Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law

Naveed Latif; Đilda Pečarić; Josip Pečarić

doi:10.1515/math-2018-0113

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Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law

Naveed Latif , Đilda Pečarić and Josip Pečarić

Published/Copyright: December 26, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper, we consider the definition of “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law associated with the real utility distribution to give the results for majorizatioQn inequalities by using monotonic sequences. We obtain the equivalent statements between continuous convex functions and Green functions via majorization inequalities, “useful” Csiszár functional and “useful” Zipf-Mandelbrot law. By considering “useful” Csiszár divergence in the integral case, we give the results for integral majorization inequality. Towards the end, some applications are given.

Keywords: “Useful” Csiszár divergence; “Useful” Zipf-Mandelbrot law; Majorization inequality; Convex functions; Green functions; Information theory

MSC 2010: 94A15; 94A17; 26A51; 26D15

1 Introduction and Preliminaries

Zipf’s law [1, 2, 3] and the power laws in general [4, 5, 6] have and continue to attract considerable attention in a wide variety of disciplines from astronomy to demographics to software structure to economics to zoology, and even to warfare [7]. Typically one is dealing with integer-valued observables (number of objects, people, cities, words, animals, corpses), with n ∈ {1, 2, 3, …}. As given in [8], sometimes the range of values is allowed to be infinite (at least in principle), sometimes a hard upper bound N is fixed (e.g., total population if one is interested in subdividing a fixed population into sub-classes). Particularly interesting probability distributions are the probability laws of the form:

Zipf’s law: p_n ∝ 1/n;
power laws: p_n ∝ 1/n^z;
hybrid geometric/power laws: p_n ∝ wⁿ/n^z.

Distance or divergence measures are of key importance in different fields like theoretical and applied statistical inference and data processing problems such as estimation, detection, classification, compression, recognition, indexation, diagnosis and model selection etc. Traditionally, the information conveyed by observing X is measured by the entropy which is defined as (see [9, p.111])

H(p):=∑i=1npilog2⁡1/pi,

and is associated with the distribution p, p_i > 0 (1 ≤ i ≤ n), where ∑i=1npi=1. A generalization of this is to attach a utility q_i > 0 to the outcome x_i (1 ≤ i ≤ n) and speak of the “useful” information measure

H(p;q):=∑i=1nqipilog2⁡1/pi,

which is associated with the utility distribution q = (q₁, …, q_n).

Bhaker and Hooda [10] (see also [9, p.112]) introduced the measures

E(p;q):=∑k=1nqkpklog2⁡1/pk∑k=1nqkpk(1)

and

Eα(p;q):=11−αlog2⁡∑k=1nqkpkα∑k=1nqkpk,0<α≠1,(2)

which have a number of useful properties. It is readily verified that these alternations leave intact the property that (2) reduces to (1) when α → 1. Also, if u ≡ 1 so that there are effectively no utilities, (1) and (2) reduce to Renyi’s entropies of order 1 and α, respectively.

Csiszár introduced the functional in [11] and later discussed it in [12]. Here, we consider “useful” Csiszár divergence (see [13, p.3], [9, 14, 15]):

Definition 1.1

(“Useful” Csiszár divergence). AssumeJ ⊂ ℝ be an interval, and letf : J → ℝ be a function with distributionp := (p₁, …, p_n), associated with the utility distributionu := (u₁, …, u_n), wherep_i, u_i ∈ ℝ for 1 ≤ i ≤ n, andq := (q₁, …, q_n) ∈ ]0, ∞[ⁿbe such that

piqi∈J,i=1,…,n,(3)

then we denote the “useful” Csiszár divergence

Ifp,q,u:=∑i=1nuiqifpiqi.(4)

Remark 1.2

One can easily seen that if we substituteu = 1, then (4) becomes

Ifp,q,1:=Ifp,q=∑i=1nqifpiqi.

One can see the various results in information theory in [3, 16, 17].

The following theorem is a generalization of the Classical Majorization Theorem known as Weighted Majorization Theorem and was proved by Fuchs in [19] (see also [20], [21, p.323]):

Theorem 1.3

(Weighted Majorization Theorem). Letx = (x₁, …, x_n), y = (y₁, …, y_n) be two decreasing realn-tuples such thatx_i, y_i ∈ Jfori = 1, …, n. Letw = (w₁, …, w_n) be a realn-tuple such that

∑i=1jwiyi≤∑i=1jwixi,(5)

forj = 1, 2, …, n − 1 and

∑i=1nwiyi=∑i=1nwixi.(6)

Then for every continuous convex functionf : J → ℝ, we have the following inequality

∑i=1nwifyi≤∑i=1nwifxi.(7)

The following theorem is valid ([22, p.32]):

Theorem 1.4

Letf : J → Rbe a continuous convex function on an intervalJ, wbe a positiven-tuple andx, y ∈ Jⁿsatisfying

∑i=1kwiyi≤∑i=1kwixifork=1,…,n−1,(8)

and

∑i=1nwiyi=∑i=1nwixi.(9)

Ifyis a decreasingn-tuple, then
∑i=1nwifyi≤∑i=1nwifxi.(10)
Ifxis an increasingn-tuple, then
∑i=1nwifxi≤∑i=1nwifyi.(11)

Iffis strictly convex andx ≠ y, then(10)and(11)are strict.

One can see the various generalizations of the majorization inequality and bounds for Zipf-Mandelbrot entropy in [23, 24, 25].

Benoit Mandelbrot in [26] gave generalization of Zipf’s law, now known as the Zipf-Mandelbrot law which gave improvement in account for the low-rank words in corpus where k < 100 [27]:

f(k)=C(k+q)s,

and when q = 0, we get Zipf’s law.

For n ∈ ℕ, q ≥ 0, s > 0, k ∈ {1, 2, …, n}, in a more clear form, the Zipf-Mandelbrot law (probability mass function) is defined with

fk,n,q,s:=1/(k+q)sHn,q,s,whereHn,q,s:=∑i=1n1(i+q)s,

n ∈ ℕ, q ≥ 0, s > 0, k ∈ {1, 2, …, n}.

Application of the Zipf-Mandelbrot law can also be found in linguistics [27], information sciences [28, 29] and ecological field studies [30].

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x and we often denote CDF as the following ratio:

CDF:=Hk,t,sHn,t,s.(12)

The cumulative distribution function is an important application of majorization.

We consider the following definition of “useful” Zipf-Mandelbrot law (see [9, 11, 12, 14, 15]):

Definition 1.5

(“Useful” Zipf-Mandelbrotl law). AssumeJ ⊂ ℝ be an interval, andf : J → ℝ be a function withn ∈ {1, 2, 3, …}, t₁ ≥ 0. Let also distributionq_i > 0 and associated with the utility distributionu_i ∈ ℝ for (i = 1, …, n) such that

1qi(i+t1)s1Hn,t1,s1∈J,i=1,…,n,(13)

then we denote “useful” Zipf-Mandelbrot law as

Ifi,n,t1,s1,q,u:=∑i=1nuiqif1qi(i+t1)s1Hn,t1,s1.

Remark 1.6

One can easily seen that foru = 1, then

Ifi,n,t1,s1,q,1=Ifi,n,t1,s1,q:=∑i=1nqif1qi(i+t1)s1Hn,t1,s1.

If we substituteqi=1(i+t3)s3Hn,t3,s3,then

Ifi,n,t1,t3,s1,s3:=∑i=1n1(i+t3)s3Hn,t3,s3f(i+t3)s3Hn,t3,s3(i+t1)s1Hn,t1,s1.

This paper is oragnised as follows. In section 2, we give the results as the connection between useful Csisár divergence, useful Zipf-Mandelbrot law and majorization inequality for one monotonic sequence or both of them. We obtain some corollaries for our obtained results. In section 3, we present the equivalent statements between continuous convex functions and defined Green functions. In section 4, we give the results for integral majorization inequality for considering the integral form of useful Csisár divergence. Finally, in section 5 we give some applications for obtained results.

2 Main results

Assume p and q be n-tuples such that q_i > 0 (i = 1, …, n) and define

pq:=p1q1,p2q2,…,pnqn.

We start with the following theorem which provides the connection between “useful” Csiszár divergence and weighted majorization as one sequence is monotonic:

Theorem 2.1

AssumeJ ⊂ ℝ be an interval, f : J → ℝ be a continuous convex function, p_i, r_i (i = 1, …, n) be real numbers andq_i, u_i (i = 1, …, n) be positive real numbers such that

∑i=1kuiri≤∑i=1kuipifork=1,…,n−1,(14)

and

∑i=1nuiri=∑i=1nuipi,(15)

withpiqi,riqi∈J(i=1,…,n).

Ifrqis decreasing, then
Ifr,q,u≤Ifp,q,u.(16)
Ifpqis increasing, then
Ifr,q,u≥Ifp,q,u.(17)

Iffis a continuous concave function, then the reverse inequalities hold in (16) and (17).

Proof

(a): We use Theorem 1.4 (a) with substitutions xi:=piqi,yi:=riqi,w_i = u_iq_i as q_i > 0, (i = 1, …, n) then we get (16).

We can prove part (b) with the similar substitutions in Theorem 1.4 (b). □

We present the following theorem as the connection between “useful” Csiszár divergence and weighted majorization theorem as both sequences are decreasing:

Theorem 2.2

AssumeJ ⊂ ℝ be an interval, f : J → ℝ be a continuous convex function, p_i, r_i, u_i (i = 1, …, n) be real numbers andq_i (i = 1, …, n) be positive real numbers such thatpqandrqbe decreasing satisfying (14) and (15) withpiqi,riqi∈J(i=1,…,n),then

Ifr,q,u≤Ifp,q,u.(18)

Proof

We use Theorem 1.3 with substitutions xi:=piqi,yi:=riqi and w_i = u_iq_i as q_i > 0 (i = 1, …, n) then we get (18). □

The following two theorem gives the connection between “useful” Zipf-Mandelbrot law and weighted majorization inequality:

Theorem 2.3

AssumeJ ⊂ ℝ be an interval, f : J → ℝ be a continuous convex function withu_i > 0, n ∈ {1, 2, 3, …}, t₁, t₂ ≥ 0 ands₁, s₂ > 0 such that satisfying

∑i=1kui(i+t2)s2≤Hn,t2,s2Hn,t1,s1∑i=1kui(i+t1)s1,k=1,…,n−1,(19)

and

∑i=1nui(i+t2)s2=Hn,t2,s2Hn,t1,s1∑i=1nui(i+t1)s1,(20)

and also letq_i > 0, (i = 1, …, n) with

1qi(i+t1)s1Hn,t1,s1,1qi(i+t2)s2Hn,t2,s2∈J(i=1,…,n).

If(i+t2)s2(i+1+t2)s2≤qi+1qi(i=1,…,n),then
Ifi,n,t2,s2,q,u:=∑i=1nuiqif1qi(i+t2)s2Hn,t2,s2≤Ifi,n,t1,s1,q,u:=∑i=1nuiqif1qi(i+t1)s1Hn,t1,s1.(21)
If(i+t1)s1(i+1+t1)s1≥qi+1qi(i=1,…,n),then
∑i=1nuiqif1qi(i+t2)s2Hn,t2,s2≥∑i=1nuiqif1qi(i+t1)s1Hn,t1,s1.(22)

Iffis continuous concave function, then the reverse inequalities hold in (21) and (22).

Proof

(a) Let us consider that pi:=1(i+t1)s1Hn,t1,s1 and ri:=1(i+t2)s2Hn,t2,s2, then

∑i=1kuipi:=∑i=1kui(i+t1)s1Hn,t1,s1=1Hn,t1,s1∑i=1kui(i+t1)s1,k=1,…,n−1,

and similarly

∑i=1kuiri:=1Hn,t2,s2∑i=1kui(i+t2)s2,k=1,…,n−1,

leading to

∑i=1kuiri≤∑i=1kuipi⇔∑i=1kui(i+t2)s2≤Hn,t2,s2Hn,t1,s1∑i=1kui(i+t1)s1,k=1,…,n−1.

One can see easily that 1(i+t1)s1Hn,t1,s1 is decreasing over i = 1, …, n and similarly r_i too. Now, we find the behaviour of rq for q_i > 0 (i = 1, 2, …, n), take

riqi=1qi(i+t2)s2Hn,t2,s2andri+1qi+1=1qi+1(i+1+t2)s2Hn,t2,s2,ri+1qi+1−riqi=1Hn,t2,s21qi+1(i+1+t2)s2−1qi(i+t2)s2≤0,⇔(i+t2)s2(i+1+t2)s2≤qi+1qi,

which shows that rq is decreasing. So, all the assumptions of Theorem 2.1 (a) are true, then by using (16) we get (21).

(b) If we switch the role of r_i to p_i in the first part (a), then by using (17) in Theorem 2.1 (b) we get (22). □

Theorem 2.4

AssumeJ ⊂ ℝ be an interval, f : J → ℝ be a continuous convex function withu_i ∈ ℝ, n ∈ {1, 2, 3, …}, t₁, t₂ ≥ 0 ands₁, s₂ > 0, such that satisfying (19), (20) and

(i+t1)s1(i+1+t1)s1≤qi+1qi(i=1,…,n),
(i+t2)s2(i+1+t2)s2≤qi+1qi(i=1,…,n),

hold and also letq_i > 0, (i = 1, …, n) with

1qi(i+t1)s1Hn,t1,s1,1qi(i+t2)s2Hn,t2,s2∈J(i=1,…,n),

then the following inequality holds

Ifi,n,t2,s2,q,u:=∑i=1nuiqif1qi(i+t2)s2Hn,t2,s2≤Ifi,n,t1,s1,q,u:=∑i=1nuiqif1qi(i+t1)s1Hn,t1,s1.(23)

Proof

Let us consider that pi:=1(i+t1)s1Hn,t1,s1 and ri:=1(i+t2)s2Hn,t2,s2, so as given in the proof of Theorem 2.3, we get y = r/q is decreasing ⇔(i+t2)s2(i+1+t2)s2≤qi+1qi, for (i = 1, …, n), similarly we can prove that x = p/q is also decreasing ⇔(i+t1)s1(i+1+t1)s1≤qi+1qi for (i = 1, …, n). So, all the assumptions of Theorem 2.2 are true, then by using (18) we get (23). □

The following two corollaries obtain form Theorem 5 and Theorem 6 respectively but we use three the Zipf-Mandelbrot laws for different parameters:

Corollary 2.5

AssumeJ ⊂ ℝ be an interval, f : J → ℝ be a continuous convex function withu_i > 0, n ∈ {1, 2, 3, …}, t₁, t₂ ≥ 0 ands₁, s₂ > 0 such that satisfying (19) and (20) and

(i+t3)s3Hn,t3,s3(i+t1)s1Hn,t1,s1,(i+t3)s3Hn,t3,s3(i+t2)s2Hn,t2,s2∈J(i=1,…,n).

If(i+1+t2)s2(i+1+t3)s3≤(i+t2)s2(i+t3)s3(i=1,…,n),then
Ifi,n,t2,s2,t3,s3,u:=∑i=1nui(i+t3)s3Hn,t3,s3f(i+t3)s3Hn,t3,s3(i+t2)s2Hn,t2,s2≤Ifi,n,t1,s1,t3,s3,u:=∑i=1nui(i+t3)s3Hn,t3,s3f(i+t3)s3Hn,t3,s3(i+t1)s1Hn,t1,s1.(24)
If(i+1+t2)s2(i+1+t3)s3≥(i+t2)s2(i+t3)s3(i=1,…,n),then
∑i=1nui(i+t3)s3Hn,t3,s3f(i+t3)s3Hn,t3,s3(i+t2)s2Hn,t2,s2≥∑i=1nui(i+t3)s3Hn,t3,s3f(i+t3)s3Hn,t3,s3(i+t1)s1Hn,t1,s1.(25)

Iffis continuous concave function, then the reverse inequalities hold in (24) and (25).

Proof

(a) Let pi:=1(i+t1)s1Hn,t1,s1,qi:=1(i+t2)s2Hn,t2,s2 and ri:=1(i+t3)s3Hn,t3,s3, here p_i, q_i and r_i are decreasing over i = 1, …, n. Now, we investigate the behaviour of rq, take

riqi=(i+t2)s2Hn,t2,s2(i+t3)s3Hn,t3,s3andri+1qi+1=(i+1+t2)s2Hn,t2,s2(i+1+t3)s3Hn,t3,s3,ri+1qi+1−riqi=(i+1+t2)s2Hn,t2,s2(i+1+t3)s3Hn,t3,s3−(i+t2)s2Hn,t2,s2(i+t3)s3Hn,t3,s3,ri+1qi+1−riqi=Hn,t2,s2Hn,t3,s3(i+1+t2)s2(i+1+t3)s3−(i+t2)s2(i+t3)s3,

the R. H. S. is non-positive by using the assumption, which shows that rq is decreasing, therefore using Theorem 5(a) we get (24).

(b) If we switch the role of rq with pq in the part (a) and using Theorem 5(b), we get (25). □

Corollary 2.6

AssumeJ ⊂ ℝ be an interval, f : J → ℝ be a continuous convex function withu_i ∈ ℝ, n ∈ {1, 2, 3, …}, t₁, t₂ ≥ 0 ands₁, s₂ > 0, such that satisfying (19) and (20) and

(i+t1)s1(i+1+t1)s1≤(i+t3)s3(i+1+t3)s3(i=1,…,n),
(i+t2)s2(i+1+t2)s2≤(i+t3)s3(i+1+t3)s3(i=1,…,n),

hold with

(i+t3)s3Hn,t3,s3(i+t1)s1Hn,t1,s1,(i+t3)s3Hn,t3,s3(i+t2)s2Hn,t2,s2∈J(i=1,…,n),

then the following inequality holds

Ifi,n,t2,s2,t3,s3,u:=∑i=1nui(i+t3)s3Hn,t3,s3f(i+t3)s3Hn,t3,s3(i+t2)s2Hn,t2,s2≤Ifi,n,t1,s1,t3,s3,u:=∑i=1nui(i+t3)s3Hn,t3,s3f(i+t3)s3Hn,t3,s3(i+t1)s1Hn,t1,s1.(26)

Proof

(a) Let us consider that pi:=1(i+t1)s1Hn,t1,s1 and ri:=1(i+t2)s2Hn,t2,s2, so as given in the proof of Corollary 2.5 for q_i > 0 where (i = 1, 2, … ., n), we get y = r/q is decreasing ⇔(i+t2)s2(i+1+t2)s2≤(i+t3)s3(i+1+t3)s3, for (i = 1, …, n), similarly we can prove that x = p/q is also decreasing ⇔(i+t1)s1(i+1+t1)s1≤(i+t3)s3(i+1+t3)s3 for (i = 1, …, n). Therefore, all the assumptions of Theorem 2.4 are true, then by using (23) we get (26). □

Remark 2.7

We can give Theorem 2.1, Theorem 2.2, Theorem 2.3, Theorem 2.4, Corollary 2.5 and Corollary 2.6 foru := 1as special case, some of them has been given in [14].

3 "Useful” information measure via Green functions

Consider the Green function G₁ defined on [ϑ₁, ϑ₂] × [ϑ₁, ϑ₂] by

G1(u,v)=(u−ϑ2)(v−ϑ1)ϑ2−ϑ1,ϑ1≤u≤v;(v−ϑ2)(u−ϑ1)ϑ2−ϑ1,u≤v≤ϑ2.(27)

The function G₁ is convex in v, it is symmetric, so it is also convex in u. The function G₁ is continuous in v and continuous in u.

For any function f : [ϑ₁, ϑ₂] → ℝ, f ∈ C²([ϑ₁, ϑ₂]), we can easily show by integrating by parts that the following is valid

f(u)=ϑ2−uϑ2−ϑ1f(ϑ1)+u−ϑ1ϑ2−ϑ1f(ϑ2)+∫ϑ1ϑ2G(u,v)f″(v)dv,

where the function G₁ is defined as above in (27) ([31]).

Let [ϑ₁, ϑ₂] ⊂ ℝ and d = 2, 3, 4, 5. Recently in (2017), Mehmood et al. [32] (also see [33]) introduced some new types of Green functions, G_d: [ϑ₁, ϑ₂] × [ϑ₁, ϑ₂] → ℝ and give Lemma 1, which are defined as follows:

G2(u,v)=(ϑ1−v),ϑ1≤v≤u,(ϑ1−u),u≤v≤ϑ2,(28)

G3(u,v)=(u−ϑ2),ϑ1≤v≤u,(v−ϑ2),u≤v≤ϑ2,(29)

G4(u,v)=(u−ϑ1),ϑ1≤v≤u,(v−ϑ1),u≤v≤ϑ2,(30)

G5(u,v)=(ϑ2−v),ϑ1≤v≤u,(ϑ2−u),u≤v≤ϑ2.(31)

Lemma 3.1

Letf : [ϑ₁, ϑ₂] → ℝ such thatf ∈ C²([ϑ₁, ϑ₂]) andG_d (d = 2, 3, 4, 5) be Green functions as defined in(28), (29), (30)and(31), then we have the following identities.

f(u)=f(ϑ1)+(u−ϑ1)f′(ϑ2)+∫ϑ1ϑ2G2(u,v)f″(v)dv,(32)

f(u)=f(ϑ2)+(u−ϑ2)f′(ϑ1)+∫ϑ1ϑ2G3(u,v)f″(v)dv,(33)

f(u)=f(ϑ2)−(ϑ2−ϑ1)f′(ϑ2)+(u−ϑ1)f′(ϑ1)+∫ϑ1ϑ2G4(u,v)f″(v)dv,(34)

f(u)=f(ϑ1)+(ϑ2−ϑ1)f′(ϑ1)−(ϑ2−u)f′(ϑ2)+∫ϑ1ϑ2G5(u,v)f″(v)dv.(35)

The following theorem gives the equivalent statements between continuous convex functions and Green functions via majorization inequality and “useful” Csiszár divergence.

Theorem 3.2

AssumeJ ⊂ ℝ be an interval, p_i, r_i (i = 1, …, n) be real numbers andq_i, u_i (i = 1, …, n) be positive real numbers such that satisfying

∑i=1nuiri=∑i=1nuipi,(36)

withpiqi,riqi∈J(i=1,…,n).Ifrqis decreasing andG_d (d = 1, 2, 3, 4, 5) be defined as in (27)-(31), then we have following equivalent statements.

For every continuous convex functionf : [ϑ₁, ϑ₂] → ℝ, we have
Ifp,q,u−Ifr,q,u≥0.(37)
For allv ∈ [ϑ₁, ϑ₂], we have
IGdp,q,u−IGdr,q,u≥0,d=1,2,3,4,5.(38)

Moreover, if we change the sign of inequality in both inequalities(37)and(38), then the above result still holds.

Proof

The scheme of proof is similar for each d = 1, 2, 3, 4, 5, therefore we will only give the proof for d = 5.

(i) ⇒ (ii): Let statement (i) holds. As the function G₅: [ϑ₁, ϑ₂] × [ϑ₁, ϑ₂] → ℝ is convex and continuous, so it will satisfy the condition (37), i.e.,

IG5p,q,u−IG5r,q,u≥0.

(ii) ⇒ (i): Let f : [ϑ₁, ϑ₂] → ℝ be a convex function such that f ∈ C²([ϑ₁, ϑ₂]), and further, assume that the statement (ii) holds. Then by Lemma 3.1, we have

f(xi)=f(ϑ1)+(ϑ2−ϑ1)f′(ϑ1)−(ϑ2−xi)f′(ϑ2)+∫ϑ1ϑ2G5(xi,v)f″(v)dv,(39)

f(yi)=f(ϑ1)+(ϑ2−ϑ1)f′(ϑ1)−(ϑ2−yi)f′(ϑ2)+∫ϑ1ϑ2G5(yi,v)f″(v)dv.(40)

From (39) and (40), we get

Ifp,q,u−Ifr,q,u=∑i=1nuiqifpiqi−∑i=1nuiqifriqi=−∑i=1nuiqiϑ2−piqif′(ϑ2)+∑i=1nuiqiϑ2−riqif′(ϑ2)+∫ϑ1ϑ2∑i=1nuiqiG5piqi,v−∑i=1nuiqiG5riqi,vf″(v)dv.(41)

Using (36), we have

Ifp,q,u−Ifr,q,u=∫ϑ1ϑ2∑i=1nuiqiG5piqi,v−∑i=1nuiqiG5riqi,vf″(v)dv.(42)

As f is convex function, therefore f″(v) ≥ 0 for all v ∈ [ϑ₁, ϑ₂]. Hence using (38) in (42), we get (37).

Note that the condition for the existence of second derivative of f is not necessary ([21, p.172]). As it is possible to approximate uniformly a continuous convex function by convex polynomials, so we can directly eliminate this differentiability condition. □

The following theorem gives equivalent statements between continuous convex functions and Green functions via majorization inequality and “useful” Zipf-Mandelbrot law.

Theorem 3.3

Assumen ∈ {1, 2, 3, …}, t₁, t₂ ≥ 0 ands₁, s₂ > 0 such that satisfying

∑i=1nui(i+t2)s2=Hn,t2,s2Hn,t1,s1∑i=1nui(i+t1)s1,(43)

with

1qi(i+t1)s1Hn,t1,s1,1qi(i+t2)s2Hn,t2,s2∈J(i=1,…,n).

If(i+t2)s2(i+1+t2)s2≤qi+1qi(i=1,…,n)andG_d (d = 1, 2, 3, 4, 5) be defined as in (27)-(31), then we have following equivalent statements.

For every continuous convex functionf : [ϑ₁, ϑ₂] → ℝ, we have
Ifi,n,t1,s1,q,u−Ifi,n,t2,s2,q,u≥0.(44)
For allv ∈ [ϑ₁, ϑ₂], we have
IGdi,n,t1,s1,q,u−IGdi,n,t2,s2,q,u≥0,d=1,2,3,4,5.(45)

Moreover, if we change the sign of inequality in both inequalities(44)and(45), then the above result still holds.

Proof

(i) ⇒ (ii): The proof is similar to the proof of Theorem 3.2.

(ii) ⇒ (i): Let f : [ϑ₁, ϑ₂] → ℝ be a convex function such that f ∈ C²([ϑ₁, ϑ₂]), and further, assume that the statement (ii) holds. Then by Lemma 3.1, we have (39) and (40).

From (39) and (40), we get

Ifi,n,t1,s1,q,u−Ifi,n,t2,s2,q,u=∑i=1nuiqifλi−∑i=1nuiqifμi=−∑i=1nuiqiϑ2−λif′(ϑ2)+∑i=1nuiqiϑ2−μif′(ϑ2)+∫ϑ1ϑ2∑i=1nuiqiG5λi,v−∑i=1nuiqiG5μi,vf″(v)dv,

where,

λi:=1qi(i+t1)s1Hn,t1,s1,andμi:=1qi(i+t2)s2Hn,t2,s2.

Using (43), we have

Ifi,n,t1,s1,q,u−Ifi,n,t2,s2,q,u=∫ϑ1ϑ2∑i=1nuiqiG5λi,v−∑i=1nuiqiG5μi,vf″(v)dv.(46)

As f is convex function, therefore f″(v) ≥ 0 for all v ∈ [ϑ₁, ϑ₂]. Hence using (45) in (46), we get (44). □

4 “Useful” information measure in integral form

The following theorem is a slight extension of Lemma 2 in [34] which is proved by Maligranda et al. (also see [35]):

Theorem 4.1

Letw, xandybe positive functions on [a, b]. Suppose thatf : [0, ∞) → ℝ is a convex function and that

∫aνy(t)w(t)dt≤∫aνx(t)w(t)dt,ν∈[a,b]and∫aby(t)w(t)dt=∫abx(t)w(t)dt.

Ifyis a decreasing function on [a, b], then
∫abfy(t)w(t)dt≤∫abfx(t)w(t)dt.(47)
Ifxis an increasing function on [a, b], then
∫abfx(t)w(t)dt≤∫abfy(t)w(t)dt.(48)

Iffis strictly convex function andx ≠ y (a. e.), then(47)and(48)are strict.

We consider “useful” Csiszár functional [11, 12] in integral form:

Definition 4.2

(“Useful” Csiszár divergence as integral form). AssumeJ := [α, β] ⊂ ℝ be an interval, and letf : J → ℝ be a function with densitiesp : [a, b] → J, q : [a, b] → (0,∞) and associated with the utility densityu : [a, b] → Jsuch that

p(x)q(x)∈J,∀x∈[a,b],

then we denote “useful” Csiszár divergence in integral form as

I^f(p,q,u):=∫abu(t)q(t)fp(t)q(t)dt.(49)

Remark 4.3

One can easily seen that if we substituteu(t) = 1 for allt ∈ [a, b], then (49) becomes

I^fp,q,1:=I^fp,q=∫abq(t)fp(t)q(t)dt.

Theorem 4.4

AssumeJ := [0, ∞) ⊂ ℝ be an interval, f : J → ℝ be a convex function andp, q, r, u : [a, b] → (0, ∞) such that

∫aυu(t)r(t)dt≤∫aυu(t)p(t)dt,υ∈[a,b](50)

and

∫abu(t)r(t)dt=∫abu(t)p(t)dt,(51)

with

p(t)q(t),r(t)q(t)∈J,∀t∈[a,b].

Ifr(t)q(t)is a decreasing function on [a, b], then
I^f(r,q,u)≤I^f(p,q,u).(52)
Ifp(t)q(t)is an increasing function on [a, b], then the inequality is reversed, i.e.
I^f(r,q,u)≥I^f(p,q,u).(53)

Iffis strictly convex function andp(t) ≠ r(t) (a. e.), then strict inequality holds in(52)and(53).

Iffis concave function then the reverse inequalities hold in (52) and (53). Iffis strictly concave andp(t) ≠ r(t) (a. e.), then the strict reverse inequalities hold in (52) and (53).

Proof

(i): We use Theorem 4.1 (i) with substitutions x(t):=p(t)q(t),y(t):=r(t)q(t),w(t):=u(t)q(t)>0∀t∈[a,b] and also using the fact that r(t)q(t) is a decreasing function then we get (52).

(ii) We can prove with the similar substitutions as in the first part by using Theorem 4.1 (ii) that is the fact that p(t)q(t) is an increasing function. □

Remark 4.5

We can give Theorem 4.4 foru(t) := 1 for allt ∈ [a, b] as special case which has been given in [36].

5 Applications

Here, we present several special cases of the previous results as applications.

The first case corresponds to the entropy of a continuous probability density (see [18, p.506]):

Definition 5.1

(Shannon Entropy). Letp : [a, b] → (0, ∞) be a positive probability density, then the Shannon entropy ofp(x) is defined by

Hp(x),u(x):=−∫abu(x)p(x)log⁡p(x)dx,(54)

and is associated with the utility densityu : [a, b] → ℝ, whenever the integral exists.

Note that there is no problem with the definition in the case of a zero probability, since

limx→0xlog⁡x=0.(55)

Corollary 5.2

Assumep, q, r, u : [a, b] → (0, ∞) be functions such that satisfying (50) and (51) with

p(t)q(t),r(t)q(t)∈J:=(0,∞),∀t∈[a,b].

Ifr(t)q(t)is a decreasing function and the base of log is greater than 1, then we have estimates for the Shannon entropy ofq(t) associatedy with utility densityu(t)
∫abu(t)q(t)logr(t)q(t)≥H(q(t),u(t)).(56)
If the base of log is in between 0 and 1, then the reverse inequality holds in (56).
Ifp(t)q(t)is an increasing function and the base of log is greater than 1, then we have estimates for the Shannon entropy ofq(t) associated with utility densityu(t)
Hq(t),u(t)≤∫abu(t)q(t)logp(t)q(t).(57)
If the base of log is in between 0 and 1, then the reverse inequality holds in (57).

Proof

(i): Substitute f(x) := − log x and p(t) := 1, ∀ t ∈ [a, b] in Theorem 4.4 (i) then we get (56).

(ii) We can prove by switching the role of p(t) with r(t) i.e., r(t) := 1 ∀ t ∈ [a, b] and f(x) := − log x in Theorem 4.4 (ii) then we get (57). □

The second case corresponds to the relative entropy or the Kullback-Leibler divergence between two probability densities associated with the utility density u(t):

Definition 5.3

(Kullback-Leibler Divergence). Letp, q : [a, b] → (0, ∞) be a positive probability densities, then the Kullback-Leibler (K-L) divergence betweenp(t) andq(t) is defined by

Lp(t),q(t),u(t):=∫abu(t)p(t)logp(t)q(t)dt,

and is associated with the utility densityu : [a, b] → ℝ.

Corollary 5.4

Assumep, q, r, u : [a, b] → (0, ∞) be functions such that satisfying (50) and (51) with

p(t)q(t),r(t)q(t)∈J:=(0,∞),∀t∈[a,b].

Ifr(t)q(t)is a decreasing function and the base of log is greater than 1, then
I^(−log⁡x)(r,q,u)≥I^(−log⁡x)(p,q,u).(58)
If the base of log is in between 0 and 1, then the reverse inequality holds in (58).
Ifp(t)q(t)is an increasing function and the base of log is greater than 1, then
I^(−log⁡x)(r,q,u)≤I^(−log⁡x)(p,q,u).(59)
If the base of log is in between 0 and 1 then the reverse inequality holds in (59).

Proof

(i): Substitute f(x) := − log x in Theorem 4.4 (i) then we get (58).

(ii) We can prove with substitution f(x) := − log x in Theorem 4.4 (ii). □

In Information Theory and Statistics, various divergences are applied in addition to the Kullback-Leibler divergence.

Definition 5.5

(Variational Distance). Letp, q : [a, b] → (0, ∞) be a positive probability densities, then variation distance betweenp(t) andq(t) is defined by

I^vp(t),q(t),u(t):=∫abu(t)|p(t)−q(t)|dt,

and associated with the utility densityu : [a, b] → ℝ.

Corollary 5.6

Assumep, q, r, u : [a, b] → (0, ∞) be functions such that satisfying (50) and (51) with

p(t)q(t),r(t)q(t)∈J:=(0,∞),∀t∈[a,b].

Ifr(t)q(t)is a decreasing function, then
I^vr(t),q(t),u(t)≤I^vp(t),q(t),u(t).(60)
Ifp(t)q(t)is an increasing function, then the inequality is reversed, i.e.
I^vr(t),q(t),u(t)≥I^vp(t),q(t),u(t).(61)

Proof

(i): Since f(x) := ∣x − 1∣ be a convex function for x ∈ ℝ⁺, therefore substitute f(x) := ∣x − 1∣ in Theorem 4.4 (i) then

∫abu(t)q(t)r(t)q(t)−1dt≤∫abu(t)q(t)p(t)q(t)−1dt,∫abu(t)q(t)r(t)−q(t)q(t)dt≤∫abu(t)q(t)p(t)−q(t)q(t)dt,

since q(t) > 0 then we get (60).

(ii) We can prove with substitution f(x) := ∣x − 1 ∣ in Theorem 4.4 (ii). □

Definition 5.7

(Hellinger Distance). Letp, q : [a, b] → (0, ∞) be a positive probability densities, then the Hellinger distance betweenp(t) andq(t) is defined by

I^Hp(t),q(t),u(t):=∫abu(t)p(t)−q(t)2dt,

and is associated with the utility densityu : [a, b] → ℝ.

Corollary 5.8

Assumep, q, r, u : [a, b] → (0, ∞) be functions such that satisfying (50) and (51) with

p(t)q(t),r(t)q(t)∈J:=(0,∞),∀t∈[a,b].

Ifr(t)q(t)is a decreasing function, then
I^Hr(t),q(t),u(t)≤I^Hp(t),q(t),u(t).(62)
Ifp(t)q(t)is an increasing function, then the inequality is reversed, i.e.
I^Hr(t),q(t),u(t)≥I^Hp(t),q(t),u(t).(63)

Proof

(i): Since f(x):=x−12 is a convex function for x ∈ ℝ⁺, therefore substituting f(x):=x−12 in Theorem 4.4 (i)

∫abu(t)q(t)r(t)q(t)−12dt≤∫abu(t)q(t)p(t)q(t)−12dt,

since q(t) > 0 then we get (62).

(ii) We can prove with substitution f(x):=x−12 in Theorem 4.4 (ii). □

Definition 5.9

(Bhattacharyya Distance). Letp, q : [a, b] → (0, ∞) be a positive probability densities, then the Bhattacharyya distance betweenp(t) andq(t) is defined by

I^Bp(t),q(t),u(t):=∫abu(t)p(t)q(t)dt,

and associated with the utility densityu : [a, b] → ℝ.

Corollary 5.10

Assumep, q, r, u : [a, b] → (0, ∞) be functions such that satisfying (50) and (51) with

p(t)q(t),r(t)q(t)∈J:=(0,∞),∀t∈[a,b].

Ifr(t)q(t)is a decreasing function, then
I^Bp(t),q(t),u(t)≤I^Br(t),q(t),u(t).(64)
Ifp(t)q(t)is an increasing function, then the inequality is reversed, i.e.
I^Bp(t),q(t),u(t)≥I^Br(t),q(t),u(t).(65)

Proof

(i): Since f(x):=−x be a convex function for x ∈ ℝ⁺, therefore substitute f(x):=−x in Theorem 4.4 (i) then

∫abu(t)q(t)−r(t)q(t)dt≤∫abu(t)q(t)−p(t)q(t)dt,

we get (64).

(ii) We can prove with substitution f(x):=−x in Theorem 4.4 (ii). □

Definition 5.11

(Jeffreys Distance). Letp, q : [a, b] → (0, ∞) be a positive probability densities, then the Jeffreys distance betweenp(t) andq(t) is defined by

I^Jp(t),q(t),u(t):=∫abu(t)p(t)−q(t)lnp(t)q(t)dt,

and associated with the utility densityu : [a, b] → ℝ.

Corollary 5.12

Assumep, q, r, u : [a, b] → (0, ∞) be functions such that satisfying (50) and (51) with

p(t)q(t),r(t)q(t)∈J:=(0,∞),∀t∈[a,b].

Ifr(t)q(t)is a decreasing function, then
I^Jr(t),q(t),u(t)≤I^Jp(t),q(t),u(t).(66)
Ifp(t)q(t)is an increasing function, then the inequality is reversed, i.e.
I^Jr(t),q(t),u(t)≥I^Jp(t),q(t),u(t).(67)

Proof

(i): Since f(x) := (x − 1) ln x be a convex function for x ∈ ℝ⁺, therefore substituting f(x) := (x − 1) ln x in Theorem 4.4 (i)

∫abu(t)q(t)r(t)q(t)−1lnr(t)q(t)dt≤∫abu(t)q(t)p(t)q(t)−1lnp(t)q(t)dt,

we get (66).

(ii) We can prove with substitution f(x) := (x − 1) ln x in Theorem 4.4 (ii). □

Definition 5.13

(Triangular Discrimination). Letp, q : [a, b] → (0, ∞) be a positive probability densities, then the triangular discrimination betweenp(t) andq(t) is defined by

I^Δp(t),q(t),u(t):=∫abu(t)p(t)−q(t)2p(t)+q(t)dt,

and is associated with the utility densityu : [a, b] → ℝ.

Corollary 5.14

Assumep, q, r, u : [a, b] → (0, ∞) be functions such that satisfying (50) and (51) with

p(t)q(t),r(t)q(t)∈J:=(0,∞),∀t∈[a,b].

Ifr(t)q(t)is a decreasing function, then
I^Δr(t),q(t),u(t)≤I^Δp(t),q(t),u(t).(68)
Ifp(t)q(t)is an increasing function, then the inequality is reversed, i.e.
I^Δr(t),q(t),u(t)≥I^Δp(t),q(t),u(t).(69)

Proof

(i): Since f(x):=(x−1)2x+1 be a convex function for x ≥ 0, therefore substitute f(x):=(x−1)2x+1 in Theorem 4.4 (i) then

∫abu(t)q(t)r(t)/q(t)−12r(t)/q(t)+1dt≤∫abu(t)q(t)p(t)/q(t)−12p(t)/q(t)+1dt,∫abu(t)q(t)(r(t)−q(t))/q(t)2(r(t)+q(t))/q(t)dt≤∫abu(t)q(t)(p(t)−q(t))/q(t)2(p(t)+q(t))/q(t)dt,

we get (68).

(ii) We can prove with substitution f(x):=(x−1)2x+1 in Theorem 4.4 (ii). □

Remark 5.15

We can give all the results of section 5 foru(t) = 1 for allt ∈ [a, b] as a special case, which has been given in [36].

Author’s contribution All authors contributed equally. All authors read and approved the final manuscript.
Competing interest The authors declare that they have no competing interests.

Acknowledgement

The publication was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008.) This publication is partially supported by Royal Commission (RC) Jubail Industrial College, Jubail, Kingdom of Saudi Arabia.

References

[1] Saichev A., Malevergne Y., Sornette D., Theory of Zipf’s law and beyond, in: Lecture notes in Economics and Mathematical systems 362, Berlin: Springer, 2009.10.1007/978-3-642-02946-2Search in Google Scholar

[2] Zipf G. K., The psychobiology of language, Cambridge, MA: Houghton-Mifflin, 1935.Search in Google Scholar

[3] Zipf G. K., Human behavior and the principle of least effort, Reading, MA: Addison-Wesley, 1949.Search in Google Scholar

[4] Baxter G., Frean M., Noble J., Rickerby M., Smith H., Visser M., Melton H., Tempero E., Understanding the shape of Java software, OOPSLA Proc. 21st Annual ACM SIGPLAN Conf. on Object-Oriented Programming Systems, Languages and Applications, Eds. Tarr P.L., Cook W.R., New York: ACM, 2006, 379-412.10.1145/1167473.1167507Search in Google Scholar

[5] Clauset A., Shalizi C. R., Newman M. E. J., Power-law distributions in empirical data, SIAM Rev., 2009, 51, 661–703.10.1137/070710111Search in Google Scholar

[6] Newmann M. E. J., Power laws, Pareto distributions and Zipf’s law, Contemp. Phys., 2007, 46, 323-351.10.1080/00107510500052444Search in Google Scholar

[7] Richardson L. F., Statistics of deadly quarrels, Pacific Grove, CA: Boxwood Press, 1960.Search in Google Scholar

[8] Visser M., Zipf’s law, power laws and maximmum entropy, New J. Phys., 2013, 15, 1-13.10.1088/1367-2630/15/4/043021Search in Google Scholar

[9] Matić M., Pearce C. E. M., Pečarić J., Some comparison theorems for the mean-value characterization of “useful” information measures, SEA Bull. Math., 1999, 23, 111-116.Search in Google Scholar

[10] Bhaker U. S., Hooda D. S., Mean value characterization of ’useful’ information measures, Tamkang J. Math., 1993, 24, 383-394.10.5556/j.tkjm.24.1993.4510Search in Google Scholar

[11] Csiszár I., Information-type measures of difference of probability distributions and indirect observations, Studia Sci. Math. Hungar., 1967, 2, 299-318.Search in Google Scholar

[12] Csiszár I., Information measure: A critical survey, Trans. 7th Prague Conf. on Info. Th., Statist. Decis. Funct., Random Processes and 8th European Meeting of Statist., B, Academia Prague, 1978, 73-86.Search in Google Scholar

[13] Horváth L., Pečarić Ð., and Pečarić J., Estimations off- and Rényi divergences by using a cyclic refinement of the Jensen’s inequality, Bull. Malays. Math. Sci. Soc., 2017, 10.1007/s40840-017-0526-4.Search in Google Scholar

[14] Latif N., Pečarić Ð., Pečarić J., Majorization, Csisár divergence and Zipf-Mandelbrot law, J. Inequal. Appl., 2017, 1-15.10.1186/s13660-017-1472-2Search in Google Scholar PubMed PubMed Central

[15] Latif N., Pečarić Ð., Pečarić J., Majorization and Zipf-Mandelbrot law, submitted.Search in Google Scholar

[16] Matić M., Pearce C. E. M., Pečarić J., Improvements of some bounds on entropy measures in information theory, Math. Inequal. Appl., 1998, 1, 295-304, 1998.10.7153/mia-01-29Search in Google Scholar

[17] Matić M., Pearce C. E. M., Pečarić J., On an inequality for the entropy of a probability distribution, Acta Math. Hungar., 1999, 85, 345-349.10.1023/A:1006711805568Search in Google Scholar

[18] Matić M., Pearce C. E. M., Pečarić J., Shannon’s and related inequalities in information theory, Survey on classical inequalities, editor Themistocles M. Rassias, Kluwer Academic Publishers, 2000, 127-164.10.1007/978-94-011-4339-4_5Search in Google Scholar

[19] Fuchs L., A new proof of an inequality of Hardy-Littlewood-Polya, Math. Tidsskr, 1947, 53-54.Search in Google Scholar

[20] Marshall A. W., Olkin I., Arnold B. C., Inequalities: Theory of Majorization and Its Applications (Second Edition), Springer Series in Statistics, New York, 2011.10.1007/978-0-387-68276-1Search in Google Scholar

[21] Pečarić J., Proschan F., Tong Y. L., Convex functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992.Search in Google Scholar

[22] Niculescu C. P., Persson L. E., Convex functions and their applications, A contemporary Approach, CMS Books in Mathematics, 23, Springer-Verlag, New York, 2006.10.1007/0-387-31077-0Search in Google Scholar

[23] Adil Khan M., Latif N., and Pečarić J., Generalization of majorization theorem, J. Math. Inequal., 2015, 9(3), 847-872.10.7153/jmi-09-70Search in Google Scholar

[24] Adil Khan M., Khalid S., Pečarić J., Refinements of some majorization type inequalities, J. Math. Inequal., 2013, 7(1), 73-92.10.7153/jmi-07-07Search in Google Scholar

[25] Adil Khan M., Pečarić D., Pečarić J., Bounds for Shannon and Zipf-mandelbrot entropies, Math. Methods Appl. Sci., 2017, 40(18), 7316-7322.10.1002/mma.4531Search in Google Scholar

[26] Mandelbrot B., Information Theory and Psycholinguistics: A Theory of Words Frequencies, In Reading in Mathematical Social Science, (ed.) P. Lazafeld, N. Henry Cambridge MA, MIT Press, 1966.Search in Google Scholar

[27] Montemurro M. A., Beyond the Zipf-Mandelbrot law in quantitative linguistics, 2001, URL: arXiv:cond-mat/0104066v2.Search in Google Scholar

[28] Egghe L., Rousseau R., Introduction to Informetrics. Quantitative Methods in Library, Documentation and Information Science, Elsevier Science Publishers, New York, 1990.Search in Google Scholar

[29] Silagadze Z. K., Citations and the Zipf-Mandelbrot Law, Complex Systems, 1997, (11), 487-499.Search in Google Scholar

[30] Mouillot D., Lepretre A., Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams (RFD), to assess changes in community diversity, Environmental Monitoring and Assessment, Springer, 2000, 63 (2), 279-295.10.1023/A:1006297211561Search in Google Scholar

[31] Widder D. V., Completely convex function and Lidstone series, Trans. Am. Math. Soc., 1942, 51, 387-398.10.1090/S0002-9947-1942-0006356-4Search in Google Scholar

[32] Mehmood N., Agarwal R. P., Butt S. I., Pečarić J., New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity, J. Inequl. Appl., 2017, 108, 1-21.10.1186/s13660-017-1379-ySearch in Google Scholar PubMed PubMed Central

[33] Butt S. I., Khan K. A., Pečarić J., Popoviciu type inequalities via Green function and Generalized Montgomery Identity, Math. Inequal. Appl., 2015, 18(4), 1519-1538.10.7153/mia-18-118Search in Google Scholar

[34] Maligranda L., Pečarić J., Persson L. E., Weighted Favard’s and Berwald’s inequalities, J. Math. Anal. Appl., 1995, 190, 248-262.10.1006/jmaa.1995.1075Search in Google Scholar

[35] Latif N., Pečarić J., Perić I., On Majorization, Favard and Berwald’s Inequalities, Annals of Functional Analysis, 2011, 2, no. 1, 31-50, ISSN: 2008-8752.10.15352/afa/1399900260Search in Google Scholar

[36] Latif N., Pečarić Ð., Pečarić J., Majorization in Information Theory, JIASF, 2017, (8) 4, 42-56.10.1186/s13660-017-1472-2Search in Google Scholar PubMed PubMed Central

[37] Matić M., Pearce C. E. M., Pečarić J., Some refinements of Shannon’s inequalities, ANZIAM J. (formerly J. Austral. Math. Soc. Ser. B), 2002, 43, 493-511.10.1017/S1446181100012104Search in Google Scholar

Received: 2017-12-08

Accepted: 2018-06-20

Published Online: 2018-12-26

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Articles in the same Issue

https://doi.org/10.1515/math-2018-0113

Keywords for this article

“Useful” Csiszár divergence; “Useful” Zipf-Mandelbrot law; Majorization inequality; Convex functions; Green functions; Information theory

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BY-NC-ND 4.0