Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

A RAB J OURNAL OF M ATHEMATICAL S CIENCES

Arab J Math Sci xxx (xxx) (xxxx) xxx

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals ˇ ´ c , J OSIP P E CARI ˇ ´ d,1 C C K HURAM A LI K HAN a , TASADDUQ N IAZ a , b , ∗, Ð ILDA P E CARI a Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan b Department of Mathematics, The University of Lahore, Sargodha-Campus, Sargodha 40100, Pakistan c Catholic University of Croatia, Ilica 242, Zagreb, Croatia d RUDN University, Miklukho-Maklaya str. 6, 117198 Moscow, Russia

Received 7 November 2018; revised 15 December 2018; accepted 18 December 2018 Available online xxxx

Abstract. In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel– Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function. Keywords: m-convex function; Jensen’s inequality; Shannon entropy; f - and Rényi divergence; Fink’s identity; Abel–Gontscharoff Green function; Entropy

1. I NTRODUCTION AND PRELIMINARY RESULTS In recent years many researchers generalized different inequalities using different identities involving green functions, for example in [24] Nasir et al. generalized the Popoviciu inequality using Mongomery identity along with the new green function. Also in [25] Niaz ∗ Corresponding author at: Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan

E-mail addresses: [email protected] (K.A. Khan), [email protected] (T. Niaz), [email protected] (Ð. Peˇcari´c), [email protected] (J. Peˇcari´c). 1 The research of 4th author was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008). Peer review under responsibility of King Saud University.

Production and hosting by Elsevier https://doi.org/10.1016/j.ajmsc.2018.12.002 c 2018 The Authors. Production and Hosting by Elsevier B.V. on behalf of King Saud University. This is an open access 1319-5166 ⃝ article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

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K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

et al. used Fink’s identity along with new Abel–Gontscharoff type Green functions for ‘two point right focal’ to generalize the refinement of Jensen inequality. The most commonly used words, the largest cities of countries, income of billionaire can be described in terms of Zipf’s law. The f -divergence means the distance between two probability distributions by making an average value, which is weighted by a specified function. As f -divergence, there are other probability distributions like Csiszár f divergence [11,12], some special case of which is Kullback–Leibler-divergence used to find the appropriate distance between the probability distributions (see [20,21]). The notion of distance is stronger than divergence because it gives the properties of symmetry and triangle inequalities. Probability theory has application in many fields and the divergence between probability distribution has many applications in these fields. Many natural phenomena like distribution of wealth and income in a society, distribution of face book likes, distribution of football goals follow power law distribution (Zipf’s Law). Like above phenomena, distribution of city sizes also follows Power Law distribution. Auerbach [3] first time gave the idea that the distribution of city size can be well approximated with the help of Pareto distribution (Power Law distribution). This idea was well refined by many researchers but Zipf [32] worked significantly in this field. The distribution of city sizes is investigated by many scholars of the urban economics, like Rosen and Resnick [29] , Black and Henderson [4], Ioannides and Overman [19], Soo [30], Anderson and Ge [2] and Bosker et al. [5]. Zipf’s law states that: “The rank of cities with a certain number of inhabitants varies proportional to the city sizes with some negative exponent, say that is close to unit”. In other words, Zipf’s Law states that the product of city sizes and their ranks appear roughly constant. This indicates that the population of the second largest city is one half of the population of the largest city and the third largest city equal to the one third of the population of the largest city and the population of nth city is n1 of the largest city population. This rule is called rank, size rule and also named as Zipf’s Law. Hence Zip’s Law not only shows that the city size distribution follows the Pareto distribution, but also shows that the estimated value of the shape parameter is equal to unity. In [18] L. Horváth et al. introduced some new functionals based on the f -divergence functionals and obtained some estimates for the new functionals. They obtained f -divergence and Rényi divergence by applying a cyclic refinement of Jensen’s inequality. They also construct some new inequalities for Rényi and Shannon entropies and used Zipf–Mandelbrot law to illustrate the results. The inequalities involving higher order convexity are used by many physicists in higher dimension problems since the founding of higher order convexity by T. Popoviciu (see [27, p. 15]). It is quite interesting fact that there are some results that are true for convex functions but when we discuss them in higher order convexity they do not remain valid. In [27, p. 16], the following criteria are given to check the m-convexity of the function. If f (m) exists, then f is m-convex if and only if f (m) ≥ 0. In recent years many researchers have generalized the inequalities for m-convex functions; like S. I. Butt et al. generalized the Popoviciu inequality for m-convex function using Taylor’s formula, Lidstone polynomial, Montgomery identity, Fink’s identity, Abel–Gontscharoff interpolation and Hermite interpolating polynomial (see [6–10]). Since many years Jensen’s inequality has of great interest. The researchers have given the refinement of Jensen’s inequality by defining some new functions (see [16,17] ). Like many researchers L. Horváth and J. Peˇcari´c in [14,17], see also [15, p. 26], gave a refinement of Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

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Jensen’s inequality for convex function. They defined some essential notions to prove the refinement given as follows: Let X be a set, and: P(X ) := Power set of X , |X | := Number of elements of X , N := Set of natural numbers with 0. Consider q ≥ 1 and r ≥ 2 be fixed integers. Define the functions Fr,s : {1, . . . , q}r → {1, . . . , q}r −1

1 ≤ s ≤ r,

( ) Fr : {1, . . . , q}r → P {1, . . . , q}r −1 , and ( ) ( ) Tr : P {1, . . . , q}r → P {1, . . . , q}r −1 , by Fr,s (i 1 , . . . , ir ) := (i 1 , i 2 , . . . , i s−1 , i s+1 , . . . , ir ) Fr (i 1 , . . . , ir ) :=

1 ≤ s ≤ r,

r ⋃ {Fr,s (i 1 , . . . , ir )}, s=1

and Tr (I ) =

⎧ ⎨φ, ⋃

I = φ; I ̸= φ.

Fr (i 1 , . . . , ir ),

⎩

(i 1 ,...,ir )∈I

Next let the function αr,i : {1, . . . , q}r → N

1≤i ≤q

defined by αr,i (i 1 , . . . , ir ) is the number of occurrences of i in the sequence (i 1 , . . . , ir ). For each I ∈ P({1, . . . , q}r ) let ∑ α I,i := αr,i (i 1 , . . . , ir )

1 ≤ i ≤ q.

(i 1 ,...,ir )∈I

(H1 ) Let n, m be fixed positive integers such that n ≥ 1, m ≥ 2 and let Im be a subset of {1, . . . , n}m such that α Im ,i ≥ 1

1 ≤ i ≤ n.

Introduce the sets Il ⊂ {1, . . . , n}l (m − 1 ≥ l ≥ 1) inductively by Il−1 := Tl (Il )

m ≥ l ≥ 2.

Obviously the sets I1 = {1, . . . , n}, by (H1 ) and this insures that α I1 ,i = 1(1 ≤ i ≤ n). From (H1 ) we have α Il ,i ≥ 1(m − 1 ≥ l ≥ 1, 1 ≤ i ≤ n). For m ≥ l ≥ 2, and for any ( j1 , . . . , jl−1 ) ∈ Il−1 , let H Il ( j1 , . . . , jl−1 ) := {((i 1 , . . . , il ), k) × {1, . . . , l}|Fl,k (i 1 , . . . , il ) = ( j1 , . . . , jl−1 )}. Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

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K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

With the help of these sets they define the functions η Im ,l : Il → N(m ≥ l ≥ 1) inductively by η Im ,m (i 1 , . . . , i m ) := 1

(i 1 , . . . , i m ) ∈ Im ; ∑

η Im ,l−1 ( j1 , . . . , jl−1 ) :=

η Im ,l (i 1 , . . . , il ).

((i 1 ,...,il ),k)∈H Il ( j1 ,..., jl−1 )

They define some special expressions for 1 ≤ l ≤ m, as follows (m − 1)! ∑ η I ,l (i 1 , . . . , il ) Am,l = Am,l (Im , x1 , . . . , xn , p1 , . . . , pn ; f ) := (l − 1)! (i ,...,i )∈I m l l 1 ⎞ ⎞ ⎛ ∑l ⎛ pi j l x i ∑ j=1 α Im ,i j pi j ⎟ j ⎠f⎜ ×⎝ ⎝ ∑l ⎠ pi j α j=1 Im ,i j j=1 α Im ,i j

and prove the following theorem. Theorem 1.1. Assume (H1 ), and let f : I → R be a convex function where∑ I ⊂ R is an interval. If x1 , . . . , xn ∈ I and p1 , . . . , pn are positive real numbers such that ns=1 ps = 1, then ( n ) n ∑ ∑ f ps xs ≤ Am,m ≤ Am,m−1 ≤ · · · ≤ Am,2 ≤ Am,1 = ps f (xs ) . (1) s=1

s=1

We define the following functionals by taking the differences of refinement of Jensen’s inequality given in (1). ( n ) ∑ Θ1 ( f ) = Am,r − f ps xs , r = 1, . . . , m, (2) s=1

Θ2 ( f ) = Am,r − Am,k ,

1 ≤ r < k ≤ m.

(3)

Under the assumptions of Theorem 1.1, we have Θi ( f ) ≥ 0,

i = 1, 2.

(4)

Inequalities (4) are reversed if f is concave on I . In [26], the green function G : [α1 , α2 ] × [α1 , α2 ] → R is defined as ⎧ (u − α2 )(v − α1 ) ⎪ ⎨ , α1 ≤ v ≤ u; α1 (5) G(u, v) = (v −αα2 − 2 )(u − α1 ) ⎪ ⎩ , u ≤ v ≤ α2 . α2 − α1 The function G is convex with respect to v and due to symmetry also convex with respect to u. One can also note that G is continuous function. In [31] it is given that any function f : [α1 , α2 ] → R, such that f ∈ C 2 ([α1 , α2 ]) can be written as ∫ α2 α2 − u u − α1 f (u) = f (α1 ) + f (α2 ) + G(u, v) f ′′ (v)dv. (6) α2 − α1 α2 − α1 α1

Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

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2. I NEQUALITIES FOR C SISZÁR DIVERGENCE In [11,12] Csiszár introduced the following notion. Definition 1. Let f : R+ → R+ be a convex function, let r = (r1 , . . . , rn ) and q = (q1 , . . . , qn ) be positive probability distributions. Then f -divergence functional is defined by ( ) n ∑ ri . (7) I f (r, q) := qi f q i i=1 And he stated that by defining f (0) := lim f (x); x→0+

( ) 0 0f := 0; 0

0f

(a ) 0

:= lim x f

(a )

x→0+

0

, a > 0,

(8)

we can also use the nonnegative probability distributions as well. In [18], L. Hor´vath, et al. gave the following functional based on the previous definition. Definition 2. Let I ⊂ R be an interval and let f : I → R be a function, let r = (r1 , . . . , rn ) ∈ Rn and q = (q1 , . . . , qn ) ∈ (0, ∞)n such that rs ∈ I, s = 1, . . . , n. qs Then they define the sum Iˆf (r, q) as ( ) n ∑ rs qs f Iˆf (r, q) := . q s s=1

(9)

We apply Theorem 1.1 to Iˆf (r, q) Theorem 2.1. Assume (H1 ), let I ⊂ R be an interval and let r = (r1 , . . . , rn ) and q = (q1 , . . . , qn ) are in (0, ∞)n such that rs ∈ I, s = 1, . . . , n. qs (i) If f : I → R is a convex function, then ( ) n ∑ rs [1] [1] [1] = A[1] Iˆf (r, q) = qs f m,1 ≥ Am,2 ≥ · · · ≥ Am,m−1 ≥ Am,m q s s=1 ( ∑n ) n rs ∑ ≥ f ∑ns=1 qs . s=1 qs s=1

(10)

where A[1] m,l =

(m − 1)! (l − 1)! (i

∑ 1 ,...,il )∈Il

⎛ ⎞ ⎛ ∑l l ∑ j=1 qi j ⎠f⎜ η Im ,l (i 1 , . . . , il ) ⎝ ⎝ ∑l α j=1 Im ,i j

ri j α Im ,i j qi j

j=1 α Im ,i j

⎞ ⎟ ⎠

(11)

If f is a concave function, then inequality signs in (10) are reversed. Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

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K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

(ii) If f : I → R is a function such that x → x f (x)(x ∈ I ) is convex, then ( n ) ( n ∑ ∑ rs ∑n rs f s=1

=

( rs f

s=1

[2] [2] [2] ≤ A[2] m,m ≤ Am,m−1 ≤ · · · ≤ Am,2 ≤ Am,1

s=1 qs

s=1 n ∑

) (12)

rs qS

)

= Iˆid f (r, q)

where ⎛ A[2] m,l =

(m − 1)! (l − 1)! (i

∑ 1 ,...,il )∈Il

ri j

⎛∑

l

⎜ j=1 × f ⎝∑ l

α Im ,i j qi j

j=1 α Im ,i j

Proof. (i) Consider ps =

f

( n ∑ s=1

qs

η Im ,l (i 1 , . . . , il ) ⎝

rs ∑n s=1 qs qs

∑nqs

s=1 qs

j=1

j=1

qi j s=1 qs

α Im ,i j

ri j α Im ,i j qi j

j=1 α Im ,i j

⎞ ⎟ ⎠

⎟ ⎠.

and xs =

)

rs qs

in Theorem 1.1, we have

(m − 1)! ∑ η I ,l (i 1 , . . . , il ) (l − 1)! (i ,...,i )∈I m l l 1 qi ⎞ j

≤ ··· ≤

∑l ⎜ j=1 ⎠f⎜ ⎜ ⎝ ∑ l ⎞

∑n

α Im ,i j

⎞ ⎛ ∑l j=1 ⎠⎜ ⎝ ∑l

⎞

⎛ ⎛ l ∑ ×⎝

l ∑ qi j

∑n r i=1 qi i j α Im ,i j qi j qi ∑n j i=1 qi

( ) n ⎟ ∑ qs rs ⎟ ∑n . f ⎟ ≤ ··· ≤ ⎠ q q s s i=1 s=1

(13)

j=1 α Im ,i j

And taking the sum

∑n

s=1 qi

we have (10).

(ii) Using f := id f (where “id” is the identity function) in Theorem 1.1, we have n ∑

ps x s f

s=1

( n ∑

) ps x s

(m − 1)! ∑ η I ,l (i 1 , . . . , il ) (l − 1)! (i ,...,i )∈I m l l ⎞ 1 ⎛∑ ⎞ pi j pi j l x x i i j=1 j j α Im ,i j α Im ,i j ⎟ ⎜ ⎟ ⎠ f ⎝ ∑l ⎠ pi j pi j

≤ ··· ≤

s=1

⎛ ⎞ ⎛ ∑l l ∑ j=1 pi j ⎠⎜ ×⎝ ⎝ ∑l α j=1 Im ,i j

j=1 α Im ,i j

≤ ··· ≤

n ∑

ps xs f (xs ).

j=1 α Im ,i j

(14)

s=1 Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

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s = 1, . . . , n, we get ) ( n n ∑ ∑ rs rs qs (m − 1)! ∑ qs ∑n ∑n η I ,l (i 1 , . . . , il ) f ≤ ··· ≤ (l − 1)! (i ,...,i )∈I m s=1 qs qs s=1 qs qs s=1 s=1 ∑nqs

Now on using ps =

and xs =

s=1 qs

rs , qs

1

qi ∑n j r l s=1 qs i j j=1 α Im ,i qi j j qi ∑n j l s=1 qs j=1 α Im ,i j

⎛ ⎛ l ∑ ×⎝

qi j ∑n

j=1

n ∑

⎞

α Im ,i j

⎛

l

l

qi ∑n j r l s=1 qs i j j=1 α Im ,i qi j j qi ∑n j l s=1 qs j=1 α Im ,i j

⎞

∑

∑

⎜ ⎠⎜ ⎜ ⎝ ∑

s=1 qs

⎞

⎟ ⎜ ⎟ ⎜ ⎟f⎜ ⎠ ⎝ ∑

( ) rs rs ∑n ≤ f . q q q S s=1 s s s=1 ∑ On taking sum ns=1 qs on both sides, we get (12).

⎟ ⎟ ⎟ ≤ ··· ⎠

qs

(15) □

3. I NEQUALITIES FOR S HANNON E NTROPY Definition 3 (See [18]). The Shannon entropy of positive probability distribution r = (r1 , . . . , rn ) is defined by S := −

n ∑

rs log(rs ).

(16)

s=1

Corollary 3.1. Assume (H1 ). (i) If q = (q1 , . . . , qn ) ∈ (0, ∞)n , and the base of log is greater than 1, then ( )∑ n n [3] [3] [3] ∑ S ≤ A[3] ≤ A ≤ · · · ≤ A ≤ A = log qs , n m,m m,m−1 m,2 m,1 s=1 qs s=1

(17)

where A[3] m,l

⎛ ⎞ ⎛ ⎞ l l ∑ ∑ qi j qi j (m − 1)! ∑ ⎠ log ⎝ ⎠. =− η I ,l (i 1 , . . . , il ) ⎝ (l − 1)! (i ,...,i )∈I m α α j=1 Im ,i j j=1 Im ,i j 1

l

(18)

l

If the base of log is between 0 and 1, then inequality signs in (17) are reversed. (ii) If q = (q1 , . . . , qn ) is a positive probability distribution and the base of log is greater than 1, then we have the estimates for the Shannon entropy of q [4] [4] [4] S ≤ A[4] m,m ≤ Am,m−1 ≤ · · · ≤ Am,2 ≤ Am,1 = log(n),

(19)

where A[4] m,l

⎛ ⎞ ⎛ ⎞ l l ∑ ∑ qi j qi j (m − 1)! ∑ ⎠ log ⎝ ⎠. =− η I ,l (i 1 , . . . , il ) ⎝ (l − 1)! (i ,...,i )∈I m α α I ,i I ,i m m j j j=1 j=1 1

l

l

Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

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K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

Proof. (i) Using f := log and r = (1, . . . , 1) in Theorem 2.1(i), we get (17). (ii) It is the special case of (i). □ Definition 4 (See [18]). The Kullback–Leibler divergence between the positive probability distribution r = (r1 , . . . , rn ) and q = (q1 , . . . , qn ) is defined by ( ) n ∑ ri . (20) D(r, q) := ri log qi s=1 Corollary 3.2. Assume (H1 ). (i) Let r = (r1 , . . . , rn ) ∈ (0, ∞)n and q := (q1 , . . . , qn ) ∈ (0, ∞)n . If the base of log is greater than 1, then ( n ) n ∑ ∑ rs [5] [5] [5] ∑n rs log ≤ A[5] m,m ≤ Am,m−1 ≤ · · · ≤ Am,2 ≤ Am,1 q s s=1 s=1 s=1 (21) ( ) n ∑ rs rs log = = D(r, q), qs s=1 where ⎛ A[5] m,l =

(m − 1)! (l − 1)! (i

∑

η Im ,l (i 1 , . . . , il ) ⎝

1 ,...,il )∈Il

ri j

⎛∑

l

α Im ,i j

⎜ j=1 × log ⎝ ∑ l

qi j

j=1 α Im ,i j

l ∑ qi j j=1

α Im ,i j

⎞ ⎛ ∑l j=1 ⎠⎜ ⎝ ∑l

ri j α Im ,i j qi j

j=1 α Im ,i j

⎞ ⎟ ⎠

⎞ ⎟ ⎠.

If the base of log is between 0 and 1, then inequality in (21) is reversed. (ii) If r and q are positive probability distributions, and the base of log is greater than 1, then we have [6] [6] [6] D(r, q) = A[6] m,1 ≥ Am,2 ≥ · · · ≥ Am,m−1 ≥ Am,m ≥ 0,

(22)

where ⎛

A[6] m,l =

l ∑

qi j (m − 1)! ∑ η Im ,l (i 1 , . . . , il ) ⎝ (l − 1)! (i ,...,i )∈I α j=1 Im ,i j 1

⎛∑

l

⎜ j=1 × log ⎝ ∑ l

l

⎞ ⎛ ∑l j=1 ⎠⎜ ⎝ ∑l

l

ri j α Im ,i j qi j

j=1 α Im ,i j

ri j α Im ,i j qi j

j=1 α Im ,i j

⎞ ⎟ ⎠

⎞ ⎟ ⎠.

If the base of log is between 0 and 1, then inequality signs in (22) are reversed. Proof. (i) On taking f := log in Theorem 2.1(ii), we get (21). Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

(ii) Since r and q are positive probability distributions therefore the smallest term in (21) is given as ) ( n n ∑ ∑ rs ∑n = 0. rs log s=1 qs s=1 s=1

∑n

s=1 rs

=

∑n

s qs

9

= 1, so

(23)

Hence for positive probability distribution r and q the (21) will become (22). □

4. I NEQUALITIES FOR R ÉNYI D IVERGENCE AND E NTROPY The Rényi divergence and entropy come from [28]. Definition 5. Let r := (r1 , . . . , rn ) and q := (q1 , . . . , qn ) be positive probability distributions, and let λ ≥ 0, λ ̸= 1. (a) The Rényi divergence of order λ is defined by ) ( n ∑ ( r i )λ 1 Dλ (r, q) := . (24) log qi λ−1 qi i=1 (b) The Rényi entropy of order λ of r is defined by ( n ) ∑ 1 λ Hλ (r) := log ri . 1−λ i=1

(25)

The Rényi divergence and the Rényi entropy can also be extended to non-negative probability distributions. If λ → 1 in (24), we have the Kullback–Leibler divergence, and if λ → 1 in (25), then we have the Shannon entropy. In the next two results, inequalities can be found for the Rényi divergence. Theorem 4.1. Assume (H1 ), let r = (r1 , . . . , rn ) and q = (q1 , . . . , qn ) are probability distributions. (i) If 0 ≤ λ ≤ µ such that λ, µ ̸= 1, and the base of log is greater than 1, then [7] [7] [7] Dλ (r, q) ≤ A[7] m,m ≤ Am,m−1 ≤ · · · ≤ Am,2 ≤ Am,1 = Dµ (r, q),

(26)

where ⎛ A[7] m,l

⎛ ⎞ l ⎜ (m − 1)! ∑ ∑ ri j 1 ⎜ ⎠ log ⎜ η Im ,l (i 1 , . . . , il ) ⎝ = ⎝ (l − 1)! µ−1 α I ,i m j (i ,...,i )∈I j=1 1

⎛ ⎜ ⎜ × ⎜ ⎝

l

l

⎞ )λ−1 ⎞ µ−1 λ−1 ⎟ ⎟ j=1 α Im ,i qi j j ⎟ ⎟ ⎟ ⎟ ri j ∑l ⎠ ⎟ ⎠ j=1 α I ,i

∑l

ri j

(

ri j

m j

The reverse inequalities hold in (26) if the base of log is between 0 and 1. Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

10

K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

(ii) If 1 < µ and the base of log is greater than 1, then ( ) n ∑ rs [8] [8] [8] D1 (r, q) = D(r, q) = rs log ≤ A[8] m,m ≤ Am,m−1 ≤ · · · ≤ Am,2 ≤ Am,1 q s (27) s=1 = Dµ (r, q), where ⎛

A[8] m,l

⎛ ⎞ l ∑ ⎜ (m − 1)! ∑ ri j 1 ⎠ log ⎜ η Im ,l (i 1 , . . . , il ) ⎝ =≤ ⎝ (l − 1)! µ−1 α I ,i m j (i ,...,i )∈I j=1 1

⎛

(µ − 1)

l

ri j

∑l

j=1 α Im ,i j

⎜ × exp ⎜ ⎝

l

( log

ri j

∑l

j=1 α Im ,i j

ri j qi j

) ⎞⎞ ⎟⎟ ⎟⎟ ⎠⎠

here the base of exp is the same as the base of log, and the reverse inequalities hold if the base of log is between 0 and 1. (iii) If 0 ≤ λ < 1, and the base of log is greater than 1, then [9] [9] [9] Dλ (r, q) ≤ A[9] m,m ≤ Am,m−1 ≤ · · · ≤ Am,2 ≤ Am,1 = D1 (r, q),

(28)

where A[9] m,l

⎞ ⎛ l ∑ ri j 1 (m − 1)! ∑ ⎠ = η I ,l (i 1 , . . . , il ) ⎝ λ − 1 (l − 1)! (i ,...,i )∈I m α I ,i m j j=1 ⎛ ⎜ ⎜ × log ⎜ ⎝

∑l

1

l

ri j

(

j=1 α Im ,i j

l

ri j qi j

ri j

∑l

j=1 α Im ,i j

)λ−1 ⎞

(29)

⎟ ⎟ ⎟. ⎠ µ−1

Proof. By applying Theorem 1.1 with I = (0, ∞), f : (0, ∞) → R, f (t) := t λ−1 ( )λ−1 rs ps := rs , xs := , s = 1, . . . , n, qs we have ( n ∑

( n ( ) ) µ−1 ∑ rs λ−1 λ−1 qs = rs ≤ qs s=1 s=1 ⎛ ⎞ l ∑ ri j (m − 1)! ∑ ⎠ ... ≤ η I ,l (i 1 , . . . , il ) ⎝ (l − 1)! (i ,...,i )∈I m α I ,i m j j=1 (

rs qs

)λ ) µ−1 λ−1

1

l

l

Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals 11

⎛

(

ri j

∑l

j=1 α Im ,i j

⎜ ⎜ ×⎜ ⎝

ri j

)λ−1 ⎞ µ−1 λ−1

qi j

⎟ ⎟ ⎟ ⎠

ri j

∑l

j=1 α Im ,i j

≤ ··· ≤

n ∑

(( rs

s=1

rs qs

)λ−1 ) µ−1 λ−1

,

(30)

if either 0 ≤ λ < 1 < β or 1 < λ ≤ µ, and the reverse inequality in (30) holds if 0 ≤ λ ≤ β < 1. By raising to power ( n ∑

( qs

s=1

1 , µ−1

we have from all

) ) 1 rs λ λ−1 ≤ qs ⎛

⎞ ⎛ l ⎜ (m − 1)! ∑ ∑ r ij ⎜ ⎠ ... ≤ ⎜ η Im ,l (i 1 , . . . , il ) ⎝ ⎝ (l − 1)! α I m ,i j (i ,...,i )∈I j=1 1

l

l

1

⎞ µ−1 )λ−1 ⎞ µ−1 λ−1 ri j ri j ∑l ⎜ j=1 α Im ,i j qi j ⎟ ⎟ ⎟ ⎟ ⎜ × ⎜ ⎟ ⎟ ri j ∑ l ⎝ ⎠ ⎟ ⎠ j=1 α I ,i ⎛

(

m j

1 ⎛ (( ) ) µ−1 ⎞ µ−1 ( n ) 1 n λ−1 λ−1 ∑ ∑ ( rs )µ µ−1 r s ⎠ ≤ ··· ≤ ⎝ . rs = qs qs qs s=1 s=1

(31)

Since log is increasing if the base of log is greater than 1, it now follows (26). If the base of log is between 0 and 1, then log is decreasing and therefore inequality in (26) is reversed. If λ = 1 and β = 1, we have (ii) and (iii) respectively by taking limit, when λ goes to 1. □ Theorem 4.2. Assume (H1 ), let r = (r1 , . . . , rn ) and q = (q1 , . . . , qn ) are probability distributions. If either 0 ≤ λ < 1 and the base of log is greater than 1, or 1 < λ and the base of log is between 0 and 1, then n ∑

1 ∑n

s=1

qs

( )λ rs qs

s=1

( qs

rs qs

)λ

( log

rs qs

)

[10] [10] [10] [11] = A[10] m,1 ≤ Am,2 ≤ · · · ≤ Am,m−1 ≤ Am,m ≤ Dλ (r, q) ≤ Am,m

(32)

[11] [11] ≤ A[11] m,m ≤ · · · ≤ Am,2 ≤ Am,1 = D1 (r, q) Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

12

K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

where 1

A[10] m,m = (λ − 1)

∑n

s=1

qs

( )λ rs qs

(m − 1)! ∑ η I ,l (i 1 , . . . , il ) (l − 1)! (i ,...,i )∈I m 1

l

l

⎛

ri j ∑l ⎛ ( )λ−1 ⎞ l ⎜ j=1 α Im ,i j ∑ ri j ri j ⎠ log ⎜ ×⎝ ⎜ ∑l ⎝ α q ij j=1 Im ,i j

(

)λ−1 ⎞

ri j qi j

ri j

j=1 α Im ,i j

⎟ ⎟ ⎟ ⎠

and A[11] m,m

⎛ ⎞ l ∑ ri j 1 (m − 1)! ∑ ⎠ = η I ,l (i 1 , . . . , il ) ⎝ λ − 1 (l − 1)! (i ,...,i )∈I m α I ,i m j j=1 ⎛

∑l

⎜ ⎜ × log ⎜ ⎝

1

l

ri j

(

j=1 α Im ,i j

∑l

l

ri j

)λ−1 ⎞

qi j ri j

j=1 α Im ,i j

⎟ ⎟ ⎟. ⎠

The inequalities in (32) are reversed if either 0 ≤ λ < 1 and the base of log is between 0 and 1, or 1 < λ and the base of log is greater than 1. Proof. We prove only the case when 0 ≤ λ < 1 and the base of log is greater than 1 and 1 the other cases can be proved similarly. Since λ−1 < 0 and the function log is concave then ( )λ−1 in Theorem 1.1, we have choose I = (0, ∞), f := log, ps = rs , xs := qrss

( n ) ( n ( ) ) ∑ ( r s )λ ∑ 1 1 rs λ−1 Dλ (r, q) = log qs = log rs λ−1 qs λ−1 qs s=1 ⎛ s=1 ⎞ l ∑ ri j 1 (m − 1)! ∑ ⎠ ≤ ··· ≤ η Im ,l (i 1 , . . . , il ) ⎝ λ − 1 (l − 1)! (i ,...,i )∈I α I ,i m j j=1 l l 1 ⎛ ( )λ−1 ⎞ ri j ri j ∑l ⎜ j=1 α Im ,i j qi j ⎟ ⎜ ⎟ log ⎜ ⎟ ri j ∑l ⎝ ⎠ j=1 α Im ,i j n

1 ∑ ≤ ··· ≤ rs log λ − 1 s=1

((

rs qs

)λ−1 ) =

n ∑ s=1

( rs log

rs qs

) = D1 (r, q)

(33)

and this gives the upper bound for Dλ (r, q). Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals 13

Since the base of log is greater than 1, the function x ↦→ x f (x) (x > 0) is convex therefore < 0 and Theorem 1.1 gives

1 1−λ

( n ) ∑ ( rs )λ 1 Dλ (r, q) = log qs λ−1 qs s=1 ( n ) ( n ) ∑ ( r s )λ ∑ ( rs )λ 1 ( qs log qs = ( )λ ) ∑n qs qs rs s=1 s=1 λ−1 s=1 qs qs 1

≥ ··· ≥ λ−1 ⎛ ⎜ ⎜ ⎜ ⎝

j=1 α Im ,i j

∑n

q s qs ⎛ )λ−1 ⎞ s=1

(

ri j

∑l

(

ri j qi j

j=1 α Im ,i j

1 λ−1

( ∑n

s=1

qs

1

l

(

ri j

∑l

⎜ ⎟ ⎜ ⎟ ⎟ log ⎜ ⎝ ⎠

ri j

∑l

⎛ ⎞ l ∑ ri j (m − 1)! ∑ ⎠ η Im ,l (i 1 , . . . , il ) ⎝ ( )λ ) (l − 1)! α I ,i rs m j (i ,...,i )∈I j=1

j=1 α Im ,i j

ri j

l

)λ−1 ⎞

qi j ri j

∑l

j=1 α Im ,i j

(m − 1)! ∑ η Im ,l (i 1 , . . . , il ) ( )λ ) (l − 1)! rs (i ,...,i )∈I 1

qs

l

l

⎛

ri j ∑l ⎛ ( )λ−1 ⎞ l ⎜ j=1 α Im ,i j ∑ ri j ri j ⎝ ⎠ log ⎜ ⎜ ∑l ⎝ α q I ,i i m j j j=1

(

ri j

)λ−1 ⎞

qi j ri j

j=1 α Im ,i j

≥ ··· ≥ ( )λ−1 ( )λ−1 n rs 1 ∑ rs rs log ∑n λ − 1 s=1 qs qs n ∑

1 ∑n

s=1

qs

( )λ rs qs

s=1

( qs

rs qs

)λ

(

rs log qs

which give the lower bound of Dλ (r, q).

⎟ ⎟ ⎟ ⎠

1 ( )λ−1

s=1 r s

=

⎟ ⎟ ⎟= ⎠

rs qs

) (34)

□

By using Theorems 4.1, 4.2 and Definition 5, some inequalities of Rényi entropy are obtained. Let n1 = ( n1 , . . . , n1 ) be a discrete probability distribution. Corollary 4.3. Assume (H1 ), let r = (r1 , . . . , rn ) and q = (q1 , . . . , qn ) are positive probability distributions. (i) If 0 ≤ λ ≤ µ, λ, µ ̸= 1, and the base of log is greater than 1, then ( ) 1 [12] [12] [12] Hλ (r) = log(n) − Dλ r, ≥ A[12] (35) m,m ≥ Am,m ≥ · · · Am,2 ≥ Am,1 = Hµ (r), n Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

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K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

where ⎛ A[12] m,l

⎛ ⎞ ⎜ l ∑ ⎜ (m − 1)! ∑ ri j 1 ⎠ η Im ,l (i 1 , . . . , il ) × ⎝ = log ⎜ ⎜ (l − 1)! 1−µ α I ,i m ⎝ j (i ,...,i )∈I j=1 1

⎛

riλ

∑l

j

⎜ j=1 × ⎜ ⎝ ∑l

α Im ,i j ri j

j=1 α Im ,i j

l

⎞ µ−1 λ−1

⎞

⎟ ⎟ ⎠

⎟ ⎟ ⎟. ⎟ ⎠

l

The reverse inequalities hold in (35) if the base of log is between 0 and 1. (ii) If 1 < µ and base of log is greater than 1, then S=−

n ∑

[13] [13] [13] pi log( pi ) ≥ A[13] m,m ≥ Am,m−1 ≥ · · · ≥ Am,2 ≥ Am,1 = Hµ (r)

(36)

s=1

where ⎛

A[13] m,l

⎛ ⎞ l ∑ ∑ r (m − 1)! 1 i ⎜ j ⎠ log ⎝ η I ,l (i 1 , . . . , il ) ⎝ = log(n) + 1−µ (l − 1)! (i ,...,i )∈I m α I ,i j m j=1 1

⎛

(µ − 1)

⎜ × exp ⎝

ri j

∑l

j=1 α Im ,i j

∑l

ri j

j=1 α Im ,i j

l

( log nri j

l

) ⎞⎞ ⎟⎟ ⎠⎠ ,

the base of exp is the same as the base of log. The inequalities in (36) are reversed if the base of log is between 0 and 1. (iii) If 0 ≤ λ < 1, and the base of log is greater than 1, then [14] [14] [14] Hλ (r) ≥ A[14] m,m ≥ Am,m−1 ≥ · · · ≥ Am,2 ≤ Am,1 = S,

(37)

where A[14] m,m =

1 (m − 1)! 1 − λ (l − 1)! (i ,...,i )∈I l l 1 ⎛ ⎞ riλ ∑l j ⎜ j=1 α Im ,i j ⎟ ⎜ ⎟ × log ⎝ ∑ ri j ⎠ . l ∑

⎛ ⎞ l ∑ ri j ⎠ η Im ,l (i 1 , . . . , il ) ⎝ α j=1 Im ,i j

(38)

j=1 α Im ,i j

The inequalities in (37) are reversed if the base of log is between 0 and 1. Proof. (i) Suppose q = 1 Dλ (r, q) = λ−1

1 n

then from (24), we have ( n ) ∑ λ−1 λ log n rs = log(n) + s=1

( n ) ∑ 1 λ log rs , λ−1 s=1

(39)

Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals 15

therefore we have 1 Hλ (r) = log(n) − Dλ (r, ). n Now using Theorem 4.1(i) and (40), we get ) ( 1 1 ≥ · · · ≥ log(n) − Hλ (r) = log(n) − Dλ r, n µ−1 ⎛

(40)

⎜ ⎜ µ−1 (m − 1)! ∑ η I ,l (i 1 , . . . , il ) × log ⎜ ⎜n (l − 1)! (i ,...,i )∈I m ⎝ l l 1 ⎛ ⎛ ⎞ ∑l l ∑ ri j ⎜ j=1 ⎠ ⎜∑ ×⎝ ⎝ l α j=1 Im ,i j

riλ

j

α Im ,i j ri j

j=1 α Im ,i j

⎞ µ−1 λ−1

⎞

⎟ ⎟ ⎠

⎟ ⎟ ⎟ ⎟ ⎠

≥ · · · ≥ log(n) − Dµ (r, q) = Hµ (r),

(41)

□

(ii) and (iii) can be proved similarly.

Corollary 4.4. Assume (H1 ) and let r = (r1 , . . . , rn ) and q = (q1 , . . . , qn ) are positive probability distributions. If either 0 ≤ λ < 1 and the base of log is greater than 1, or 1 < λ and the base of log is between 0 and 1, then n ∑ 1 [15] [15] [15] − ∑n rsλ log(rs ) = A[15] m,1 ≥ Am,2 ≥ · · · ≥ Am,m−1 ≥ Am,m λ s=1 r s s=1 (42) [16] [16] [16] ≥ Hλ (r) ≥ A[16] m,m ≥ Am,m−1 ≥ · · · Am,2 ≥ Am,1 = H (r) ,

where A[15] m,l =

(λ − 1)

(m − 1)! λ (l − 1)! r s=1 s (i

1 ∑n

∑ 1 ,...,il )∈Il

⎛ ⎞ l riλj ∑ ⎠ η Im ,l (i 1 , . . . , il ) ⎝ α j=1 Im ,i j ⎛

∑l

⎜ λ−1 j=1 log ⎜ ⎝n ∑

riλ

⎞

j

α Im ,i j

ri j l j=1 α Im ,i j

⎟ ⎟ ⎠

and

A[16] m,1 =

1 (m − 1)! ∑ 1 − λ (l − 1)! (i ,...,i )∈I 1

l

l

⎛ ⎛ ⎞ ∑l l ∑ ⎜ j=1 ri j ⎠ log ⎜ ∑ η Im ,l (i 1 , . . . , il ) ⎝ ⎝ l α I ,i m j j=1

riλ

j

α Im ,i j ri j

j=1 α Im ,i j

⎞ ⎟ ⎟. ⎠

The inequalities in (42) are reversed if either 0 ≤ λ < 1 and the base of log is between 0 and 1, or 1 < λ and the base of log is greater than 1. Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

16

K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

Proof. The proof is similar to Corollary 4.3 by using Theorem 4.2. □ 5. I NEQUALITIES BY USING Z IPF –M ANDELBROT LAW In probability theory and statistics, the Zipf–Mandelbrot law is a distribution. It is a power law distribution on ranked data, named after the linguist G. K. Zipf who suggests a simpler distribution called Zipf’s law. The Zipf’s law is defined as follows (see [32]). Definition 6. Let N be a number of elements, s be their rank and t be the value of exponent characterizing the distribution. Zipf’s law then predicts that out of a population of N elements, the normalized frequency of element of rank s, f (s, N , t) is 1

f (s, N , t) = ∑ Ns

t

1 j=1 j t

.

(43)

The Zipf–Mandelbrot law is defined as follows (see [22]). Definition 7. Zipf–Mandelbrot law is a discrete probability distribution depending on three parameters N ∈ {1, 2, . . . , }, q ∈ [0, ∞) and t > 0, and is defined by 1 f (s; N , q, t) := , s = 1, . . . , N , (44) (s + q)t HN ,q,t where HN ,q,t =

N ∑ j=1

1 . ( j + q)t

(45)

If the total mass of the law is taken over all N, then for q ≥ 0, t > 1, s ∈ N, density function of Zipf–Mandelbrot law becomes 1 f (s; q, t) = , (46) (s + q)t Hq,t where Hq,t =

∞ ∑ j=1

1 . ( j + q)t

(47)

For q = 0, the Zipf–Mandelbrot law (44) becomes Zipf’s law (43). Conclusion 5.1. Assume (H1 ), let r be a Zipf–Mandelbrot law, by Corollary 4.3(iii), we get: If 0 ≤ λ < 1, and the base of log is greater than 1, then ( ) n 1 1 ∑ 1 Hλ (r) = log ≥ ··· ≥ 1−λ HNλ ,q,t s=1 (s + q)λs ⎛ ⎞ l ∑ ∑ 1 1 (m − 1)! ⎠ η I ,l (i 1 , . . . , il ) ⎝ 1 − λ (l − 1)! (i ,...,i )∈I m α (i + q)HN .q,t I ,i j m j j=1 l l 1 ∑l ⎛ ⎞ 1 λs j=1 α (i −q) Im ,i j j 1 ⎠ ≥ ··· ≥ × log ⎝ λ−1 ∑l 1 HN ,q,t j=1 α (i −q)s Im ,i j

j

Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals 17

t HN ,q,t

N ∑ log(s + q) s=1

(s + q)t

+ log(HN ,q,t ) = S.

(48)

The inequalities in (48) are reversed if the base of log is between 0 and 1. Conclusion 5.2. Assume (H1 ), let r1 and r2 be the Zipf–Mandelbort law with parameters N ∈ {1, 2, . . .}, q1 , q2 ∈ [0, ∞) and s1 , s2 > 0, respectively, then from Corollary 3.2(ii), we have if the base of log is greater than 1, then ( ) n ∑ (s + q2 )t2 HN ,q2 ,t2 1 ¯ 1 , r2 ) = D(r log ≥ ··· (s + q1 )t1 HN ,q1 ,t1 (s + q1 )t1 HN ,q2 ,t1 s=1 (m − 1)! ∑ η I ,l (i 1 , . . . , il ) ≥ (l − 1)! (i ,...,i )∈I m l l 1 ⎞ ⎛ 1 ⎛ ⎞ ∑ (i j +q1 )t1 H N ,q ,t l 1 1 1 l ⎟ ⎜ j=1 ∑ α Im ,i j (i j +q2 )t2 H N ,q2 ,t2 ⎟ ⎜ ⎝ ⎠ × ⎟ ⎜ 1 ⎠ ⎝∑ α Im ,i j (i +q )t2 H j=1

l j=1

⎛

1 (i j +q1 )t1 H N ,q ,t 1 1 j=1 α Im ,i j

j

2

N ,q2 ,t2

α Im ,i j

⎞

∑l ⎜ ⎜ × log ⎜ ⎝∑

1

(i j +q2 )t2 H N ,q ,t l 2 2 j=1 α Im ,i

⎟ ⎟ ⎟ ⎠

j

≥ · · · ≥ 0.

(49)

The inequalities in (49) are reversed if the base of log is between 0 and 1.

6. S HANNON ENTROPY, Z IPF –M ANDELBROT LAW AND HYBRID Z IPF –M ANDELBROT LAW Here we maximize the Shannon entropy using method of Lagrange multiplier under some equations constraints and get the Zipf–Mandelbrot law. Theorem 6.1. If J = {1, 2, . . . , N }, for a given q ≥ 0 a probability distribution that maximizes the Shannon entropy under the constraints ∑ ∑ rs = 1, rs (ln(s + q)) := Ψ , s∈J

s∈J

is Zipf–Mandelbrot law. Proof. If J = {1, 2, . . . , N }, we set the Lagrange multipliers λ and t and consider the expression ( N ) ( N ) N ∑ ∑ ∑ ˜ S=− rs ln rs − λ rs − 1 − t rs ln(s + q) − Ψ s=1

s=1

s=1

Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

18

K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

Just for the sake of convenience, replace λ by ln λ − 1, thus the last expression gives ( N ) ( N ) N ∑ ∑ ∑ ˜ S=− rs ln rs − (ln λ − 1) rs − 1 − t rs ln(s + q) − Ψ s=1

s=1

s=1

From ˜ Srs = 0, for s = 1, 2, . . . , N , we get rs =

1 , λ(s + q)t

and on using the constraint ) N ( ∑ 1 λ= (s + 1)t s=1

∑N

s=1 r s

= 1, we have

where t > 0, concluding that rs =

1 , (s + q)t HN ,q,t

s = 1, 2, . . . , N .

□

Remark 6.2. Observe that the Zipf–Mandelbrot law and Shannon Entropy can be bounded from above (see [23]). S=−

N ∑

f (s, N , q, t) ln f (s, N , q, t) ≤ −

s=1

N ∑

f (s, N , q, t) ln qs

s=1

where (q1 , . . . , q N ) is a positive N -tuple such that

∑N

s=1 qs

= 1.

Theorem 6.3. If J = {1, . . . , N }, then probability distribution that maximizes Shannon entropy under constraints ∑ ∑ ∑ rs = 1, rs ln(s + q) := Ψ , srs := η s∈J

s∈J

s∈J

is hybrid Zipf–Mandelbrot law given as rs =

ws , (s + q)k Φ ∗ (k, q, w)

s ∈ J,

where Φ J (k, q, w) =

∑ s∈J

ws . (s + q)k

Proof. First consider J = {1, . . . , N }, we set the Lagrange multiplier and consider the expression ( N ) ( N ) N ∑ ∑ ∑ S˜ = − rs ln rs + ln w srs − η − (ln λ − 1) rs − 1 s=1

−k

s=1

( N ∑

s=1

) rs ln(s + q) − Ψ .

s=1 Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals 19

On setting S˜rs = 0, for s = 1, . . . , N , we get − ln rs + s ln w − ln λ − k ln(s + q) = 0, ∑N ws , and we recognize this as the partial sum of after solving for rs , we get λ = s=1 (s+q)k Lerch’s transcendent that we will denote by Φ N∗ (k, q, w) =

N ∑ s=1

ws (s + q)k

with w ≥ 0, k > 0.

□ Remark 6.4. Observe that for Zipf–Mandelbrot law, Shannon entropy can be bounded from above (see [23]). S=−

N ∑

f h (s, N , q, k) ln f h (s, N , q, k) ≤ −

s=1

N ∑

f h (s, N , q, k) ln qs

s=1

where (q1 , . . . , q N ) is any positive N -tuple such that

∑N

s=1 qs

= 1.

Under the assumption of Theorem 2.1(i), define the non-negative functionals as follows: Θ3 ( f ) =

[1] Am,r

) n ( ∑n rs ∑ qs , − f ∑ns=1 s=1 qs s=1

[1] [1] Θ4 ( f ) = Am,r − Am,k ,

r = 1, . . . , m,

1 ≤ r < k ≤ m.

(50) (51)

Under the assumption of Theorem 2.1(ii), define the non-negative functionals as follows: ( n ) (∑ ) n ∑ s=1 r s [2] ∑ , r = 1, . . . , m, (52) Θ5 ( f ) = Am,r − rs f n s=1 qs s=1 [2] [2] Θ6 ( f ) = Am,r − Am,k ,

1 ≤ r < k ≤ m.

(53)

Under the assumption of Corollary 3.1(i), define the following non-negative functionals: n ∑ Θ7 ( f ) = A[3] + qi log(qi ), r = 1, . . ., n (54) m,r i=1

Θ8 ( f ) =

A[3] m,r

− A[3] m,k , 1 ≤ r < k ≤ m.

(55)

Under the assumption of Corollary 3.1(ii), define the following non-negative functionals as Θ9 ( f ) = A[4] m,r − S, r = 1, . . . , m

(56)

[4] Θ10 ( f ) = A[4] m,r − Am,k , 1 ≤ r < k ≤ m.

(57)

Under the assumption of Corollary 3.2(i), let us define the non-negative functionals as follows: ( n ) n ∑ ∑ rn [5] rs log log ∑n Θ11 ( f ) = Am,r − , r = 1, . . ., m (58) s=1 qs s=1 s=1 [5] Θ12 ( f ) = A[5] m,r − Am,k , 1 ≤ r < k ≤ m.

(59)

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K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

Under the assumption of Corollary 3.2(ii), define the non-negative functionals as follows: [6] Θ13 ( f ) = A[6] m,r − Am,k , 1 ≤ r < k ≤ m.

(60)

Under the assumption of Theorem 4.1(i), consider the following functionals: Θ14 ( f ) = A[7] m,r − Dλ (r, q), r = 1, . . . , m

(61)

[7] Θ15 ( f ) = A[7] m,r − Am,k , 1 ≤ r < k ≤ m.

(62)

Under the assumption of Theorem 4.1(ii), consider the following functionals: Θ16 ( f ) = A[8] m,r − D1 (r, q), r = 1, . . . , m

(63)

A[8] m,k ,

(64)

Θ17 ( f ) =

A[8] m,r

−

1 ≤ r < k ≤ m.

Under the assumption of Theorem 4.1(iii), consider the following functionals: Θ18 ( f ) = A[9] m,r − Dλ (r, q), r = 1, . . . , m

(65)

[9] Θ19 ( f ) = A[9] m,r − Am,k , 1 ≤ r < k ≤ m.

(66)

Under the assumption of Theorem 4.2 consider the following non-negative functionals: Θ20 ( f ) = Dλ (r, q) − A[10] m,r ,

r = 1, . . . , m

(67)

Θ21 ( f ) =

A[10] m,k

− A[10] m,r , 1 ≤ r < k ≤ m.

(68)

Θ22 ( f ) =

A[11] m,r

− Dλ (r, q), r = 1, . . . , m

(69)

[11] Θ23 ( f ) = A[11] m,r − Am,r , 1 ≤ r < k ≤ m.

Θ24 ( f ) = A[11] m,r −

A[10] m,k ,

r = 1, . . . , m, k = 1, . . . , m.

(70) (71)

Under the assumption of Corollary 4.3(i), consider the following non-negative functionals: Θ25 ( f ) = Hλ (r) − A[12] m,r , r = 1, . . . , m

(72)

[12] Θ26 ( f ) = A[12] m,k − Am,r , 1 ≤ r < k ≤ m.

(73)

Under the assumption of Corollary 4.3(ii), consider the following functionals: Θ27 ( f ) = S − A[13] m,r , r = 1, . . . , m

(74)

[13] Θ28 ( f ) = A[13] m,k − Am,r , 1 ≤ r < k ≤ m.

(75)

Under the assumption of Corollary 4.3(iii), consider the following functionals: Θ29 ( f ) = Hλ (r) − A[14] m,r , r = 1, . . . , m Θ30 ( f ) =

A[14] m,k

−

A[14] m,r ,

1 ≤ r < k ≤ m.

(76) (77)

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Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals 21

Under the assumption of Corollary 4.4, define the following functionals: Θ31 = A[15] m,r − Hλ (r), r = 1, . . . , m

(78)

[15] Θ32 = A[15] m,r − Am,k , 1 ≤ r < k ≤ m.

(79)

Θ33 = Hλ (r) − A[16] m,r , r = 1, . . . , m

(80)

Θ34 =

A[16] m,k

−

A[16] m,r ,

1 ≤ r < k ≤ m.

[16] Θ35 = A[15] m,r − Am,k , r = 1, . . . , m, k = 1, . . . , m.

(81) (82)

7. G ENERALIZATION OF REFINEMENT OF J ENSEN ’ S , R ÉNYI AND S HANNON TYPE INEQUALITIES F INK ’ S I DENTITY AND A BEL –G ONTSCHAROFF G REEN FUNCTION In [13], A. M. Fink gave the following result. Let f : [α1 , α2 ] → R, where [α1 , α2 ] be an interval, is a function such that f (n−1) is absolutely continuous then the following identity holds: ∫ α2 n f (ζ ) dζ f (z) = α2 − α1 α1 ( ) n−1 ∑ n − λ f (λ−1) (α2 )(z − α2 )λ − f (λ−1) (α1 )(z − α1 )λ + λ! α2 − α1 λ=1 ∫ α2 1 + (z − ζ )n−1 Fαα12 (ζ, z) f (n) (ζ ) dζ, (83) (n − 1)!(α2 − α1 ) α1 where Fαα12 (ζ, z)

{ =

ζ − α1 , ζ − α2 ,

α1 ≤ ζ ≤ z ≤ α2 ; α1 ≤ z < ζ ≤ α2 .

(84)

The complete reference about Abel–Gontscharoff polynomial and theorem for ‘two-point right focal’ problem is given in [1]. The Abel–Gontscharoff polynomial for ‘two-point right focal’ interpolating polynomial for n = 2 can be given as ∫ α2 G 1 (z, w) f ′′ (w) dw, (85) f (z) = f (α1 ) + (z − α1 ) f ′ (α2 ) + α1

where G 1 (z, w) =

{

α1 − w, α1 − z,

α1 ≤ w ≤ z; z ≤ w ≤ α2 .

In [8], S. I. Butt et al. gave some new types of Green functions defined as { α − z, α1 ≤ w ≤ z; G 2 (z, w) = 2 α2 − w, z ≤ w ≤ α2 , { z − α1 , α1 ≤ w ≤ z; G 3 (z, w) = w − α1 , z ≤ w ≤ α2 , { α − w, α1 ≤ w ≤ z; G 4 (z, w) = 2 α2 − z, z ≤ w ≤ α2 ,

(86)

(87) (88) (89)

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K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

Fig. 1. Graph of Green functions for fix w.

Fig. 1 shows the graph of Green functions G i (z, w), i = 1, 2, 3, 4 defined in (86)–(89) respectively for fixed value of w. They also introduced some new Abel–Gontscharoff type identities by using these new Green functions in the following lemma. Lemma A. Let f : [α1 , α2 ] be a twice differentiable function and G k (k = 2, 3, 4) be the ‘two-point right focal problem’-type Green functions defined by (87)–(89). Then the following identities hold: ∫ α2 ′ f (z) = f (α2 ) − (α2 − z) f (α1 ) − G 2 (z, w) f ′′ (w) dw, (90) α1 ∫ α2 f (z) = f (α2 ) − (α2 − α1 ) f ′ (α2 ) + (z − α1 ) f ′ (α1 ) + G 3 (z, w) f ′′ (w) dw, (91) ∫α1α2 ′ ′ f (z) = f (α1 ) + (α2 − α1 ) f (α1 ) − (α2 − z) f (α2 ) + G 4 (z, w) f ′′ (w) dw. (92) α1

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Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals 23

Theorem 7.1. Assume (H1 ), and let f : I = [α1 , α2 ] → R be a function such that for m ≥ 3 (an integer) f (m−1) is absolutely continuous. Also, let x1 , . . . , xn ∈ I , p1 , . . . , pn , be ∑n α positive real numbers such that i=1 pi = 1. Assume that Fα12 , G k (k = 1, 2, 3, 4) and Θi (i = 1, . . . , 35) are the same as defined in (84), (86)–(89), (2), (3), (50)–(82) respectively. Then: (i) For k = 1, 3, 4 we have the following identities: ( Θi ( f ) = (m − 2)

) ∫ α2 ( ) f ′ (α2 ) − f ′ (α1 ) Θi G k (·, w) dw α2 − α1 α1 ∫ α2 ( ) Θi G k (·, w)

1 α2 − α1 α1 m−3 ∑( m − 2 − λ )( ) × f (λ+1) (α2 )(w − α2 )λ − f (λ+1) (α1 )(w − α1 )λ dw λ! λ=1 ∫ α2 1 f (m) (ζ ) + (m − 3)!(α2 − α1 ) α1 (∫ α2 ) ( ) m−3 α2 × Θi G k (·, w) (w − ζ ) Fα1 (ζ, w) dw dζ, +

α1

i = 1, . . . , 35.

(93)

(ii) For k = 2 we have ) ∫ α2 ( ) f ′ (α2 ) − f ′ (α1 ) Θi G 2 (·, w) dw α2 − α1 α1 ∫ α2 ( ) (−1) + Θi G 2 (·, w) α2 − α1 α1 m−3 ∑( m − 2 − λ )( ) f (λ+1) (α2 )(w − α2 )λ − f (λ+1) (α1 )(w − α1 )λ dw × λ! λ=1 ∫ α2 (−1) + f (m) (ζ ) (m − 3)!(α2 − α1 ) α1 ) (∫ α2 ( ) × Θi G 2 (·, w) (w − ζ )m−3 Fαα12 (ζ, w) dw dζ, (

Θi ( f ) = (−1)(m − 2)

α1

i = 1, . . . , 35.

(94)

Proof. (i) Using Abel–Gontsharoff-type identities (85), (91), (92) in Θi ( f ), i = 1, . . . , 35, and using properties of Θi ( f ), we get ∫ Θi ( f ) =

α2

α1

( ) Θi G k (·, w) f ′′ (w) dw,

i = 1, 2.

(95)

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24

K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

From identity (83), we get ( ′ ) f (α2 ) − f ′ (α1 ) ′ f (w) = (m − 2) α2 − α1 m−3 ∑( m − 2 − λ )( f (λ) (α2 )(w − α2 )λ−1 − f (λ) (α2 )(w − α2 )λ−1 ) + λ! α2 − α1 λ=1 ∫ α2 1 (w − ζ )m−3 Fαα12 (ζ, w) f (m) (ζ ) dζ. + (m − 3)!(α2 − α1 ) α1

(96)

Using (95) and (96) and applying Fubini’s theorem we get the result (93) for k = 1, 3, 4. (ii) Substituting Abel–Gontscharoff-type inequality (90) in Θi ( f ), i = 1, . . . , 35, and following similar steps to (i), we get (94). □ Theorem 7.2. Assume (H1 ), and let f : I = [α1 , α2 ] → R be a function such that for m ≥ 3 (an integer) f (m−1) is absolutely continuous. Also, let x1 , . . . , xn ∈ I , p1 , . . . , pn are positive ∑n α real numbers such that i=1 pi = 1. Assume that Fα12 , G k (k = 1, 2, 3, 4) and Θi (i = 1, 2) are the same as defined in (84), (86)–(89) (2), (3), (50)–(82) respectively. For m ≥ 3 assume that ∫ α2 ( ) Θi G k (·, ζ ) (w − ζ )m−3 Fαα12 (ζ, w) dw ≥ 0, α1

ζ ∈ [α1 , α2 ],

i = 1, . . . , 35,

(97)

for k = 1, 3, 4. If f is an m-convex function, then (i) For k = 1, 3, 4, the following holds: ( ′ ) ∫ α2 ( ) f (α2 ) − f ′ (α1 ) Θi ( f ) ≥ (m − 2) Θi G k (·, w) dw α2 − α1 α1 ∫ α2 ( ) 1 + Θi G k (·, w) α2 − α1 α1 m−3 ∑ ( m − 2 − λ )( × f (λ+1) (α2 )(w − α2 )λ λ! λ=1 ) − f (λ+1) (α1 )(w − α1 )λ dw, i = 1, . . . , 35.

(98)

(ii) For k = 2, we have ( ′ ) ∫ α2 ( ) f (α2 ) − f ′ (α1 ) Θi G 2 (·, w) dw Θi ( f ) ≤ (−1)(m − 2) α1 ∫ α2 α2 − α1 ( ) (−1) + Θi G 2 (·, w) α2 − α1 α1 m−3 ∑ ( m − 2 − λ )( × f (λ+1) (α2 )(w − α2 )λ λ! λ=1 ) − f (λ+1) (α1 )(w − α1 )λ dw, i = 1, . . . , 35.

(99)

Proof. (i) Since f (m−1) is absolutely continuous on [α1 , α2 ], f (m) exists almost everywhere. Also, since f is m-convex therefore we have f (m) (ζ ) ≥ 0 for a.e. on [α1 , α2 ]. So, applying Theorem 7.1, we obtain (98). (ii) Similar to (i). □ Please cite this article in press as: K.A. Khan, T. Niaz, Ð. Peˇcari´c et al., Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals, Arab Journal of Mathematical Sciences (2019), https://doi.org/10.1016/j.ajmsc.2018.12.002.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals 25

Remark A. We can investigate the bounds for the identities related to the generalization of ˘ refinement of Jensen inequality using inequalities for the Ceby˘ sev functional and some results relating to the G¨russ and Ostrowski type inequalities can be constructed as given in Section 3 of [6]. Also we can construct the non-negative functionals from inequalities (98)– (99) and give related mean value theorems and we can construct the new families of m-exponentially convex functions and Cauchy means related to these functionals as given in Section 4 of [6]. ACKNOWLEDGMENTS The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. C OMPETING INTERESTS The authors declare that there is no conflict of interest regarding the publication of this paper. AUTHORS CONTRIBUTION All authors jointly worked on the results and they read and approved the final manuscript. R EFERENCES [1] R.P. Agarwal, P.J.Y. Wong, Error Inequalities in Polynomial Interpolation and their Applications, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993. [2] G. Anderson, Y. Ge, The size distribution of Chinese cities, Reg. Sci. Urban Econ. 35 (6) (2005) 756–776. [3] F. Auerbach, Das Gesetz der Bevölkerungskonzentration, Petermanns Geographische Mitteilungen 59 (1913) 74–76. [4] D. Black, V. Henderson, Urban evolution in the USA, J. Econ. Geogr. 3 (4) (2003) 343–372. [5] M. Bosker, S. Brakman, H. Garretsen, M. Schramm, A century of shocks: the evolution of the German city size distribution 1925-1999, Reg. Sci. Urban Econ. 38 (4) (2008) 330–347. [6] S.I. Butt, K.A. Khan, J. Peˇcari´c, Generaliztion of Popoviciu inequality for higher order convex function via Tayor’s polynomial, Acta Univ. Apulensis Math. Inform. 42 (2015) 181–200. [7] S.I. Butt, K.A. Khan, J. Peˇcari´c, Popoviciu type inequalities via Hermite’s polynomial, Math. Inequal. Appl. 19 (4) (2016) 1309–1318. [8] S.I. Butt, N. Mehmood, J. Peˇcari´c, New generalizations of Popoviciu type inequalities via new green functions and Fink’s identity, Trans A. Razmadze Math. Inst. 171 (3) (2017) 293–303. [9] S.I. Butt, J. Peˇcari´c, Weighted Popoviciu type inequalities via generalized Montgomery identities, Hrvat. Akad. Znan. I Umjet.: Mat. Znan. 19 (523) (2015) 69–89. [10] S.I. Butt, J. Peˇcari´c, Popoviciu’S Inequality for N -Convex Functions, Lap Lambert Academic Publishing, 2016. [11] I. Csiszár, Information measures: a critical survey, in: Tans. 7th Prague Conf. on Info. Th. Statist. Decis. Funct. Random Process and 8th European Meeting of Statist. Vol. B, Academia Prague, 1978, pp. 73–86. [12] I. Csiszár, Information-type measures of difference of probability distributions and indirect observations, Stud. Sci. Math. Hungar. 2 (1967) 299–318. [13] A.M. Fink, Bounds on the deviation of a function from its averages, Czechoslovak Math. J. 42 (2) (1992) 289–310. [14] L. Horváth, A method to refine the discrete Jense’s inequality for convex and mid-convex functions, Math. Comput. Modelling 54 (9–10) (2011) 2451–2459. [15] L. Horváth, K.A. Khan, J. Peˇcari´c, Combinatorial Improvements of Jensens Inequality / Classical and New Refinements of Jensens Inequality with Applications, in: Monographs in inequalities 8, Element, Zagreb, 2014. [16] L. Horváth, K.A. Khan, J. Peˇcari´c, Refinement of Jensen’s inequality for operator convex functions, Advances in Inequalities and Applications (2014).

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K.A. Khan, T. Niaz, Ð. Peˇcari´c et al.

[17] L. Horváth, J. Peˇcari´c, A refinement of discrete Jensen’s inequality, Math. Inequal. Appl. 14 (2011) 777–791. [18] L. Horváth, Ð. Peˇcari´c, J. Peˇcari´c, Estimations of f-and Rényi divergences by using a cyclic refinement of the Jensen’s inequality, Bull. Malays. Math. Sci. Soc. (2017) 1–14. [19] Y.M. Ioannides, H.G. Overman, Zipf’s law for cities: an empirical examination, Reg. Sci. Urban Econ. 33 (2) (2003) 127–137. [20] S. Kullback, Information Theory and Statistics, Courier Corporation, 1997. [21] S. Kullback, R.A. Leibler, On information and sufficiency, Ann. Math. Statist. 22 (1) (1951) 79–86. [22] N. Lovriˇcevi´c, Ð. Peˇcari´c, J. Peˇcari´c, Zipf-Mandelbrot law, f-divergences and the Jensen-type interpolating inequalities, J. Inequal. Appl. 2018 (1) (2018) 36. [23] M. Matic, C.E. Pearce, J. Peˇcari´c, Shannon’s and related inequalities in information theory, in: Survey on Classical Inequalities, Springer, Dordrecht, 2000, pp. 127–164. [24] N. Mehmood, R.P. Agarwal, S.I. Butt, J. Peˇcari´c, New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity, J. Inequal. Appl. 2017 (1) (2017) 108. [25] T. Niaz, K.A. Khan, J. Peˇcari´c, On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function, J. Inequal. Appl. 2017 (1) (2017) 254. [26] J. Peˇcari´c, K.A. Khan, I. Peri´c, Generalization of Popoviciu type inequalities for symmetric means generated by convex functions, J. Math. Comput. Sci. 4 (6) (2014) 1091–1113. [27] J. Peˇcari´c, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992. [28] A. Rényi, On measure of information and entropy, in: Proceeding of the Fourth Berkely Symposium on Mathematics, Statistics and Probability, 1960, pp. 547–561. [29] K.T. Rosen, M. Resnick, The size distribution of cities: an examination of the Pareto law and primacy, J. Urban Econ. 8 (2) (1980) 165–186. [30] K.T. Soo, Zipf’s Law for cities: a cross-country investigation, Reg. Sci. Urban Econ. 35 (3) (2005) 239–263. [31] D.V. Widder, Completely convex function and Lidstone series, Trans. Amer. Math. Soc. 51 (1942) (1942) 387–398. [32] G.K. Zipf, Human Behaviour and the Principle of Least-Effort, Addison-Wesley, Cambridge MA edn. Reading, 1949.

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