λ=1nθλγλ.Bhattacharya divergence also satisfies three properties of a metric like Jefferys divergence, but does not obey the triangle inequality. Bhattacharya divergence has limited range. The limited range of this divergence makes it quite attractive for a distance comparison. Hussein et al. [44] used Bhattacharyya divergence for solving track-to-track association (TTTA) problem in space surveillance.Hellinger divergence [45] is defined as(8)Hθ,γ=∑λ=1nγλ−θλ2,which corresponds to ϕκ=1−κ2, κ>0, in (3). It satisfies all the properties of metric; therefore, Hellinger divergence is a proper metric. Hellinger divergence is used widely in data analysis, particularly when the objects being analogized are high-dimensional experiential probability distribution built from information data [46].Total variational distance [47] can be deduced by choosing ϕκ=κ−1, κ>0, in (3):(9)Vθ,γ=∑λ=1nθλ−γλ.This divergence is also a proper metric. Total variational distance is a basic quantity in probability and statistics. In information theory, variational distance is utilized to characterize solid typicality and asymptotic equipartition of sequences generated by sampling from a given distribution [48].Now, by substitutingϕκ=κs, κ>0, for s>1 in (3), we can obtain Renyi divergence [49], which is given by(10)Rθ,γ=∑λ=1nθλsγλ1−s.Renyi divergence appears as an important tool to provide proofs of convergence of minimum description length and Bayesian estimators, both in parametric and nonparametric models [50]. Some other divergences are given below which can be obtained from (3).(1)Chi−square divergence (see [47]): for ϕκ=κ−12, κ>0, the χ2-divergence is given by(11)χ2θ,γ=∑λ=1nθλ−γλ2γλ.(2)Triangular discrimination (see [42]): the formula for triangular discrimination can be deduced by selecting ϕκ=κ−12/κ+1,κ>0, which is given by(12)Δθ,γ=∑λ=1nθλ−γλ2θλ+γλ.(3)Relative arithmetic−geometric divergence (see [42]): for ϕκ=κ+1/2log1+κ/2κ where κ>0, the relative arithmetic-geometric divergence is given by(13)Gθ,γ=∑λ=1nθλ+γλ2logθλ+γλ2θλ.Zipf’s law is one of the mandatory laws in information science and is often utilized in linguistics as well. In 1932, George Zipf [51] found that one can tally how often each word shows up in the text. Therefore, if p is the rank of the word and q is the frequency of occurrence of that word, then q.p=c, where c is a constant.There are several utilizations of Zipf’s law, such as in city populations [52], geology [53], and solar flare intensity [54]. For more details, see [55, 56].In 1966, Benoit Mandelbrot [57] gave generalization of Zipf’s law, known as Zipf–Mandelbrot law, which gave improvement on account of the low-rank words in corpus [58] and is given as follows:(14)gλ=cλ+τu,where λ<1000, u,c>0, and τ≥0. If τ=0, we obtain Zipf’s law.Ifn∈ℤ+, τ∈ℝ+∪0, u>0, λ∈1,2,…,n, and Wn,τ,u=∑λ=1n1/λ+τu, then the probability mass function for Zipf–Mandelbrot law is given by(15)fλ,n,τ,u=1/λ+τuWn,τ,u.The formula for Zipf–Mandelbrot entropy is given as follows:(16)ZW,τ,u=uWn,τ,u∑λ=1nlogλ+τλ+τu+logWn,τ,u.In 2017, Khan et al. [59] used some refinements of the Jensen inequality for convex functions and monotone convex functions to obtain inequalities for Zipf–Mandelbrot and Shannon entropy. They have used two parametric Zipf–Mandelbrot laws instead of different weights in the inequalities for Shannon entropy. As a result, different parametric Zipf–Mandelbrot entropies have been obtained. In 2017, Naveed et al. [60] obtained different inequalities for these entropies by using some majorization type inequalities. In 2018, Khan et al. [55] obtained new estimations for these entropies by applying some refinements of the Jensen inequality and Taylor’s formula. In 2019, Khalid et al. [61] also gave some results related to these entropies. For more recent results related to these entropies, see [61–71].The purpose of this paper is to use Jensen–Mercer’s inequality and to give several inequalities in information theory. We obtain bounds for Csiszár divergence by using Jensen–Mercer’s inequality. Also, we give some bounds for different divergences by using particular convex functions. In addition, we establish bounds for Zipf–Mandelbrot entropy by applying Zipf–Mandelbrot laws instead of probability distributions in Kullback–Leibler and Jefferys divergences. Furthermore, we deduce new estimates for Zipf–Mandelbrot entropy associated to different parametric Zipf–Mandelbrot laws.


## 2. Bounds for Csiszár and Related Divergences

Theorem 1.
Letϕ:δ,ɛ⟶ℝ be a convex function. If θλ and γλλ=1,2,…,n are positive real numbers with θλ/γλ∈δ,ϵɛ such that ∑λ=1nθλ=Θn and ∑λ=1nγλ=Γn, then(17)ϕδ+ɛ−ΘnΓn≤ϕδ+ϕɛ−1ΓnIϕθ,γ.Proof.
Replacingθλ by γλ and substituting κλ=θλ/γλ in (2), we obtain (17).Theorem 2.
Letδ and ϵ be positive real numbers with δ<ɛ. If θλ and γλλ=1,2,…,n are positive real numbers with γλ/θλ∈δ,ɛ such that ∑λ=1nθλ=Θn and ∑λ=1nγλ=Γn, then(18)logδ+ɛ−ΓnΘn−1≤log1δɛ−1ΘnKθ,γ.Proof.
Letϕκ=−logκ, κ>0, so clearly −logκ is a convex function. Therefore, using (17) and interchanging γλ and θλ, we obtain(19)−logδ+ɛ−ΓnΘn≤−logδ−logɛ+1Θn∑λ=1nθλlogγλθλ.
Now, using the definition of Kullback–Leibler divergence, we obtain (18).Corollary 1.
Letδ and ɛ be positive real numbers with δ<ϵ. If θλλ=1,2,…,n are positive real numbers such that 1/θλ∈δ,ɛ and ∑λ=1nθλ=1, then(20)logδ+ɛ−n−1≤log1δɛ+Hθ.Proof.
By choosingγλ=1, λ=1,2,…,n, in (18), we obtain (20).Theorem 3.
Letδ and ɛ be positive real numbers with δ<ɛ. If θλ and γλλ=1,2,…,n are positive real numbers such that θλ/γλ∈δ,ɛ for λ=1,2,…,n, ∑λ=1nθλ=Θn, and ∑λ=1nγλ=Γn, then(21)δ+ɛ−ΘnΓnlogδ+ɛ−ΘnΓn≤logδδɛɛ−1ΓnKθ,γ.Proof.
Letϕκ=κlogκ, κ>0, so clearly κlogκ is a convex function. Therefore, using (17), we obtain (21).Corollary 2.
Letδ and ɛ be positive real numbers with δ<ɛ. If θλλ=1,2,…,n are positive real numbers such that θλ∈δ,ɛ and ∑λ=1nθλ=1, then(22)δ+ɛ−1nlogδ+ɛ−1n≤logδδɛɛ+1nHθ.Proof.
By choosingγλ=1,λ=1,2,…,n, in (21), we obtain (22).Theorem 4.
Let all the hypotheses of Theorem3 hold, then we have the following inequality:(23)−δ+ɛ−ΘnΓn≤−δ+ɛ+1ΓnBθ,γ.Proof.
Ifϕκ=−κ, κ>0, then ϕ″κ=1/4κ3/2>0, so ϕκ is a convex function for κ>0. Therefore, using ϕκ=−κ in Theorem 1, we obtain (23).Theorem 5.
Let all the hypotheses of Theorem3 hold, then we have the following inequality:(24)δ+ɛ−Θn+ΓnΓnlogδ+ɛ−ΘnΓn≤logδδ−1ɛɛ−1−1ΓnJθ,γ.Proof.
Ifϕκ=κ−1logκ, κ>0, then ϕ″κ=κ+1/κ2>0, so ϕκ is a convex function for κ>0. Therefore, using ϕκ=κ−1logκ in Theorem 1, we obtain (24).Corollary 3.
Letδ and ϵ be positive real numbers with δ<ɛ. If θλλ=1,2,…,n are positive real numbers such that θλ∈δ,ɛ and ∑λ=1nθλ=1, then(25)δ+ɛ−1+nnlogδ+ɛ−1n≤logδδ−1ɛɛ−1+1n∑λ=1nlogθλ+1nHθ.Proof.
By choosingγλ=1, λ=1,2,…,n, in (24), we obtain(26)δ+ɛ−1+nnlogδ+ɛ−1n≤logδδ−1ɛɛ−1+1n∑λ=1nlogθλ−1n∑λ=1nθλlogθλ.
Now, using the definition of Shannon entropy, we obtain (25).Theorem 6.
Let all the hypotheses of Theorem3 hold, then we have the following inequality:(27)1−δ+ɛ−ΘnΓn2≤δ2+ɛ2+21−δ+ɛ−1ΓnHθ,γ.Proof.
Ifϕκ=1−κ2, κ>0, then ϕ″κ=1/2κ−κ−1/2κ3/2≥0, so ϕκ is a convex function for κ>0. Therefore, using ϕκ=1−κ2 in Theorem 1, we obtain (27).Theorem 7.
Let all the hypotheses of Theorem3 hold, then we have the following inequality:(28)δ+ɛ−ΘnΓns≤δs+εs−1ΓnRθ,γ,where s>1.Proof.
Fors>1, the function ϕκ=κs, κ>0, is clearly a convex function. Therefore, using ϕκ=κs in Theorem 1, we obtain (28).Theorem 8.
Let all the hypotheses of Theorem3 hold, then we have the following inequality:(29)δ+ɛ−Θn+ΓnΓn≤δ−1+ɛ−1−1ΓnVθ,γ.Proof.
Ifϕκ=κ−1, κ∈ℝ, and we know that absolute function is always convex on ℝ. Therefore, using ϕκ=κ−1 in Theorem 1, we obtain (29).Theorem 9.
Let all the hypotheses of Theorem3 hold, then we have the following inequality:(30)δ+ɛ−Θn+ΓnΓn2≤δ2+ɛ2−2δ+ɛ−1−1Γnχ2θ,γ.Proof.
Ifϕκ=κ−12, κ>0, then ϕ″κ=2>0, so ϕz is a convex function for κ>0. Therefore, using ϕκ=κ−12 in Theorem 1, we obtain (30).Theorem 10.
Let all the hypotheses of Theorem3 hold, then we have the following inequality:(31)δ+ɛ−1/ΓnΘn+Γn2δ+ɛ−1/ΓnΘn−Γn≤δ−12δ+1+ε−12ε+1−1ΓnΔθ,γ.Proof.
Ifϕκ=κ−12/κ+1, κ>0, then ϕ″κ=8/κ+13≥0, so ϕκ is a convex function for κ>0. Therefore, using ϕκ=κ−12/κ+1 in Theorem 1, we obtain (31).Theorem 11.
Let all the hypotheses of Theorem3 hold, then we have the following inequality:(32)12δ+ɛ−1ΓnΘn−Γnlogδ+ɛ−1/ΓnΘn−Γn2δ+ε−Θn/Γn≤δ+12log1+δ2δ+ε+12log1+ɛ2ɛ−1ΓnGθ,γ.Proof.
Ifϕκ = κ+1/2log1+κ/2κ, κ>0, then ϕ″κ=1/2κ2κ+1>0, so ϕκ is a convex function for κ>0. Therefore, using ϕκ=κ−12/κ+1 in Theorem 1, we obtain (32).


## 3. Bounds for Zipf–Mandelbrot Entropy

In this section, we present some bounds for Zipf–Mandelbrot entropy.Theorem 12.
Letδ and ɛ be positive real numbers with δ<ɛ. If n∈1,2,3,…, τ≥0, u>0, and γλ>0, λ=1,2,…,n, such that γλλ+τuWn,τ,u∈δ,ɛ and ∑λ=1nγλ=Γn, then(33)logδ+ɛ−Γn−1≤log1δɛ+∑λ=1nlogγλλ+τuWn,τ,u+ZW,τ,u.Proof.
Letθλ=1/λ+τuWn,τ,u, λ=1,2,…,n, then(34)∑λ=1nθλlogθλγλ=∑λ=1n1λ+τuWn,τ,ulog1γλ1λ+τuWn,τ,u=−∑λ=1nlogγλλ+τuWn,τ,u−∑λ=1nlogλ+τuWn,τ,uλ+τuWn,τ,u=−∑λ=1nlogγλλ+τuWn,τ,u−∑λ=1nuλ+τuWn,τ,ulogλ+τ−∑λ=1nlogWn,τ,uλ+τuWn,τ,u=−∑λ=1nlogγλλ+τuWn,τ,u−ZW,τ,u.
AsWn,τ,u=∑λ=1n1/λ+τu, ∑λ=1nθλ=∑λ=1n1/λ+τuWn,τ,u=1. Hence, using (18) for θλ=1/λ+τuWn,τ,u,λ=1,2,…,n, we obtain (33).
The following corollary is the special case of Theorem12.Corollary 4.
Letδ and ɛ be positive real numbers with δ<ɛ. If n∈1,2,3,…, τ≥0, u>0, and λ+τuWn,τ,u∈δ,ɛ, then(35)logδ+ε−n−1≤log1δɛ+ZW,τ,u.Proof.
By choosingγλ=1,λ=1,2,…,n, in (33), we obtain (35).
The next result gives the inequality for Zipf–Mandelbrot entropy using two different parameters.Theorem 13.
Letδ and ɛ be positive real numbers with δ<ɛ. Let n∈1,2,…,,τ1,τ2≥0, u1,u2>0, and λ+τ1u1Wn,τ1,u1/λ+τ2u2Wn,τ2,u2∈δ,ɛ, then(36)logδ+ɛ−1−1≤log1δɛ+ZW,τ1,u1−∑λ=1nlogλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1.Proof.
Letθλ=1/λ+τ1u1Wn,τ1,u1 and γλ=1/λ+τ2u2Wn,τ2,u2, λ=1,2,…,n, then(37)∑λ=1nθλlogθλγλ=∑λ=1n1λ+τ1u1Wn,τ1,u1logλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1=−∑λ=1nlogλ+τ1u1Wn,τ1,u1λ+τ1u1Wn,τ1,u1+∑λ=1nlogλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1=−ZW,τ1,u1+∑λ=1nlogλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1.
Also,∑λ=1nθλ=∑λ=1n1/λ+τ1u1Wn,τ1,u1=1 and ∑λ=1nγλ=∑λ=1n1/λ+τ2u2Wn,τ2,u2=1; therefore, using (18) for θλ=1/λ+τ1u1Wn,τ1,u1 and γλ=1/λ+τ2u2Wn,τ2,u2, λ=1,2,…,n, we obtain (36).Theorem 14.
Letδ and ɛ be positive real numbers with δ<ɛ. If n∈1,2,3,…, τ≥0, u>0, and γλ>0, λ=1,2,…,n, such that 1/γλλ+τuWn,τ,u∈δ,ɛ and ∑λ=1nγλ=Γn, then(38)δ+ɛ−1Γnlogδ+ɛ−1Γn≤logδδɛɛ+1Γn∑λ=1nlogγλλ+τuWn,τ,u+1ΓnZW,τ,u.Proof.
Substitutingθλ=1/λ+τuWn,τ,u, λ=1,2,…,n, in (21) and using similar method which we used in the proof of Theorem 12, we obtain (38).
The following corollary is the special case of Theorem14.Corollary 5.
Letδ and ɛ be positive real numbers with δ<ɛ. If n∈1,2,3,…, τ≥0, u>0, and 1/λ+τuWn,τ,u∈δ,ε, then(39)δ+ɛ−1nlogδ+ɛ−1n≤logδδɛɛ+1nZW,τ,u.Proof.
By choosingγλ=1,λ=1,2,…,n, in (38), we obtain (39).
The next result gives the inequality for Zipf–Mandelbrot entropy using two different parameters.Theorem 15.
Letδ and ϵ be positive real numbers with δ<ϵ. Let n∈1,2,…,,τ1,τ2≥0, u1,u2>0, and λ+τ2u2Wn,τ2,u2/λ+τ1u1Wn,τ1,u1∈δ,ɛ, then(40)δ+ɛ−1logδ+ɛ−1≤logδδɛɛ+ZW,τ1,u2−∑λ=1nlogλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1.Proof.
Substitutingθλ=1/λ+τ1u1Wn,τ1,u1 and γλ=1/λ+τ2u2Wn,τ2,u2, λ=1,2,…,n, in (21) and using similar method which we used in the proof of Theorem 13, we obtain (40).Theorem 16.
Let all the hypotheses of Theorem14 hold, then we have the following inequality:(41)δ+ɛ−1+ΓnΓnlogδ+ɛ−1Γn≤logδδ−1ɛɛ−1+1Γn∑λ=1nlogγλλ+τuWn,τ,u+ZW,τ,u−∑λ=1nγλlogγλ−u∑λ=1nγλlogλ+τ−logWn,τ,u.Proof.
Letθλ=1/λ+τuWn,τ,u, λ=1,2,…,n, then(42)∑λ=1nθλlogθλγλ=−∑λ=1nlogγλλ+τuWn,τ,u−ZW,τ,u,(43)∑λ=1nγλlogθλγλ=−∑λ=1nγλlogγλλ+τWn,τ,u=−∑λ=1nγλlogγλ−u∑λ=1nγλlogλ+τ−ΓnlogWn,τ,u.
Also,∑λ=1nθλ=∑λ=1n1/λ+τuWn,τ,u=1; therefore, using (42) and (43) in (24) for θλ=1/λ+τuWn,τ,u, λ=1,2,…,n, we obtain (41).
The following corollary is the particular case of Theorem16.Corollary 6.
Letδ and ɛ be positive real numbers with δ<ɛ. If n∈1,2,3,…, τ≥0, u>0, and 1/λ+τuWn,τ,u∈δ,ɛ, then(44)δ+ɛ−1+nnlogδ+ɛ−1n≤logδδ−1ɛɛ−1+1nZW,τ,u−u∑λ=1nlogλ+τ−logWn,τ,u.Proof.
By choosingγλ=1,λ=1,2,…,n, in (41), we obtain (44).Theorem 17.
Let all the hypotheses of Theorem15 hold, then we have the following inequality:(45)δ+ɛ−2logδ+ɛ−1≤logδδ−1ɛɛ−1+ZW,τ1,u1−ZW,τ2,u2+∑λ=1nlogλ+τ1u1Wn,τ1,u1λ+τ2u2Wn,τ2,u2−logλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1.Proof.
Letθλ=1/λ+τ1u1Wn,τ1,u1 and γλ=1/λ+τ2u2Wn,τ2,u2, λ=1,2,…,n, then(46)∑λ=1nθλlogθλγλ=−ZW,τ1,u1+∑λ=1nlogλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1,(47)∑λ=1nγλlogθλγλ=∑λ=1n1λ+τ2u2Wn,τ2,u2logλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1=∑λ=1nlogλ+τ2u2Wn,τ2,u2λ+τ2u2Wn,τ2,u2−∑λ=1nlogλ+τ2u2Wn,τ2,u2λ+τ1u1Wn,τ1,u1=ZW,τ2,u2−∑λ=1nlogλ+τ1u1Wn,τ1,u1λ+τ2u2Wn,τ2,u2.
Also,∑λ=1nθλ=∑λ=1n1/λ+τ1u1Wn,τ1,u1=1 and ∑λ=1nγλ=∑λ=1n1/λ+τ2u2Wn,τ2,u2=1; therefore, using (46) and (47) in (24) for θλ=1/λ+τ1u1Wn,τ1,u1 and γλ=1/λ+τ2u2Wn,τ2,u2, λ=1,2,…,n, we obtain (45).



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