On New Generalized Dunkel Type Integral Inequalities with Applications

In this paper, by applying majorization theory, we study the Schur convexity of functions related to Dunkel integral inequality. We establish some new generalized Dunkel type integral inequalities and their applications to inequality theory.


Introduction and Preliminaries
Over the last half a century, rapid developments in inequality theory and its applications have contributed greatly to many branches of mathematics such as linear and nonlinear analysis, differential equations, finance, statistics, physics, fractional calculus, and so on; for more details, one can refer to [1][2][3][4] and the references therein.
The original Dunkel integral inequality can be stated as follows.
In fact, if f (x) is a nonnegative continuous real-valued function on [a, b] (here, f is allowed to be a zero function), then from (1) one deduces the following fascinating concise inequality: (2) In 1923, Professor Issai Schur first systematically studied the functions preserving the ordering of majorization. In Schur's honor, such functions are named to have "Schurconvexity". During the previous more than four decades, majorization theory and Schurconvexity have been applied widely in many areas of mathematics including integral inequality, stochastic matrices, rearrangement theory, analytic inequalities, information theory, quantum correlations, quantum cryptography, combinatorial optimization, and other related fields (see, e.g., [7][8][9][10][11][12]).
Let us recall some basic definitions and notation that will be needed in this paper.
(i) Let x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) ∈ R n . x is said to be majorized by y (in symbols and y [1] ≥ · · · ≥ y [n] are rearrangements of x and y in a descending order; (ii) Ω is called convex if αx + βy ∈ Ω for any x, y ∈ Ω and α, β ≥ 0 with α + β = 1; (iii) Ω is called symmetric if x ∈ Ω implies Px ∈ Ω for every n × n permutation matrix P; (iv) A function ϕ : Ω → R is called symmetric if for every permutation matrix P, for all x ∈ Ω; ϕ is said to be Schur concave on Ω if and only if −ϕ is Schur convex.
The paper is divided into five sections. In Sections 2 and 3, by applying majorization theory, we present some new generalized Dunkel type integral inequalities and new Dunkel (p)-type integral inequalities for p ≥ 2. As applications of our new results, some new integral inequalities are established in Section 4. Finally, some summary and conclusions are given in Section 5.

Some Generalizations of Dunkel Integral Inequality
The following two known results are important for proving our new theorem.

Remark 1.
It is worth noticing that Lemma 2 is equivalent to the following: ϕ is a Schur convex (resp. Schur concave) function, if and only if it is symmetric on Ω and for all x ∈ D ∩ Ω, where D = {x : x 1 ≥ · · · ≥ x n }.
With the help of Lemmas 1 and 2, we can establish the following crucial result.

Theorem 2.
Let I be an interval of R. Assume that f (x) and g(x) are two nonnegative continuous real-valued functions on I, and κ(x) and λ(x) are two continuous real-valued functions on I. Define L : for any (a, b) ∈ I × I. Then the following holds: Schur concave on I × I.

Proof.
Obviously, L(a, b) is a symmetric operator for a, b ∈ I. So, without loss of generality, we may assume that b ≥ a. Since The proof is completed.
We now present the following generalized Dunkel type integral inequality which is one of the main results of this paper. Theorem 3. Let I be an interval of R. Assume that f (x) and g(x) are two nonnegative continuous real-valued functions on I, and κ(x) and λ(x) are two continuous real-valued functions on I.
Then the following holds: Proof. We only show case (i) and a similar argument could be made for the case (ii). Define L : x, a, b ∈ I, by applying Theorem 2 (i), we show that L is Schur convex on I × I. On the other hand, by using Lemma 1, we get The proof is completed.
As a direct consequence of Theorem 3, we can obtain the following generalized Dunkel integral inequality.
Thus, all the assumptions of Theorem 3 (i) are satisfied. Therefore the desired conclusion follows immediately from Theorem 3.
The following generalized Dunkel integral inequality is an immediate consequence of Theorem 4.

Corollary 1 (Generalized Dunkel integral inequality). Let f (x) be a continuous nonnegative real-valued function on [a, b] and m be any real number. Then
(4)

Remark 2.
It is worth noticing that inequality (3) in Theorem 4 and inequality (4) in Corollary 1 are real generalizations of inequality (2).

A New Dunkel (p)-Type Integral Inequality for p ≥ 2
In this section, we will present a new Dunkel (p)-type integral inequality for p ≥ 2. In order to prove our results, we need the following important auxiliary lemma. Lemma 3. Let k ∈ N ∪ {0}. Denote I k := 2kπ, 2kπ + π 2 . Assume that f (x) is a nonnegative continuous real-valued function on I k . Define M : Proof. It is obvious that M(a, b) is symmetric for a, b. Hence, without loss of generality, we may assume that b ≥ a. By Corollary 1, we have By Lemma 2, M is Schur concave on I k × I k . The proof is completed.
The following result is a new Dunkel (p)-type integral inequality for p ≥ 2.
for (a, b) ∈ I k × I k . By Lemmas 1 and 3, we obtain The following result is immediate from Theorem 5.

Some New Integral Inequalities
In this section, we will provide some new integral inequalities by applying our main results. Lemma 4 (Bessel inequality; see [1]). Let f (x) be a continuous or a piecewise continuous nonnegative function on [0, 2π]. The Fourier series of f (x) is Lemma 5 (see [1]). Let f (x) be a nonnegative integrable concave function on [a, b].
Proof. Using the notations in Lemma 4 and applying Theorem 4, we get By combining (5) with Bessel inequality (see Lemma 4), we obtain Let n = 1. By applying Lemma 5, we obtain The proof is completed.
In the same way, we can also show that κ(x) · κ(a) + λ(x) · λ(a) ≤ 1 for x ∈ [a, b]. Therefore, the desired inequality (6) follows immediately from Theorem 2 (i). Proof. From Theorem 8, we know that the right side of the desired inequality (7) holds. Next, we verify that the left side of desired inequality (7) also holds. By the AM-GM inequality, we have The proof is completed.

Conclusions
In this paper, we establish the following two important main results for the generalized Dunkel type integral inequality: •