Bounds for the Jensen Gap in terms of Power Means with Applications

Jensen’s and its related inequalities have attracted the attention of several mathematicians due to the fact that Jensen’s inequality has numerous applications in almost all disciplines of mathematics and in other fields of science. In this article, we propose new bounds for the difference of two sides of Jensen’s inequality in terms of power means. An example has been presented for the importance and support of the main results. Related results have been given in quantum calculus. As consequences, improvements of quantum integral version of Hermite-Hadamard inequality have been derived. The obtained inequalities have been applied for some well-known inequalities such as Hermite-Hadamrd, Hölder, and power mean inequalities. Finally, some applications are given in information theory. The tools performed for obtaining the main results may be applied to obtain more results for other inequalities.


Introduction
There is no doubt that one of the most important classes of functions is the class of convex functions. The beauty of convex functions is due to its unique graphical representation, geometrical interpretation, and developments in the theory of inequalities. There are numerous applicable inequalities which have been established for this class of functions such as Jensen's, the Jensen-Steffensen, and majorization inequalities [1,2]. One of the most important and widely applicable inequalities which has attracted the attention of many mathematicians is the Jensen inequality [2 , p. 43]. According to this inequality, if ϕ : ½α 1 , α 2 ⟶ ℝ is a convex function and x i ∈ ½α 1 , α 2 , p i ≥ 0 for each i ∈ f1, 2, ⋯, ng with P n ≔ ∑ n i=1 p i > 0, then The inequality in (1) flips when the function is concave.
In the literature, discrepancy between the two sides of Jensen's inequality has been studied by several mathematicians in different directions which provides error bounds for certain approximations. In 2015, Costarelli and Spigler [11] studied the discrepancy between the right and left sides of Jensen's inequality by using convex functions from the class of C 2 functions as well as the class of merely Lipschitz continuous functions. Some illustrative examples are presented and compared the bounds with some existing bounds. In 2018, Pecarić et al. [12] focused to find the bounds for the difference of two sides of Jensen's inequality. They considered some Green convex functions and their related identities and derived bounds for the Jensen gap for the class of C 2 functions without using convexity condition. In particular, they have applied the Hölder inequality. Related discrete results have also been obtained, and several applications in information theory are presented. In 2020, Khan et al. [3] introduced a new method for the derivation of bounds using higher-order convex functions that is 4-convexity. First of all, they obtained an identity for the Jensen difference in terms of Green convex functions and double derivative of a function. Further, they obtained bounds by using 4-convexity and some properties of absolute function. Some examples are considered for their main results and compared with earlier bounds. Also, several applications for some well-known inequalities such as Hölder and Hermite-Hadmard inequalities have been given. At the end, several applications in information theory also presented. Related discrete results are given in [13]. In 2021, Khan et al. [14] further modified the method given in [3,13] and derived several results for Jensen and related inequalities. In this method, they have used real weights and found the integrals of some functions which pertaining Green functions, in a very simple way with the help of the obtained identity for the Jensen gap. By virtue of this procedure, they were able to obtain bounds for the Jensen-Steffensen and converse of Jensen's inequalities. For more interesting results related to the celebrated Jensen's inequality, we recommend [15][16][17].
The main results of this manuscript utilize power means and its related inequalities; therefore, we want to mention them in the following part of this section. The following well-known power means and their monotonicity are given in [18, p.19]: For two positive real n-tuples x = ðx 1 , ⋯, x n Þ and p = ðp 1 , ⋯, p n Þ, the power mean of order κ ∈ ℝ is defined by where P n = ∑ n i=1 p i . For the above n-tuples, the quasiarithmetic mean is defined byM where h is a strictly monotone and continuous function.
The integral power mean can be defined as follows: If p, g : ½a, b ⟶ R + and g are integrable functions, then the integral power mean of order κ ∈ ℝ is defined by If The main aim of this paper is to obtain new interesting bounds for the discrepancy of the two sides of the Jensen inequality using new tools. The bounds pertain power means. We give an example, which shows that the bounds obtained in this paper are better than the earlier ones. We proved Jensen's inequality for quantum integrals and also derived its improvements. As applications, Hermite-Hadamard inequality and its improvements have been deduced for q-integrals. We also give applications for some well-known inequalities such as Hermite-Harmard, Hölder, power, and quasiarithmetic mean inequalities. At the end, we focused to give applications for Shannon-entropy, Csiszár, and Zipf-Mandelbrot entropy etc.

Main Results
We begin by presenting our first major finding. Theorem 2. Let ϕ : ½a, b ⟶ ℝ be a convex function, x i ∈ ½a, b, p i > 0 for i = 1, 2, ⋯, n with P n = ∑ n i=1 p i . Then, for s ≥ 1 and r ≤ 1, the following inequalities hold: where x = ð1/P n Þ∑ n i=1 p i x i and y = ðy 1 , y 2 , ⋯, y n Þ with y i = ϕð Proof. It is obvious that Since the function ϕ is convex, therefore, we have
In the theorem below, we give an integral version of the above theorem. Theorem 3. Consider the interval I and the convex function ϕ : I ⟶ ℝ and let g, p : ½a, b ⟶ ℝ be integrable functions such that gðℓÞ ∈ I, pðℓÞ ∈ ℝ + for all ℓ ∈ ½a, b and P = Ð b a pðℓÞdℓ . Then, for s ≥ 1 and r ≤ 1, we have where hðℓÞ = ϕðgðℓÞÞ − ϕð gÞ − ϕ′ + ð gÞðgðℓÞ − gÞ with The next main result is presented in the following theorem.
where y = ðy 1 , Proof. By simple calculation, we can write Since the function ϕ is convex, therefore, we have As y ≔ ∑ n i=1 y i , so from (15), we can write Now, by considering the weights y i and applying power mean inequality (6), we have and similarly, Using (18) and (19) in (17), we obtain (14).
In the forthcoming theorem, we present the integral version of the above theorem.
Theorem 5. Consider ϕ : I ⟶ ℝ be a convex function and p, g : ½a, b ⟶ ℝ be integrable functions such that gðωÞ ∈ I , pðωÞ ∈ ℝ + for all ω ∈ ½a, b and P = Ð b a pðωÞdω. Then, for s ≥ 1 and r ≤ 1, the following inequalities hold: where In the following example, we compare our new bound with earlier bounds of the Jensen gap. Example 1. Let the functions ϕ, g : ½0, 1 ⟶ ℝ be defined by ϕðxÞ = x 4 , gðxÞ = x and pðxÞ = 1 for all x ∈ ½0, 1. Then, and Now, we calculate the right hand side of (13) for s = 2: For the same functions, the bound for the Jensen gap from inequality (5) in [3] is a 1 ≔ 0:25. Also, in bounds for the Jensen gap from the inequalities (6) and (11) in [11], we have a 2 = 0:8748 and a 3 = 0:4998, respectively.

Journal of Function Spaces
Hence, we concluded that M 2 ðh, pÞ = 0:2101 < a i for i = 1, 2, 3: That is, for these functions, the bound obtained in (13) is better than the earlier bound obtained in [3,11].
As M s ðh ; pÞ is increasing with respect to s, therefore for more better estimate, we can find M s ðh, pÞ for 1 < s < 2. For example, M 1:5 ðh ; pÞ is given by For inequality (20), we consider the above functions with pðxÞ = x. For these functions, the value of the Jensen difference is 0:1358, and the value of the bound for the Jensen gap from the inequality (5) in [3] is: b 1 ≔ 0:2074. Now, we calculate the right hand side of (20).
Hence, in this case, we concluded that the bound obtained in (20) is better than the earlier bound obtained in the inequality (5) in [3].

Jensen's Type Inequalities in Quantum Calculus
The q-calculus or quantum calculus deals with the study of calculus without utlizing the idea of limits. The popular mathematician Euler proposed the ponder q-calculus within the 18 th century, while he introduced the term q in Newton's work of infinite series. Jackson has begun a symmetric study of q-calculus and presented q-definite integrals in twentieth century [19]. The field of q-calculus has various interesting applications in several branches of Physics, Mathematics, and in other areas [20,21]. This field has gotten extraordinary attention by numerous researchers, and a lot of research is devoted to this field. In [22], the authors defined q-analogue operator of Ruscheweyh type involving multivalent functions and derived several properties. By using the newly presented Harmonic q-Starlike class of functions, some important problems such as distortion limits, neces-sary and sufficient conditions, convolutions and convexity, and problems with partial sums have been studied in [23]. Srivastava et al. [24] applied the idea of a particular advanced convolution q-operator together with the concept of convolution and analyzed two new classes of meromorphically harmonic functions. First, we give some preliminaries which are useful in our results. Throughout this section, q belongs to ð0, 1Þ. Also, we assume that all the series which are used in this section are convergent.
Definition 6 (see [25]). Let ψ : ½a, b ⟶ ℝ be a continuous function and y ∈ ½a, b: Then, the q -derivative of the function ψ at y is denoted by a D q ψðyÞ and defined by We say that ψ is q-differentiable on ½a, b if a D q ψðyÞ exists for all y ∈ ½a, b.
Definition 7 (see [25]). Let ψ : ½a, b ⟶ ℝ be a continuous function. Then, the q -integral on ½a, b is defined as for y ∈ ½a, b. Moreover, if c ∈ ða, yÞ, then the q-integral on ½ a, b is defined as Remark 8. From Definitions 6 and 7, we make the following remarks: (1) By taking a = 0, the expression in (26) becomes the well-known q-derivative, D q ψðwÞ, of the function ψ defined by (2) Also, if a = 0, then (27) reduces to the classical q -integral of a function ψ : ½0, ∞Þ ⟶ ℝ defined by Now, we present Jensen's inequality for q-integrals.

Journal of Function Spaces
Theorem 9. Let ϕ : I ⟶ ℝ be a continuous convex function defined on the interval I and ψ : ½a, b ⟶ I be a continuous function. Then, Proof. From the convexity of ϕ, we have Taking (32), we obtain Multiplying both sides of (33) by ð1 − qÞq k and then taking summation over k, we get Since y = ð1 − qÞ∑ ∞ k=0 q k ψðq k b + ð1 − q k ÞaÞ, therefore, from (34), we have which is equivalent to (31).
As a consequence of the above theorem, we deduce Hermite-Hadamard inequality for quantum integral. This inequality has been proved in [26,27].

Now, we give improvement of quantum integral version of Jensen's inequality in terms of means.
Theorem 11. Let ϕ : I ⟶ ℝ be a continuous convex function defined on the interval I and ψ : ½a, b ⟶ I be a continuous function. Then, for s ≥ 1 and r ≤ 1, the following inequalities hold: where z k ≔ ϕðψðq k b + ð1 − q k ÞaÞÞ − ϕð yÞ − ϕ′ + ð yÞðψðq k b + ð1 − q k ÞaÞ − yÞ with y = ð1/ðb − aÞb − aÞ Ð b a ψðxÞ a d q x.
As an application of the above theorem, we deduce improvement of quantum integral version of Hermite-Hadamard inequality.
In the following theorem, we present another improvement of quantum integral version of Jensen's inequality in terms of means. Theorem 13. Let ϕ : I ⟶ ℝ be a continuous convex function defined on the interval I and ψ : ½a, b ⟶ I be a continuous function. Then, for s ≥ 1 and r ≤ 1, the following inequalities hold: where Proof. The proof can be given by a similar way as the proof of Theorem 11.
As an application of Theorem 13, we deduce improvement of quantum integral version of Hermite-Hadamard inequality.

Applications for Some Well-Known Inequalities
In this section, we demonstrate improvements of some wellknown inequalities. We start with the Hermite-Hadamard inequality.
In the corollary below, we give improvements of power means inequality. Corollary 16. Let x = ðx 1 , ⋯, x n Þ and p = ðp 1 , ⋯, p n Þ be positive n-tuples with P n = ∑ n i=1 p i and s, r, k 1 , k 2 ∈ ℝ such that s ≥ 1, r ≤ 1 and k 1 ≤ k 2 .
Therefore, applying Theorem 4 for this function and x i ⟶ x k 2 i , we obtain (45) (ii) If k 1 > 0, then ϕðxÞ = x k 1 /k 2 is a concave function, so applying Theorem 4 for concave function, we deduce the reverse inequality in (45) (iii) If k 1 ≠ 0, k 2 ≥ 0 and ϕðxÞ = x k 2 /k 1 , then ϕ is convex. Therefore, applying Theorem 4 for this function and Similarly, we can prove the reverse inequality in (46).

Corollary 17.
Let p = ðp 1 , ⋯, p n Þ and x = ðx 1 , ⋯, x n Þ be positive n-tuples with P n = ∑ n i=1 p i and s, r, ∈ℝ such that s ≥ 1, r ≤ 1. Let the function h be monotone continuous function and f ∘ h −1 be convex function, then where Proof. The inequality (47) can be obtained by using Theorem 2 for ϕ ⟶ f ∘ h −1 and x i ⟶ hðx i Þ for i = 1, 2, ⋯, n.

Journal of Function Spaces
Remark 20. Similarly, we can present applications of the second main results. Also, we can give integral version of the above results.

Applications in Information Theory
Information theory studies how to measure, store, and transmit digital information. The field of information theory was initially entrenched by Hartley's work in 1920 and got worldwide attention by the work of Shannon in 1940s. The field of information theory is in close collaboration with probability theory, statistical mechanics, information, and electrical engineering. A fundamental measure in information theory is entropy that measures the uncertainty involved in the occurrence of a random process. After Shannon's work, the field attracted the concentration of various scientists and got inflated. Different entropy functional have been introduced that can be elaborated as generalized entropies.

Conclusion
In the literature, there are several results which are devoted to Jensen's and its related inequalities. In this manuscript, we have used a new approach for the derivation of improvements of Jensen's inequality. We have given Jensen's inequality and its improvements in quantum calculus. In particular, we have deduced Hermite-Hadamard type inequalities for q-integrals. We have also given applications of main results for some well-known inequalities and in information theory. The results of this manuscript which are initiated and given in quantum calculus may stimulate further research.

Data Availability
No data were used to support this study.