Estimations of the Slater Gap via Convexity and Its Applications in Information Theory

Convexity has played a prodigious role in various areas of science through its properties and behavior. Convexity has booked record developments in the eld of mathematical inequalities in the recent few years. e Slater inequality is one of the inequalities which has been acquired with the help of convexity. In this note, we obtain some estimations for the Slater gap while dealing with the notion of convexity in an extensive manner. We acquire the deliberated estimations by utilizing the denition of convex function, Jensen’s inequality for concave functions, and triangular, power mean, and Hölder inequalities. We discuss several consequences of the main results in terms of inequalities for the power means. Moreover, by utilizing the main results, we give estimations for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and the Bhattacharyya coecient. Furthermore, we present some estimations for the Zipf–Mandelbrot entropy as additional applications of the acquired results. e perception and approaches adopted in this note may pretend more research in this direction.


Introduction
In the diverse elds of science, convex functions are of the greatest importance due to their dominant manners and wealthy structure [1][2][3]. In recent years, the abundant applicability of convex functions has been observed in engineering [4], di erential equations [5], epidemiology [6], information theory [7], statistics [8], optimization [9], and many others. Moreover, convex functions have some unique properties and, due to such properties, it became a focus point for researchers [10,11]. Furthermore, convex functions have also been generalized, re ned, and extended in di erent directions while utilizing their characteristics and behavior [12]. In a formal way, the convex function can be de ned as follows.
De nition 1. Let [a, b] be an interval in R and Ψ be a realvalued function de ned on [a, b]. en, the function Ψ is said to be convex, if the inequality holds, for all x, y ∈ [a, b] and β ∈ [0, 1]. If the inequality (1) is valid in the opposite direction, then the function Ψ is said to be concave. e eld of mathematical inequalities is one of the favorable areas for the class of convex functions, where it has been employed extensively. Many inequalities would not be conceivable to prove without convex functions [13][14][15][16][17]. Majorization [18], Favard [19], Jensen-Mercer [20], and Hermite-Hadamard [21] inequalities are some of the important inequalities which have been acquired with the use of convex functions. e Jensen inequality [22] is one of the most important inequalities among the aforementioned inequalities for the class of convex functions. Jensen's inequality has a very strong relationship with ordinary convexity in the sense that it generalizes the definition of convex function. Another important fact about this inequality is that it is the origin of many other classical inequalities [23]. e mathematical form of Jensen's inequality is stated in the next theorem.
If Ψ is a concave function, then (2) holds in the opposite direction.
e continuous form of (2) is verbalized in the following theorem.
If Ψ is a concave function, then (3) holds in the opposite direction.
e Jensen inequality has a variety of interesting properties and also has a very desirable structure [24]. Furthermore, there are a huge number of applications of this inequality in the different fields of science [4,7,25,26]. Due to the huge importance and applicability of this inequality, a lot of work has been carried out on it [8,27]. In 1980, Slater [28] presented a companion inequality to the aforementioned inequality while using a convex function, which is well known as Slater's inequality in the literature. e formal form of the Slater inequality is given below. Theorem 3. Let Ψ: (a, b) ⟶ R be any function and q i ≥ 0, If the function Ψ is convex and increasing, then In 1985, Pečarić [29] generalized the Slater inequality by relaxing the monotonicity condition of the function. (4) is true. Inequality (4) can also be written in the following form: roughout the article, by the Slater gap (difference), we shall mean that the left side of the above inequality.
In the recent decades, a lot of work has been dedicated to Slater's inequality by numerous researchers from different angles by following different techniques and methods. In 1985, Pečarić [30] acquired some extensions of Slater's inequality by taking convex functions of several real variables. In 2000, Matić and Pečarić [31] established a couple of companion inequalities to Jensen's inequality in both discrete and integral versions by utilizing convex functions. en, they used these generalized inequalities and deduced Slater's as well as some related inequalities. In 2010, Khan and Pečarić [32] obtained an improvement and a reversion of Slater's inequality by utilizing some earlier established results. In 2013, Khan et al. [33] presented refinements of Slater's and other related inequalities of the Slater type for convex functions defined on linear spaces. Moreover, they also provided refinements for majorization type inequalities. In 2018, Song et al. [34] established Slater's inequality for strongly convex functions and also presented some more results of the Jensen type for the aforementioned class of convex functions.

Estimations of the Slater Gap
In this section, we are going to establish some new estimations for the Slater gap. e proposed estimations will be acquired by utilizing the definition of convex function, Jensen's inequality for concave functions, Hölder, power mean, and triangular inequalities. First, we state a lemma in which a general inequality is constructed while using a twice differentiable function.
Mathematical Problems in Engineering Proof. Without loss of generality, let Utilizing the integration by parts rule, we have From this, we can write that Instantly, taking absolute of (8) and then applying the triangular inequality, we acquire (6).

Mathematical Problems in Engineering
In the following theorem, we obtain an estimate for the Slater gap by applying the definition of the convex function. □ Theorem 5. Assume that the hypotheses of Lemma 1 are true, and further let the function |Ψ ″ | p be convex for p > 1. en, Proof. Applying Hölder inequality on the right side of (6), we obtain By utilizing the convexity of |Ψ ″ | p on the right side of (10), we acquire Now, evaluating the integrals in (11), we receive (9). Utilizing Jensen's inequality for concave functions, we receive an estimate for the Slater gap stated in the following theorem. □ Theorem 6. Suppose that the assumptions of Lemma 1 are true. Furthermore, if |Ψ ″ | p is concave for p〉1, then Proof. By utilizing (10), we acquire As the function |Ψ ″ | p is concave, therefore by applying Jensen's inequality to (13), we obtain Inequality (12) can easily be deduced by finding the integrals in (14).
In the following theorem, we construct an inequality which provides an estimate for the Slater gap while using the definition of the convex function and the renowned Hölder inequality. □ Theorem 7. Let all the assumptions of eorem 5 be valid.
Proof. From (6), we can write that Now, using Hölder inequality on the right side of (16), we obtain

Mathematical Problems in Engineering
Instantly, applying the definition of convex function on the right side of (17), we get Finding integrals in (18), we obtain (15). By utilizing the Hölder inequality and Jensen's inequality for concave functions, we obtain an estimate for the Slater gap given in the following theorem.
Proof. Utilizing the Jensen inequality on the right side of (17), we deduce Now, evaluating integrals in (20), we obtain (19).

Mathematical Problems in Engineering
We receive another estimate for the Slater gap by applying the power mean inequality and the definition of convex function which states the following. □ Theorem 9. Let the conditions of eorem 5 be true. en, Proof. Applying the power mean inequality on the right side of (16), we receive Now, utilizing the convexity of |Ψ ″ | p on the right side of (22), we acquire By finding the integrals in (23), we get (21). In the following theorem, we acquire an estimate for the Slater gap by utilizing the power mean inequality and the famous Jensen's inequality for concave functions.

Mathematical Problems in Engineering
Proof. Since the function |Ψ ″ | p is concave, therefore, by applying Jensen's inequality on the right side of (22), we arrive Instantly, simplifying (25), we deduce (24).

Applications for the Power Means
In this section of the note, we will establish some new relations for the power means. e intended relations will be acquired by putting some particular convex functions in the main results. Now, we recall the definition of the power mean.
e following corollary is the consequence of eorem 5 for the power means.

Mathematical Problems in Engineering
(iii) Obviously, the functions Ψ and |Ψ ″ | p are both convex on (0, ∞) for the specified values of t, u, and p. erefore, the inequality (28) can easily be deduced by adopting the procedure of (i).
For the case, when u > 0 and t < 0, follow the procedure of (31). □ We utilized eorem 8 and obtained a relation for power means, which is stated in the next corollary.
Proof. (i) Let Ψ(y) � y t/u , y > 0. en surely, the functions Ψ and |Ψ ″ | p are concave and convex, respectively, with respect to the given conditions. erefore, using Ψ(y) � y t/u , q i � c i , and y i � ζ u i in (21), we arrive (33). (ii) − (iii) For the specified conditions on t, u, and p given in the cases (ii) and (iii), respectively, both the functions Ψ and |Ψ ″ | p are convex on (0, ∞). erefore, to deduce (3.32), follow the procedure of (i).
We receive the following relation for the power means as a consequence of eorem 10.

Corollary 6. Let all the hypotheses of Corollary 2 be true. en,
Proof. Consider Ψ(y) � y t/u defined on (0, ∞), then undoubtedly, the functions Ψ and |Ψ ″ | p are convex and concave for the stated conditions. erefore, to get (35), assume Ψ(y) � y t/u , q i � c i , and y i � ζ u i in (24). □ e following are some more relations for the power means as a consequence of eorem 5.

Applications in Information Theory
In this section, we are going to present applications of the main results in the information theory. e applications will provide different estimations for Csiszár and Kullback-Leibler divergences, Shannon entropy, and for the Bhattacharyya coefficient. To proceed to the desired goals, first we state the definition of Csiszár divergence.
e following theorem is an application of eorem 5 for the Csiszár divergence.
Proof. To receive (42), put q i � c i and y i � ζ i /c i in (9). e following theorem provides a bound for the Csiszár divergence. □ Theorem 12. Assume that s 1 � (c 1 , c 2 , . . . , c m ) and , and |Ψ ″ | p is a concave function on (0, ∞) for p > 1, then Proof. Inequality (44) can easily be acquired by assuming q i � c i and y i � ζ i /c i in (12). Utilizing eorem 7, we acquire an estimate for the Csiszár divergence.
Proof. Utilizing (15) by taking q i � c i and y i � ζ i /c i , we arrive (45). e following inequality gives an estimate for the Csiszár divergence, which can be acquired from eorem 8.
Proof. Putting q i � c i and y i � ζ i /c i in (19), we obtain (46).

Mathematical Problems in Engineering
With the help of eorem 9, we establish a relation for the Csiszár divergence, which is stated in the next theorem. □ Theorem 15. Let all the assumptions of eorem 11 be satisfied. en, Proof. To obtain (47), use q i � c i and y i � ζ i /c i in (21). □ e following estimate for the Csiszár divergence is deduced from eorem 10.
Proof. Use q i � c i and y i � ζ i /c i in (24), we get (48). Now, recall the definition of Shannon entropy.
e following inequality provides an estimate for the Shannon entropy.
Proof. To get (54), we need to just put Ψ(y) � − logy, y > 0 in (43). An application of eorem 7 for the Kullback-Leibler divergence is given in the next corollary.
Another estimate for the Bhattacharyya coefficient is stated in the following corollary.

Applications for the Zipf-Mandelbrot Entropy
e present section of the note is dedicated to the Zipf-Mandelbrot entropy. Here, we establish a number of relations for the aforementioned entropy by using the main results. First, we discuss some basics about the aforementioned entropy.

Conclusions
e mathematical inequalities have a wealthy history with the diverse type of applications in several areas of science. It has been observed that a lot of inventive concepts about mathematical inequalities and their applications can be received with the support of convex functions. e Slater inequality is one of the inequalities which has been established with the help of convex functions.
is inequality is actually a companion inequality to the famous Jensen's inequality. In this note, we obtained some estimations for the Slater inequality by utilizing the definition of the convex function, Jensen's inequality for concave functions, Hölder, power mean, and triangular inequalities. We gave different relations for the power means as consequences of the acquired estimations. Furthermore, we presented some applications of the received estimations in the form of inequalities for Csiszár and Kullback-Leibler divergence, Shannon entropy, and the Bhattacharyya coefficient. Moreover, we gave some additional applications of the established estimations for Zipf-Mandelbrot entropy.

Data Availability
is study was not supported by any data.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.