#### Abstract

We generalize Jensen’s integral inequality for real Stieltjes measure by using Montgomery identity under the effect of 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 from Montgomery identity in calculus and fractional integrals. Also, we give applications in information theory for our obtained results, especially for Zipf and Hybrid Zipf–Mandelbrot entropies.

#### 1. 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 [2–4] and the references therein).

The well-known Jensen’s inequality asserts that for the function it holds thatif is a convex function on interval , where are positive real numbers and , while .

However, the well-known integral analogue of Jensen’s inequality is as follows.

Theorem 1. *Let be a continuous function and be an increasing and bounded function with . Then, for every continuous convex function , the following inequality holds:where*

There are several inequalities coming from Jensen’s inequality both in integral and discrete cases which can be obtained by varying conditions on the function and measure defined in Theorem 1.

Montgomery identity is used in quantum calculus or calculus. There 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 [7–21]. The 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. The readers can find some motivated findings related to convex functions and some new integral inequalities in [22–27].

In [28], Khan et al. have mentioned about -convex functions as follows.

*Definition 1. *A function is called convex of order or -convex if for all choices of distinct points we have .

If -th order derivative exists, then is convex if and only if . For , a function is convex if and only if exists and is convex.

In the present paper, we will use Montgomery identity that is presented as following.

Theorem 2. *Let , be such that is absolutely continuous, is an open interval, and , . Then, the following identity holds:where*

#### 2. 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: Let be continuous function. Let be a continuous function or the functions of bounded variation such that .

##### 2.1. New Generalization of Jensen’s Integral Inequality

In our first main result, we employ Montgomery identity to obtain the following real Stieltjes measure’s theoretical representations of Jensen’s inequality.

Theorem 3. *Let be as defined in such that . Also, let be such that for , is absolutely continuous. If is convex such thatwith and as defined in (3) and (5), respectively, then we have*

*Proof. *As is absolutely continuous for , we can use the representation of using Montgomery identity (4) and can calculateThe integration of the composition of functions for the real measure on givesNow computing the difference , we get the following generalized identity involving real Stieltjes measure:Finally, by our assumption, is absolutely continuous on ; as a result, exists almost everywhere. Moreover, is supposed to be convex, so we have almost everywhere on . Therefore, by taking into account the last term in generalized identity and integral analogue of Jensen’s inequality that is given in (6), we get (7).

In the later part of this section, we will vary our conditions on functions and Stieltjes measure 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 convex functions.

Theorem 4. *Let defined in Theorem 3 be convex and defined in be monotonic. Then, the following results hold.*(i)*If defined in satisfies* *then for even , (6) is valid.*(ii)*Moreover, if (6) is valid and the function* *is convex, then we get inequality (2) which is called generalized Jensen–Steffensen inequality for convex function.*

*Proof. *(i)By applying second derivative test, we can show that the function is convex for even . Now using the assumed conditions, one can employ Jensen–Steffensen inequality given by Boas (see [29] or [30], p. 59) for convex function to obtain (6).(ii)Since we can rewrite the R.H.S. of (7) in the differencefor convex function and by our assumed conditions on functions and , 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 convex function.

Now, we give similar results related to Jensen–Boas inequality [30], p. 59], which is a generalization of Jensen–Steffensen inequality.

Corollary 1. *Let defined in Theorem 3 be convex function. Also, let be as defined in with and be monotonic in each of the intervals . Then, the following results hold.*(i)*If as defined in satisfies* * and , then for even , (6) is valid.*(ii)*Moreover, if (6) is valid and the function defined in (18) is convex, then again inequality (2) holds and is called Jensen–Boas inequality for convex function.*

*Proof. *We follow the similar argument as in the proof of Theorem 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 defined in Theorem 3 be convex and defined in be an increasing function. Then, the following results hold.*(i)*If defined in with and* *and* * holds, then for even , (6) is valid.*(ii)*Moreover, if (6) is valid and the function defined in (18) is convex, then again inequality (2) holds and is called Jensen–Brunk inequality for convex function.*

*Proof. *We proceed with the similar idea as in the proof of Theorem 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. *The similar result in Corollary 2 is also valid provided that the function is decreasing. Also, assuming that the function is monotonic, one can replace the conditions in Corollary 2 by

*Remark 2. *It is interesting to see that by employing similar method as in Theorem 4, we can also get the generalization of classical Jensen’s inequality (2) for convex functions by assuming the functions and along with the respective conditions in Theorem 1.

Another important consequence of Theorem 3 can be given by setting the function as . This form is the generalized version of L.H.S. inequality of the Hermite-Hadamard inequality.

Corollary 3. *Let be a function of bounded variation such that with and . Under the assumptions of Theorem 3, if is convex such thatthen we have**If the inequality (18) holds in reverse direction, then (19) also holds reversely.*

The special case of above corollary can be given in the form of following remark.

*Remark 3. *It is interesting to see that substituting gives and . Using these substitutions in (2) and by following remark (20), we get the L.H.S. inequality of renowned Hermite–Hadamard inequality for convex functions.

##### 2.2. 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 , such that , 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: Let be such that for all where is defined in .

For a given function , we consider the differencewhere is defined in (3).

Using Montgomery identity, we obtain the following representation of the converse of Jensen’s inequality.

Theorem 5. *Let be as defined in and let be such that for , is absolutely continuous. If is convex such thatorthen we get the following extension of the converse of Jensen’s difference:where is defined in (5).*

*Proof. *As is absolutely continuous for , we can use the representation of using Montgomery identity (4) in the difference :After simplification and following the fact that is zero for to be constant or linear, we get the following generalized identity:Now using characterizations of convex functions like in the proof of Theorem 3, we get (23).

The next result gives converse of Jensen’s inequality for higher-order convex functions.

Theorem 6. *Let defined in Theorem 5 be convex and be as defined in . Then, the following results hold.*(i)*If is non-negative measure on , then for even , (22) is valid.*(ii)*Moreover, if (22) is valid and the function defined in (12) is convex, then we get the following inequality for convex function to be valid:*

*Proof. *The idea of the proof is similar to that of (6), but we use converse of Jensen’s inequality (see [32] or [30], p. 98).

##### 2.3. Applications of Jensen’s Integral Inequality

In this section, we give applications of Jensen’s integral inequality.

Another important consequence of Theorem 3 is by setting the function as gives generalized version of L. H. S. inequality of the Hermite–Hadamard inequality.

Corollary 4. *Let be a function of bounded variation such that with and . Under the assumptions of Theorem 5, if is convex such thatthen we have**If the inequality (27) holds in reverse direction, then (28) also holds reversely.*

The special case of above corollary can be given in the form of following remark.

*Remark 4. *It is interesting to see that substituting and by following Theorem 6, we get the R.H.S. inequality of renowned Hermite–Hadamard inequality for convex functions.

#### 3. 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. The proofs are similar to those of continuous case as given in previous section; therefore, we give results directly.

##### 3.1. Generalization of Jensen’s Discrete Inequality for Real Weights

In discrete case, we have that for all . Here we give generalizations of results allowing to be negative real numbers. Also, with usual notations for , we notateto be tuples.and

Using Montgomery identity (4), we obtain the following representations of Jensen’s discrete inequality.

Theorem 7. *Let be such that for , is absolutely continuous. Also, let , be such that and .*(i)*Then, the following generalized identity holds:* *where is defined in (5).*(ii)*Moreover, if is convex and the inequality**holds, then we have the following generalized inequality:**If inequality (33) holds in reverse direction, then (34) also holds reversely.*

*Proof. *Similar to that of Theorem 3.

In the later part of this section, we will vary our conditions on 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 convex functions.

Theorem 8. *Let be as defined in Theorem 7. Also, let be monotonic tuple, , and be a real tuple such thatis satisfied.*(i)*If is convex, then for even , (33) is valid.*(ii)*Moreover, if (33) is valid and the function defined in (12) is convex, then we get the following generalized Jensen–Steffensen discrete inequality:*

*Proof. *It is interesting to see that under the assumed conditions on tuples and , we have that . For ,This shows that . Also, , since we haveFor further details, see the proof of Jensen–Steffensen discrete inequality ([24], p. 57). The idea of the rest of the proof is similar to that of Theorem 3, but here we employ Theorem 7 and Jensen–Steffensen discrete inequality.

Corollary 5. *Let be as defined in Theorem 7 and let with being a positive tuple.*(i)*If is convex, then for even , (34) is valid.*(ii)*Moreover, if (33) is valid and the function defined in (12) is convex, then again we get (36) which is called Jensen’s inequality for convex functions.*

*Proof. *For ensures that . So, by applying classical Jensen’s discrete inequality (1) and idea of Theorem 8, we will get the required results.

*Remark 5. *Under the assumptions of Corollary 5, if we choose , then Corollary 5 gives the following inequality for convex functions:Now we give following reverses of Jensen–Steffensen and Jensen-type inequalities.

Corollary 6. *Let be as defined in Theorem 7. Also, let be monotonic tuple, , and be a real tuple such that there exist such thatwhere and .*(i)*If is convex, then for even , then reverse of inequality (33) holds.*(ii)*Moreover if (33) holds reversely and the function defined in (12) is convex, then we get reverse of generalized Jensen–Steffensen inequality (36) for convex functions.*

*Proof. *We follow the idea of Theorem 8, but according to our assumed conditions, we employ reverse of Jensen–Steffensen inequality to obtain results.

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

Corollary 7. *Let be as defined in Theorem 7 and let such that . Let be a real tuple such thatis satisfied.*(i)*If is convex, then for even , the reverse of inequality (33) is valid.*(ii)*Also, if reverse of (33) is valid and the function defined in (12) is convex, then we get reverse of (36).*

*Proof. *We follow the idea of Theorem 8, but according to our assumed conditions, we employ reverse of Jensen inequality to obtain results.

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. The proof goes the same way as in the previous corollaries.

Corollary 8. *Let be as defined in Theorem 7 and let be such that . Let be a real tuple with such thatis satisfied. Then, we get the equivalent results given in Theorem 8 and , respectively.*

*Remark 6. *Under the assumptions of Corollary 8, if there exist such thatand , then we get the equivalent results for reverse Jensen–Steffensen inequality given in Corollary 6 and , respectively.

*Remark 7. *It is interesting to see that the conditions on given in Corollary 8 and Remark 6 are coming from Jensen–Petrović inequality which become equivalent to conditions for for Jensen–Steffensen results given in Theorem 8 and Corollary 6, respectively, when .

Now we give results for Jensen and its reverses for tuples and when is an odd number.

Corollary 9. *Let be as defined in Theorem 7 and let for be such that , be real tuples, and for all . If for every , we have* * * * ,* *then we have the following statements to be valid.*(i)*If is convex, then for even , the inequality*(ii)*Also if (44) is valid and the function defined in (12) is convex, then we get the following generalized inequality:*

*Proof. *We employ the idea of the proofs of Theorems 7 and 8 for 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 hold and the reverse inequalities in condition hold. Then, again we can give inequalities (44) and (45), respectively, given in Corollary 9. (**Case 2**) If in case of conditions and , the following are valid: , 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 holds and the reverse inequalities in condition hold.The result given in is type of generalization of inequality by Szegö [35].

##### 3.2. 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 , , be such that . Then, we have the following difference of converse of Jensen’s inequality for :

Similarly, we assume the Giaccardi difference [36] given aswhere

Theorem 9. *Let be such that for , is absolutely continuous. Also, let , , be such that .*(i)*Then, the following generalized identity holds:* *where is defined in (5).*(ii)*Moreover, if is convex and the inequality**holds, then we have the following generalized inequality:**If inequality (50) holds in reverse direction, then (51) also holds reversely.*

Theorem 10. *Let be such that for , is absolutely continuous. Also, let , , be such that and .*(i)*Then, the following generalized Giaccardi identity holds:* *where is defined in (5).*(ii)*Moreover, if is 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 to obtain generalized converse discrete variants of Jensen’s inequality and Giaccardi inequality for convex functions.

Theorem 11. *Let be as defined in Theorem 9. Also, let and be a positive tuple.*(i)*If is convex, then for even , (50) is valid.*(ii)*Moreover, if (50) is valid and the function defined in (12) is convex, then we get the following generalized converse of Jensen’s inequality:*

*Proof. *We follow the idea of Theorem 8, but according to our assumed conditions, we employ converse of Jensen’s inequality (see [32] or [30], p. 98) to obtain results.

Finally, in this section, we give Giaccardi inequality for higher-order convex functions.

Theorem 12. *Let be as defined in Theorem 9. Also, let and be a positive tuple such that*(i)*If is convex, then for even , (53) is valid.*(ii)*Moreover, if (53) is valid and the function defined in (12) is convex, then we get the following generalized Giaccardi inequality:**where and are defined in (47).*

*Proof. *We follow the idea of Theorem 8, but according to our assumed conditions, we employ Giaccardi inequality (see [36] or [37], p. 11) to obtain results.

##### 3.3. Applications in Information Theory for Jensen’s Discrete Inequality

Jensen’s inequality plays a key role in information theory to construct lower bounds for some notable inequalities, but here we will use it to make connections between inequalities in information theory.

Let be a convex function and let and be positive probability distributions; then, -divergence functional is defined (in [38]) as follows:

Horváth et al. in [39] defined the generalized Csiszár divergence functional as follows.

*Definition 2. *Let be an interval in and be a function. Also, let and such thatThen, letIn this section, we write Jensen’s difference here that we use in upcoming results:

Theorem 13. *Under the assumptions of Theorem 9 (ii), let (51) hold and be convex. Also, let and ; then, we have the following results:*

*Proof. *From Theorem 9 by following Jensen’s difference (61), we can rearrange (34) asNow replace with and with , and we get (62).

For positive -tuple such that , the *Shannon entropy* is defined by

Corollary 10. *Under the assumptions of Theorem 9 (ii), let (51) hold and be convex.*(i)*If , then*(ii)*We can get bounds for the Shannon entropy of , if we choose to be a positive probability distribution.*

*Proof. *(i)Substituting and using in Theorem 13, we get (65).(ii)Since we have , by multiplying on both sides of (65) and taking into account (64), we get (66).The Kullback–Leibler distance [40] between the positive probability distributions and is defined by

Corollary 11. *Under the assumptions of Corollary 10,*(i)*If *