Uniform Treatment of Jensen’s Inequality by Montgomery Identity

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect of n−convex functions; also, we give different versions of Jensen’s discrete inequality along with its converses for real weights. As an application, we give generalized variants of Hermite–Hadamard inequality. Montgomery identity has a great importance as many inequalities can be obtained fromMontgomery identity in q−calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf–Mandelbrot entropies.


Introduction
Convex functions have a great importance in mathematical inequalities, and the well-known Jensen's inequality is the characterization of convex functions. Jensen's inequality for differentiable convex functions plays a significant role in the field of inequalities as several other inequalities can be seen as special cases of it. One can find the application of Jensen's discrete inequality in discrete-time delay systems in [1].
Taking into consideration the tremendous applications of Jensen's inequality in various fields of mathematics and other applied sciences, the generalizations and improvements of Jensen's inequality have been a topic of supreme interest for the researchers during the last few decades as evident from a large number of publications on the topic (see [2][3][4] and the references therein). e well-known Jensen's inequality asserts that for the function Γ it holds that if Ψ is a convex function on interval I ⊂ R, where p J are positive real numbers and x J ∈ I(J � 1, . . . , m), while P m � m J�1 p J . However, the well-known integral analogue of Jensen's inequality is as follows. where ere are several inequalities coming from Jensen's inequality both in integral and discrete cases which can be obtained by varying conditions on the function Z and measure λ defined in eorem 1.
Montgomery identity is used in quantum calculus or q−calculus. ere are different identities of Montgomery, and several inequalities of Ostrowski type were formulated by using these identities. Budak and Sarikaya established the generalized Montgomery-type identities for differential mappings in [5]. Applications of Montgomery identity can be found in fractional integrals as well as in quantum integral operators. Here we utilize Montgomery's identity for the generalization of Jensen's inequality. In [6], Cerone and Dragomir developed a systematic study which produced some novel inequalities. Several interesting results related to inequalities and different types of convexity can be found in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. e class of convex functions is a very useful concept that has become a focus of interest for researchers in statistics, convex programming, and many other applied disciplines, as well as in inequality theory. e readers can find some motivated findings related to convex functions and some new integral inequalities in [22][23][24][25][26][27].
In [28], Khan et al. have mentioned about n-convex functions as follows.
Definition 1. A function f: I ⟶ R is called convex of order n or n-convex if for all choices of (n + 1) distinct points In the present paper, we will use Montgomery identity that is presented as following.

Generalization of Jensen's Integral Inequality by Using Montgomery Identity
Before giving our main results, we consider the following assumptions that we use throughout our paper: with Z and R n (x, s) as defined in (3) and (5), respectively, then we have Proof. As Ψ (n−1) is absolutely continuous for (n ≥ 1), we can use the representation of Ψ using Montgomery identity (4) and can calculate a dλ(ζ), we get the following generalized identity involving real Stieltjes measure: Finally, by our assumption, Ψ (n−1) is absolutely continuous on [α, β]; as a result, Ψ (n) exists almost everywhere. Moreover, Ψ is supposed to be n−convex, so we have erefore, by taking into account the last term in generalized identity (GI.1) and integral analogue of Jensen's inequality that is given in (6), we get (7).

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In the later part of this section, we will vary our conditions on functions g and Stieltjes measure dλ to obtain generalized variants of Jensen-Steffensen, Jensen-Boas, Jensen-Brunk, and Jensen-type inequalities. We start with the following generalization of Jensen-Steffensen inequality for n−convex functions.  (11) then for even n ≥ 3, (6) is valid.
(ii) Moreover, if (6) is valid and the function Journal of Mathematics is convex, then we get inequality (2) which is called generalized Jensen-Steffensen inequality for n−convex function.
Proof. (i) By applying second derivative test, we can show that the function R n (x, s) is convex for even n > 3. Now using the assumed conditions, one can employ Jensen-Steffensen inequality given by Boas (see [29] or [30], p. 59) for convex function R n (x, s) to obtain (6).
(ii) Since we can rewrite the R.H.S. of (7) in the difference for convex function H and by our assumed conditions on functions Z and λ, this difference is non-positive by using Jensen-Steffensen inequality difference [29]. As a result, the R.H.S. of inequality (7) is non-positive and we get generalized Jensen-Steffensen inequality (2) for n−convex function.

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Now, we give similar results related to Jensen-Boas inequality [30], p. 59], which is a generalization of Jensen-Steffensen inequality.
. en, the following results hold.

is valid. (ii) Moreover, if (6) is valid and the function H(·) defined in (18) is convex, then again inequality (2) holds and is called Jensen-Boas inequality for n−convex function.
Proof. We follow the similar argument as in the proof of eorem 4, but under the conditions of this corollary, we utilize Jensen-Boas inequality (see [29] or [24], p. 59) instead of Jensen-Steffensen inequality.
Next, we give results for Jensen-Brunk inequality. □ Corollary 2. Let Ψ defined in eorem 3 be n−convex and Z defined in M 1 be an increasing function. en, the following results hold.
∀x ∈ [a, b] holds, then for even n ≥ 3, (6) is valid. (6) is valid and the function H(·) defined in (18) is convex, then again inequality (2) holds and is called Jensen-Brunk inequality for n−convex function.
Proof. We proceed with the similar idea as in the proof of eorem 4, but under the conditions of this corollary, we employ Jensen-Brunk inequality (see [31] or [30], p. 59]) instead of Jensen-Steffensen inequality.

Remark 1.
e similar result in Corollary 2 is also valid provided that the function Z is decreasing. Also, assuming that the function Z is monotonic, one can replace the conditions in Corollary 2(i) by Remark 2. It is interesting to see that by employing similar method as in eorem 4, we can also get the generalization of classical Jensen's inequality (2) for n−convex functions by assuming the functions Z and λ along with the respective conditions in eorem 1.
Another important consequence of eorem 3 can be given by setting the function Z as Z(ζ) � ζ. is form is the generalized version of L.H.S. inequality of the Hermite-Hadamard inequality.
If the inequality (18) holds in reverse direction, then (19) also holds reversely. e special case of above corollary can be given in the form of following remark.
Using these substitutions in (2) and by following remark (20), we get the L.H.S. inequality of renowned Hermite-Hadamard inequality for n−convex functions.

New Generalization of Converse of Jensen's Integral
Inequality. In this section, we give the results for the converse of Jensen's inequality to hold, giving the conditions on the real Stieltjes measure dλ, such that λ(a) ≠ λ(b), allowing that the measure can also be negative, but employing Montgomery identity.
To start with we need the following assumption for the results of this section: For a given function Ψ: where Z is defined in (3). Using Montgomery identity, we obtain the following representation of the converse of Jensen's inequality.
then we get the following extension of the converse of Jensen' where R n (·, s) is defined in (5).

Journal of Mathematics
Proof. As Ψ (n−1) is absolutely continuous for (n ≥ 1), we can use the representation of Ψ using Montgomery identity (4) in the difference CJ(Ψ, Z m,M { } ; λ): After simplification and following the fact that CJ(Ψ, Z m,M { } ; λ) is zero for Ψ to be constant or linear, we get the following generalized identity:   (12) is convex, then we get the following inequality for n−convex function to be valid: Proof. e idea of the proof is similar to that of (6), but we use converse of Jensen's inequality (see [32] or [30], p. 98). □

Applications of Jensen's Integral Inequality.
In this section, we give applications of Jensen's integral inequality. Another important consequence of eorem 3 is by setting the function Z as Z(ζ) � ζ gives generalized version of L. H. S. inequality of the Hermite-Hadamard inequality.
then we have If the inequality (27) holds in reverse direction, then (28) also holds reversely. e special case of above corollary can be given in the form of following remark.

Remark 4.
It is interesting to see that substituting λ(ζ) � ζ and by following eorem 6, we get the R.H.S. inequality of renowned Hermite-Hadamard inequality for n−convex functions.

Generalization of Jensen's Discrete Inequality by Using Montgomery Identity
In this section, we give generalizations for Jensen's discrete inequality by using Montgomery identity. e proofs are similar to those of continuous case as given in previous section; therefore, we give results directly.

Generalization of Jensen's Discrete Inequality for Real
Weights. In discrete case, we have that p J > 0 for all J � 1, 2, . . . , m. Here we give generalizations of results allowing p J to be negative real numbers. Also, with usual notations for p J x J (J � 1, 2, . . . , n), we notate x � x 1 , x 2 , . . . , x m and p � p 1 , p 2 , . . . , p m (29) to be m−tuples. and Using Montgomery identity (4), we obtain the following representations of Jensen's discrete inequality.
(i) en, the following generalized identity holds: where R n (·, s) is defined in (5)

. (ii) Moreover, if Ψ is n−convex and the inequality
holds, then we have the following generalized inequality:
In the later part of this section, we will vary our conditions on p J x J (J � 1, 2, . . . , n) to obtain generalized discrete variants of Jensen-Steffensen, Jensen's, and Jensen-Petrovic type inequalities. We start with the following generalization of Jensen-Steffensen discrete inequality for n−convex functions.
is satisfied. (12) is convex, then we get the following generalized Jensen-Steffensen discrete inequality:

(i) If Ψ is n−convex, then for even n ≥ 3, (33) is valid. (ii) Moreover, if (33) is valid and the function H(·) defined in
Proof. It is interesting to see that under the assumed conditions on tuples x and p, we have that x ∈ [a, b]. For is shows that x 1 ≥ x. Also, x ≥ x n , since we have For further details, see the proof of Jensen-Steffensen discrete inequality ( [24], p. 57). e idea of the rest of the proof is similar to that of eorem 3, but here we employ eorem 7 and Jensen-Steffensen discrete inequality. □ (i) If Ψ is n−convex, then for even n ≥ 3, (34) is valid.
(ii) Moreover, if (33) is valid and the function H(·) defined in (12) is convex, then again we get (36) Now we give following reverses of Jensen-Steffensen and Jensen-type inequalities.
(i) If is n−convex, then for even n ≥ 3, then reverse of inequality (33)

holds. (ii) Moreover if (33) holds reversely and the function H(·)
defined in (12) is convex, then we get reverse of generalized Jensen-Steffensen inequality (36) for n−convex functions.
Proof. We follow the idea of eorem 8, but according to our assumed conditions, we employ reverse of Jensen-Steffensen inequality to obtain results.

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In the next corollary, we give explicit conditions on real tuple p such that we get reverse of classical Jensen inequality.

Corollary 7. Let Ψ be as defined in eorem 7 and let
is satisfied.
(i) If Ψ is n−convex, then for even n ≥ 3, the reverse of inequality (33)

is valid. (ii) Also, if reverse of (33) is valid and the function H(·)
defined in (12) is convex, then we get reverse of (36).
Proof. We follow the idea of eorem 8, but according to our assumed conditions, we employ reverse of Jensen inequality to obtain results.

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In [33] (see also [30]), one can find the result which is equivalent to the Jensen-Steffensen and the reverse Jensen-Steffensen inequality together. It is the so-called Jensen-Petrović inequality. Here, without the proof, we give the adequate corollary which uses that result. e proof goes the same way as in the previous corollaries. 8 Journal of Mathematics Corollary 8. Let Ψ be as defined in eorem 7 and let is satisfied. en, we get the equivalent results given in eorem 8 (i) and (ii), respectively. Remark 6. Under the assumptions of Corollary 8, if there exist m ∈ 1, 2, . . . , n { } such that and x ∈ [α, β], then we get the equivalent results for reverse Jensen-Steffensen inequality given in Corollary 6 (i) and (ii), respectively.

Remark 7.
It is interesting to see that the conditions on p J , J � 1, 2, . . . , m given in Corollary 8 and Remark 6 are coming from Jensen-Petrović inequality which become equivalent to conditions for p J , J � 1, 2, . . . , m for Jensen-Steffensen results given in eorem 8 and Corollary 6, respectively, when P m � 1. Now we give results for Jensen and its reverses for n−tuples x and p when n is an odd number.
) ≥ 0, then we have the following statements to be valid. (i) If Ψ is n−convex, then for even n ≥ 3, the inequality (12) is convex, then we get the following generalized inequality:

(ii) Also if (44) is valid and the function H(·) defined in
Proof. We employ the idea of the proofs of eorems 7 and 8 for n � odd along with inequality of Vasić and Janic [34].
□ Remark 8. We can also discuss the following important cases by considering the explicit conditions given in [34]. We conclude this section by giving the following important cases: (Case 1) Let the condition (i * ) hold and the reverse inequalities in condition (ii * ) hold. en, again we can give inequalities (44) and (45), respectively, given in Corollary 9.
(Case 2) If in case of conditions (i * ) and (ii * ), the following are valid: ) ≤ 0, then we can give reverses of inequalities (44) and (45), respectively, given in Corollary 9. (Case 3) Finally, we can also give reverses of inequalities (44) and (45), respectively, given in Corollary 9 provided that the condition (iii * ) holds and the reverse inequalities in condition (iv * ) hold. e result given in (Case 3) is type of generalization of inequality by Szegö [35].

Generalization of Converse Jensen's Discrete Inequality for
Real Weights. In this section, we give the results for converse of Jensen's inequality in discrete case by using the Montgomery identity. Let . . , n) be such that P m ≠ 0. en, we have the following difference of converse of Jensen's inequality for Ψ: [α, β] ⟶ R: Similarly, we assume the Giaccardi difference [36] given as where Journal of Mathematics 9 Theorem 9. Let Ψ: [α, β] ⟶ R be such that for n ≥ 1, (i) en, the following generalized identity holds: where R n (·, s) is defined in (5).
(ii) Moreover, if Ψ is n−convex and the inequality holds, then we have the following generalized inequality: If inequality (50) holds in reverse direction, then (51) also holds reversely. (i) en, the following generalized Giaccardi identity holds: where R n (·, s) is defined in (5).
(ii) Moreover, if Ψ is n−convex and the inequality holds, then we have the following generalized Giaccardi inequality: If inequality (53) holds in reverse direction, then (54) also holds reversely.
In the later part of this section, we will vary our conditions on p J x J (J � 1, 2, . . . , m) to obtain generalized Proof. From eorem 9 by following Jensen's difference (61), we can rearrange (34) as Now replace p J with q J and x J with p J /q J , and we get (62).
(ii) We can get bounds for the Shannon entropy of q, if we choose q ≔ (q 1 , . . . , q n ) to be a positive probability distribution.
(i) Using Ψ(x) ≔ − ln x (which is n-convex for even n) in eorem 13, we get (68) after simplification. (ii) It is a special case of (i).

Results for Zipf and Hybrid Zipf-Mandelbrot Entropy.
One of the basic laws in information science is Zipf's law [41,42] which is highly applied in linguistics. Let c ≥ 0, d > 0, and N ∈ 1, 2, . . . where Consider q J � Ψ(J; N, c, d) � 1 where Ψ(J; m, c, d) is discrete probability distribution known as Zipf-Mandelbrot law. Zipf-Mandelbrot law has many application in linguistics and information sciences. Some of the recent study about Zipf-Mandelbrot law can be seen in the listed references (see [39,43]). Now we state our results involving entropy introduced by Mandelbrot law by establishing the relationship with Shannon and relative entropies.  (80) Finally, substituting this q J � ω J /(J + c) d H * c,d,ω in Corollary 10 (ii), we get the desired result.