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Difference equations related to majorization theorems via Montgomery identity and Green’s functions with application to the Shannon entropy
Advances in Difference Equations volume 2020, Article number: 430 (2020)
Abstract
In this paper we give generalized results of a majorization inequality by using extension of the Montgomery identity and newly defined Green’s functions (Mehmood et al. in J. Inequal. Appl. 2017(1):108, 2017). We obtain a generalized majorization theorem for the class of nconvex functions. We use Csiszár fdivergence and generalized majorizationtype inequalities to obtain new generalized results. We further discuss our obtained generalized results in terms of the Shannon entropy and the Kullback–Leibler distance.
Introduction
The theory of majorization is perhaps most remarkable for its simplicity. It is a powerful, easytouse, and flexible mathematical tool which can be applicable to a wide number of fields. The key contributors in majorization are Dalton [14], Hardy et al. [16], Lorenz [30], Muirhead [36], and Schur [43]. Many important contributions were also made by other authors. Particularly, the comprehensive survey by Ando [8] gives alternative derivations, generalizations, and a different viewpoint. For an elementary discussion of majorization, see Marshall and Olkin’s monograph [32].
In 2018, Latif et al. [29] studied generalized results related to the majorization inequality by using Taylor’s polynomial in combination with newly introduced Green’s functions. In the same year, Siddique et al. [44] gave generalized majorization results via Lidstone’s polynomial and newly defined Green’s functions. The theory of majorization is widely used in many fields of application. In [21], Khan et al. presented significant material on majorization along with its applications in the field of information theory.
In this paper, our main goal is to obtain generalized results about majorization via new Green’s functions and an extension of the Montgomery identity. We further make connection of majorization with information theory and discuss our generalized majorization inequality in terms of divergences and entropies. The results we obtain in this paper are closely related to the contents given in [1–5]. Moreover, some related results with the present topic can also be found in [10, 11, 27, 41, 42].
The following definition of majorization is from [39, page 319].
Definition 1
Let \(\mathbf {x}= (x_{1},\ldots, x_{m} )\), \(\mathbf {y} = (y_{1},\ldots, y_{m} )\) be two real mtuples. Then we say that x majorizes y (denoted by \(\mathbf{x}\succ \mathbf{y}\)) if, for \(\lambda =1,2,\ldots,m1\),
holds and
where \(x_{[i]}\) and \(y_{[i]}\) denote their nonincreasing order.
Note that, in the definition of majorization, the original order of \(x_{i}\)s and \(y_{i}\)s plays no role because real mtuples can always be reordered nonincreasingly.
The following theorem is famed in literature as classical majorization theorem and is given in [33, page 11] (see also [39, page 320]).
Theorem 1
Let \(I=[\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\)and \(\mathbf {x}= (x_{1},\ldots, x_{m} )\), \(\mathbf {y} = (y_{1},\ldots, y_{m} )\in I^{m}\)be two nonincreasing mtuples. Then x majorizes y if and only if the following inequality holds:
where \(f:[\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\)is a continuous convex function.
A generalization of the aforementioned theorem is regarded as weighted majorization theorem and is proved by Fuchs in [15] (see also [39, page 323]).
Theorem 2
Let \(I=[\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\)and \(\mathbf {x}= (x_{1},\ldots, x_{m} )\), \(\mathbf {y} = (y_{1},\ldots, y_{m} )\in I^{m}\)be two nonincreasing mtuples. Let \(\mathbf{p}=(p_{1},\ldots, p_{m})\in \mathbb{R}^{m}\)be such that
and
Then the following inequality holds:
where \(f:[\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\)is a continuous convex function.
The following theorem represents an integral form of Theorem 2 and is in fact a simple consequence of Theorem 1 given in [37] (see also [39, page 328]).
Theorem 3
Let \(\phi , \psi : [a, b]\rightarrow [\zeta _{1},\zeta _{2}]\)be two continuous nonincreasing functions and \(p : [a, b]\rightarrow \mathbb{R}\)be continuous. If
and
hold, then
where \(f:[\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\)is a continuous convex function.
For other forms of an integral version and generalization of the majorization theorem, see [33, page 583], [9, 22, 24–26, 28, 31]. In this paper, we present our results for nonincreasing functions ϕ and ψ which satisfy the conditions of Theorem 3, but those results hold too for nondecreasing ϕ and ψ satisfying the following inequality:
and condition (6). For instance, see example in [33, page 584].
Definition 2
Let \(I=[\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\) and \(f:[\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\) be a function. Then nth order divided difference of f at distinct points \(x_{0},\ldots,x_{n}\in {}[ \zeta _{1},\zeta _{2}]\) is defined recursively (see [6, 39]) by
and
Note that nth order divided difference of a function f does not depend on the order of points.
We can extend this definition by considering the condition that some (or all) points coincide. Assuming that \(f^{(k1)}\) exists, we define
Popoviciu [40] initially discussed the notion of nconvexity. We follow the definition given by Karlin [20].
Definition 3
A function \(f:[\zeta _{1},\zeta _{2}]\rightarrow \mathbb{R}\) is nconvex, \(n\geq 0\) if
holds for all choices of \((n+1)\) distinct points \(x_{0},\ldots,x_{n}\in {}[ \zeta _{1},\zeta _{2}]\).
Aljinović et al. in [7] proved the following proposition which gives an extension of the Montgomery identity via Taylor’s formula.
Proposition 1
Let \(f: I \rightarrow \mathbb{R}\)be such that \(f^{(n1)}\)is absolutely continuous, where \(n\in \mathbb{N}\)and \(I\subset \mathbb{R}\)is an open interval. Then, for \(\zeta _{1}, \zeta _{2}\in I\)with \(\zeta _{1}<\zeta _{2}\), the following identity holds:
where
As a special case, for \(n=1\), the sum \(\sum_{k=0}^{n2}\cdots \) in (10) is empty, so (10) reduces to the following famous Montgomery identity (see [35]):
where \(P(x, s)\) is the Peano kernel given by
As stated in [34], the complete reference about Abel–Gontscharoff polynomial and a theorem for ‘twopoint right focal problem’ is given in [6].
Remark 1
Abel–Gontscharoff polynomial as a special choice for ‘twopoint right focal’ interpolating polynomial for \(n=2\) is as follows:
where \(G_{\varOmega , 2} (z, w): [\zeta _{1}, \zeta _{2}]\times [\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\) is Green’s function for ‘twopoint right focal problem’ given by
Motivated by Abel–Gontscharoff Green’s function for ‘twopoint right focal problem’, Mehmood et al. (see [34]) presented some new types of Green’s functions which are continuous as well as convex, as follows:
Let \([\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\). Define new types of Green’s functions \(G_{d}: [\zeta _{1}, \zeta _{2}]\times [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\), where \(d= 2, 3, 4\), as follows:
The following lemma, given by Mehmood et al. [34], will help us to obtain the new generalizations of majorization inequality.
Lemma 1
Let \(f: [\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\)be such that \(f \in C^{2}([\zeta _{1}, \zeta _{2}])\)and \(G_{d}\), (\(d=1,2,3,4\)) be Green’s functions given in (15)–(18) respectively. Then along with identity (14) the following identities hold:
We organize this paper in the following way:
In Sect. 2, we give generalized results of the majorization inequality and related bounds by using an extension of the Montgomery identity and new Green’s functions. In Sect. 3, we use Csiszár fdivergence and generalized majorizationtype inequalities to obtain new generalized results. We further discuss our obtained generalized results in terms of the Shannon entropy and the Kullback–Leibler distance.
Generalized majorized identities and related bounds via Montgomery identity and new Green’s functions
Before starting this section, we first define some notations which will be used throughout this article.
Majorization difference for a continuous convex function f is denoted as follows:
where x, y, and p are as defined in Theorem 2. Similarly, the integral majorization difference for a continuous convex function f is denoted as follows:
where ϕ, ψ, and p are as defined in Theorem 3.
The following theorem gives two equivalent statements between the weighted majorization inequality for a continuous convex function and the inequality involving newly defined Green’s functions.
Theorem 4
Let \(I=[\zeta _{1}, \zeta _{2}]\subset \mathbb{R}\)and \(\mathbf {x}= (x_{1},\ldots, x_{m} )\), \(\mathbf {y} = (y_{1},\ldots, y_{m} )\in I^{m}\)be two nonincreasing mtuples. Let \(\mathbf{p}=(p_{1},\ldots, p_{m})\in \mathbb{R}^{m}\)be such that it satisfies (3) and \(G_{d} \) (\(d=1,2,3,4\)) be as defined in (15)–(18) respectively. Then the following two assertions are equivalent:

(i)
If \(f : [\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\)is a continuous convex function, we have
$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, f( \cdot ) \bigr) \geq 0. \end{aligned}$$(24) 
(ii)
For \(s \in [\zeta _{1}, \zeta _{2}]\), the following inequality holds:
$$\begin{aligned} \mathbb{D} \bigl(\mathbf{x}, \mathbf{y}, \mathbf{p}, G_{d}(\cdot , s) \bigr)\geq 0, \quad d= 1, 2, 3, 4. \end{aligned}$$(25)
Proof
Let assertion (i) hold. Then \(G_{d}(\cdot , s)\) (\(s \in [\zeta _{1}, \zeta _{2}]\)), being continuous and convex, for fixed \(d=1, 2, 3, 4\) satisfies inequality (24), i.e., inequality (25) holds.
On the other hand, let assertion (ii) hold and \(f: [\zeta _{1}, \zeta _{2}]\rightarrow \mathbb{R}\) be a convex function such that \(f\in C^{2} ([\zeta _{1}, \zeta _{2}] )\). Then we can write the function f in the forms (14), (19), (20), and (21) for Green’s functions \(G_{d}\), \(d=1,2,3,4\), respectively. Hence using (3) and performing simple calculations, for all \(s \in [\zeta _{1}, \zeta _{2}]\), we have
Since f is convex, \(f^{\prime \prime }(s)\geq 0\) for all \(s \in [\zeta _{1}, \zeta _{2}]\). Also, inequality (25) holds, so from (26) we get inequality (24).
One must note that in this proof, the demand for the existence of the second derivative of f is not necessary ([39], page 172). We can directly eliminate this condition because it is possible to approximate uniformly continuous convex functions by convex polynomials. □
The following theorem gives weighted majorization difference by using extension of the Montgomery identity and newly defined Green’s functions.
Theorem 5
Let all the assumptions of Theorem 2hold. Let \(f: I \rightarrow \mathbb{R}\)be such that \(f^{(n1)}\)is absolutely continuous, where \(n\in \mathbb{N} \) (\(n\geq 3\)) and \(I\subset \mathbb{R}\)is an open interval. Then, for \(\zeta _{1}, \zeta _{2}\in I\)with \(\zeta _{1}<\zeta _{2}\)and for all \(s\in [\zeta _{1}, \zeta _{2}]\), we have the following identities:
where
and \(G_{d}\) (\(d=1,2,3,4\)) are Green’s functions defined in (15)–(18) respectively. Moreover, we have
where \(T_{n2}\)is as defined in (11).
Proof
Using identities (14), (19), (20), and (21), for fixed \(d=1,2,3,4\), into weighted majorization difference (22), we get
Now, using an extension of the Montgomery identity given in (10) for the function \(f(s)\) and after differentiating it twice with respect to s, we get
Applying Fubini’s theorem in the last term of (32), we get (27).
Also, replacing f by \(f''\) and n by \(n2\) (\(n\geq 3\)) in (10) and then rearranging indices, we have
which can also be written as
Using (34) in (30) and then applying Fubini’s theorem, we obtain (29). □
An integral version of Theorem 5 is as follows.
Theorem 6
Let all the assumptions of Theorem 3hold. Let \(f: I \rightarrow \mathbb{R}\)be such that \(f^{(n1)}\)is absolutely continuous, where \(n\in \mathbb{N} \) (\(n\geq 3\)) and \(I\subset \mathbb{R}\)is an open interval. Then, for \(\zeta _{1}, \zeta _{2}\in I\)with \(\zeta _{1}<\zeta _{2}\)and for all \(s\in [\zeta _{1}, \zeta _{2}]\), we have the following identities:
where \(\widehat{T}_{n}\)is as defined in (28)and \(G_{d}\) (\(d=1,2,3,4\)) are the Green’s functions defined in (15)–(18) respectively. Moreover,
where \(T_{n2}\)is as defined in (11).
Proof
Using identities (14), (19), (20), and (21), for fixed \(d=1,2,3,4\), into the integral weighted majorization difference (23) and following similar steps as in the proof of Theorem 5, we get required results. □
A refinement of the weighted majorizationtype inequality is presented in the following theorem.
Theorem 7
Let all the assumptions of Theorem 5hold. Let \(f: I\rightarrow \mathbb{R}\)be an nconvex function. If, for \(d=1,2,3,4\),
then
Moreover, if
then
If we reverse the sign of inequalities in (37) and (39), then inequalities (38) and (40) are also reversed.
Proof
As f is an nconvex function, it follows that \(f^{(n)}\geq 0\) (see [39], page 19 and page 293). Using this fact and substituting (37) and (39) in (27) and (29), respectively, we get the desired results. □
An integral version of Theorem 7 is as follows.
Theorem 8
Let all the assumptions of Theorem 6hold. Let \(f: I\rightarrow \mathbb{R}\)be an nconvex function. If, for \(d=1,2,3,4\),
then
Moreover, if
then
If we reverse the sign of inequalities in (41) and (43), then inequalities (42) and (44) are also reversed.
Proof
Using (41) and (43) in (35) and (36) respectively and following similar steps as in the proof of Theorem 7, we get required results. □
Theorem 9
Let all the assumptions of Theorem 5be true. If f is nconvex, where n is even, then inequalities (38) and (40) hold.
Proof
Since \(G_{d}\) is continuous as well as convex for \(d= 1, 2, 3, 4\), therefore from Theorem 2 we can write
Note that, when \(n2\) is even, \(\widehat{T}_{n2}(s, t)\) and \(T_{n2}(s, t)\) are nonnegative, so (37) and (39) hold. Now, using Theorem 7, we get the required results. □
An integral version of Theorem 9 is as follows.
Theorem 10
Let all the assumptions of Theorem 6be true. If f is nconvex, where n is even, then inequalities (42) and (44) hold.
Proof
Similar to the proof of Theorem 9. □
The following corollary gives a generalized majorization theorem, i.e., Fuchs’s theorem for nconvex functions.
Corollary 1
Let all the assumptions of Theorem 9be true. If the functions \(\mathcal{F}_{1}, \mathcal{F}_{2}:[\zeta _{1}, \zeta _{2}] \rightarrow \mathbb{R}\), given by
and
are convex, then the righthand sides of (42) and (44) are nonnegative, i.e., (4) is satisfied.
Proof
Note that inequalities (38) and (40) can be written as follows:
Now, the use of convex functions \(\mathcal{F}_{i}\), \(i=1, 2\), in (4) lead us to the nonnegativity of the righthand side of (48), which gives the required result. □
Remark 2
As given for previous theorems, we can obtain an integral version of Corollary 1, which is a generalization of the integral majorization theorem.
Remarks 1

(i)
We can obtain upper bounds like Grüss and Ostrowskitype inequalities for our obtained generalized identities. We can also present Lagrange and Cauchytype mean value theorems by using linear functionals deduced from our generalized results (see for example [29, 38, 44]).

(ii)
We can use an elegant method introduced by Jakšetić and Pečarć [18, 19] (see also [23, 34]) to give nexponential convexity, exponential convexity, and logconvexity, with the help of linear functionals deduced from our generalized results, on a given family with the same property for both discrete and integral cases. For more details, see [38].
Csiszár fdivergence for majorization
This section belongs to the study of generalized majorizationtype inequality (38) in the form of divergences and entropies. We use Csiszár fdivergence and generalized majorizationtype inequalities to obtain new generalized results. Moreover, results related to the Shannon entropy and the Kullback–Leibler (K–L) distance are also discussed.
The following notion of fdivergence was introduced by Csiszár in [12]. For more details, see [13].
Definition 4
Let \(f:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) be a convex function. If \(\mathbf {r}= (r_{1},\ldots, r_{m} )\) and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\) are two positive probability distributions, then the fdivergence functional is
Note that in the fdivergence functional, nonnegative probability distributions can also be used by defining
In [17], Horváth et al. considered the following functionality based on the previous definition.
Definition 5
Let \(J\subset \mathbb{R}\) be an interval and \(f: J \rightarrow \mathbb{R}\) be an nconvex function. Let \(\mathbf {r}= (r_{1},\ldots, r_{m} )\in \mathbb{R}^{m}\) and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\in \mathbb{R}_{+}^{m}\) such that \(\frac{r_{i}}{w_{i}}\in J\), \(i=1, 2,\ldots, m\). Then
Let \(\mathbf {r}= (r_{1},\ldots, r_{m} )\) and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\) be two mtuples. Onwards now, we use the following notations in this article, i.e.,
The following theorem connects the generalized majorizationtype inequality given in Theorem 9 and Csiszár fdivergence.
Theorem 11
Let \(f: I \rightarrow \mathbb{R}\)be such that \(f^{(n1)}\)is absolutely continuous, where \(n\in \mathbb{N}\) (\(n> 3\)) and \(I\subset \mathbb{R}\)is an open interval. Let \(G_{d}\) (\(d=1,2,3,4\)) be as defined in (15)–(18) respectively. Also, let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\), \(\mathbf {r}= (r_{1},\ldots, r_{m} )\in \mathbb{R}^{m}\), and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\in \mathbb{R}_{+}^{m}\). Let
for \(\lambda =1, 2,\ldots, m1\)and
with \(\frac{q_{i}}{w_{i}}, \frac{r_{i}}{w_{i}}\in I\) (\(i=1, 2,\ldots, m\)). If \(\frac{\mathbf {q}}{\mathbf {w}}\)and \(\frac{\mathbf {r}}{\mathbf {w}}\)are decreasing and f is an nconvex function for \(n=\mathrm{even}\) (\(n>3\)), then
Proof
Take \(x_{i}=\frac{q_{i}}{w_{i}}\), \(y_{i}=\frac{r_{i}}{w_{i}}\), and \(p_{i}=w_{i}>0 \) (\(i=1, 2,\ldots, m\)), then conditions (49) and (50) imply that conditions (2) and (3) hold. So, using these substitutions in (38), we get (51). □
Theorem 12
Let \(g:I\rightarrow \mathbb{R}\)be a function. If, for \(f(x):=xg(x)\), \(x\in I\), all the conditions of Theorem 11hold, then
Proof
Following the proof of Theorem 11 for \(f(x):=xg(x)\), we get (52). □
The notion of entropic measure of disorder and the theory of majorization are closely related. Next we present two special cases for majorization relations with the connection to entropic inequalities.
In the first case we discuss a generalized majorizationtype inequality with the entropy of a discrete probability distribution.
Definition 6
Let \(\mathbf {r}= (r_{1},\ldots, r_{m} )\) be a positive probability distribution. Then the Shannon entropy of r is defined as follows:
Note that the definition does not provide any problem for the zero probability case, because \(\lim_{x\rightarrow 0}x\log x=0\).
Corollary 2
Let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\), \(\mathbf {r}= (r_{1},\ldots, r_{m} )\in \mathbb{R}_{+}^{m}\), and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\)be a positive probability distribution such that conditions (49) and (50) hold with \(\frac{q_{i}}{w_{i}}, \frac{r_{i}}{w_{i}}\in I\) (\(i=1, 2,\ldots, m\)). If log has base b greater than 1 and \(\frac{\mathbf {q}}{\mathbf {w}}\)and \(\frac{\mathbf {r}}{\mathbf {w}}\)are decreasing, then for the Shannon entropy of w, the following estimate holds:
If log has base b between 0 and 1, then inequality (53) is reversed.
Proof
Take \(f(x):=\log x\), which is an nconvex function for \(n=\mathrm{even}\) (\(n>3\)) and \(q_{i}=1\) (\(i=1, 2,\ldots, m\)). Then, by using Theorem 11, we get (53). Moreover, for \(n=\mathrm{odd}\) (\(n>3\)), the inequality in (53) is reversed. □
Corollary 3
Let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\)and \(\mathbf {r}= (r_{1},\ldots, r_{m} )\)be two positive probability distributions such that conditions (49) and (50) hold with \(q_{i}\), \(r_{i}\in I\) (\(i=1, 2,\ldots, m\)). If log has base b greater than 1 and q and r are decreasing, then the relation between the Shannon entropies of q and r is given by the following estimate:
where for \(u=1, 2\), \((x\log x)'(\zeta _{u})=\frac{1}{\ln b}(1+\ln \zeta _{u})\)and \((x\log x)^{(k)}(\zeta _{u})= \frac{(1)^{k}(k2)!}{\zeta _{u}^{k1}\ln b}\), \(k\geq 2\). If log has base b between 0 and 1, then inequality (54) is reversed.
Proof
Take \(g(x):=\log x\) so that \(xg(x):=x\log x\) is an nconvex function for \(n=\mathrm{even}\) (\(n>3\)) and \(w_{i}=1 \) (\(i=1, 2,\ldots, m\)). Then, by using Theorem 12, we get (54). Moreover, for \(n=\mathrm{odd}\) (\(n>3\)), the inequality in (54) is reversed. □
In the second case we study a generalized majorizationtype inequality in terms of the K–L distance or relative entropy between two probability distributions.
Definition 7
Let \(\mathbf {r}= (r_{1},\ldots, r_{m} )\) and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\) be two positive probability distributions. Then the K–L distance between them is defined by
Corollary 4
Let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\), \(\mathbf {r}= (r_{1},\ldots, r_{m} )\), \(\mathbf {w}= (w_{1},\ldots, w_{m} ) \in \mathbb{R}_{+}^{m}\)such that conditions (49) and (50) hold with \(\frac{q_{i}}{w_{i}}, \frac{r_{i}}{w_{i}}\in I \) (\(i=1, 2,\ldots, m\)). If log has base b greater than 1 and \(\frac{\mathbf {q}}{\mathbf {w}}\)and \(\frac{\mathbf {r}}{\mathbf {w}}\)are decreasing, then
If log has base b between 0 and 1, then inequality (55) is reversed.
Proof
Take \(f(x):=\log x\), which is an nconvex function for \(n=\mathrm{even}\) (\(n>3\)). Then, by Theorem 11, we get (55). Moreover, for \(n=\mathrm{odd}\) (\(n>3\)), the inequality in (55) is reversed. □
Corollary 5
Let \(\mathbf {q}= (q_{1},\ldots, q_{m} )\), \(\mathbf {r}= (r_{1},\ldots, r_{m} )\), and \(\mathbf {w}= (w_{1},\ldots, w_{m} )\)be positive probability distributions such that conditions (49) and (50) hold with \(\frac{q_{i}}{w_{i}}, \frac{r_{i}}{w_{i}}\in I\) (\(i=1, 2,\ldots, m\)). If log has base b greater than 1 and \(\frac{\mathbf {q}}{\mathbf {w}}\)and \(\frac{\mathbf {r}}{\mathbf {w}}\)are decreasing, then the relation between the K–L distance of \((\mathbf {r}, \mathbf {w})\)and \((\mathbf {q}, \mathbf {w})\)is given by the following estimate:
where for \(u=1, 2\), \((x\log x)'(\zeta _{u})=\frac{1}{\ln b}(1+\ln \zeta _{u})\)and \((x\log x)^{(k)}(\zeta _{u})= \frac{(1)^{k}(k2)!}{\zeta _{u}^{k1}\ln b}\), \(k\geq 2\). If log has base b between 0 and 1, then inequality (56) is reversed.
Proof
Take \(g(x):=\log x\) so that \(xg(x):=x\log x\) is an nconvex function for \(n=\mathrm{even}\) (\(n>3\)). Then, by using Theorem 12, we get (56). Moreover, for \(n=\mathrm{odd}\) (\(n>3\)), the inequality in (56) is reversed. □
Remark 3
In Sect. 3, we use generalized majorizationtype inequality (38) to obtain results in terms of the Shannon entropy and the K–L distance. Following the same way, we can also give all these results related to the Shannon entropy and the K–L distance by using the generalized majorizationtype inequality given in (40).
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The authors are thankful to the anonymous referees for reading the manuscript and for giving fruitful comments and suggestions. The Ministry of Education and Science of Russian Federation has supported the research of the 4th author (Agreement No. 02.a03.21.0008).
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Siddique, N., Imran, M., Khan, K.A. et al. Difference equations related to majorization theorems via Montgomery identity and Green’s functions with application to the Shannon entropy. Adv Differ Equ 2020, 430 (2020). https://doi.org/10.1186/s13662020028847
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Keywords
 Majorization inequality
 Fuchs’s theorem
 Montgomery identity
 New Green’s functions
 Čebyšev functional
 Csiszár fdivergence
 Kullback–Leibler divergence
 Shannon entropy