Some majorization integral inequalities for functions defined on rectangles.

In this paper, we first prove an integral majorization theorem related to integral inequalities for functions defined on rectangles. We then apply the result to establish some new integral inequalities for functions defined on rectangles. The results obtained are generalizations of weighted Favard's inequality, which also provide a generalization of the results given by Maligranda et al. (J. Math. Anal. Appl. 190:248-262, 1995) in an earlier paper.

Introduction

There is a certain intuitive appeal to the vague notion that the components of an n-tuple x are less spread out, or more nearly equal, than the components of an n-tuple y. The notion arises in a variety of contexts, and it can be made precise in a number of ways. In remarkably many cases, the appropriate statement is that x is majorized by y (or y majorizes x). Namely, for two n-tuples and , x is said to be majorized by y (denoted ) if for and , where and are rearrangements of x and y in descending order. A mathematical origin of majorization is illustrated by the work of Schur [35] on Hadamard’s determinant inequality. Many mathematical characterization problems are known to have solutions that involve majorization. Complete and superb references on the subject are the books [9, 28]. The comprehensive survey by Ando [7] provides alternative derivations, generalizations, and a different viewpoint.

The following theorem is known in the literature as the majorization theorem (see [20, 22, 23, 33, 35]).

Theorem 1.1

Let be a continuous convex function on the interval I, and let and be two n-tuples such that (). If x is majorized by y, then

The inequality asserted by Theorem 1.1 is also called majorization inequality. It is an inequality in elementary algebra for convex real-valued functions defined on an interval of the real line, and it generalizes the finite form of Jensen’s inequality. This majorization ordering is equivalently described in Kemperman’s review [25]. An extension of this fact for arbitrary real weights and decreasing n-tuples x and y can be found in [19]. General results of this type are obtained by Dragomir [17] and Niezgoda [30]. In recent years, many formulas such as Taylor formula, Hermite interpolating, Montgommery identities and inequalities for means, etc. have been used and generalized by majorization inequalities for n-convex functions; see [1–8, 10–15, 21, 24, 29, 31, 36, 37, 41–45] and references therein.

Recently, it has come to our attention that certain integral majorization theorems, we begin with recalling some relevant results. In 1947, Fuchs [19] gave the following integral majorization theorem for convex functions and two monotonic sequences.

Theorem 1.2

([19])

Let be continuous and increasing functions, and let be a function of bounded variation.

If and then for every continuous convex function ϕ, we have

If then for every continuous increasing convex function ϕ, we have

In 1995, Maligranda, Pečarić, and Persson [27] established the following analogue of the Fuchs inequality.

Theorem 1.3

([27])

Let w be a weight function, and let f and g be positive integrable functions on . Suppose that is a convex function and that and

If f is a decreasing function on , then

If g is an increasing function on , then

In 1933, Favard [18] proved the following results.

Theorem 1.4

Let Φ be a nonnegative continuous concave function on , not identically zero, and let ϒ be a convex function on , where Then

As a consequence of Theorem 1.4, the following inequality was also established in [18].

Theorem 1.5

Let Φ be a concave nonnegative function on . If , then

Maligranda, Pečarić, and Persson [27] gave the following generalization of the Favard inequality.

Theorem 1.6

([27])

Let Φ be a positive increasing concave function on , and let ϒ be a convex function on , where Then

Let Φ be a positive decreasing concave function on , and let ϒ be a convex function on , where Then

For further results related to generalizations, extensions, and refinements of the integral inequalities of majorization type, we refer the reader to [1–6, 26, 28, 32, 34, 38–40, 46–48].

In this paper, we extend majorization and Favard inequalities from functions defined on intervals to functions defined on rectangles. The results presented in this paper generalize the results of Maligranda, Pečarić, and Persson [27].

Preliminaries

In this section, we introduce some notions and lemmas.

Definition 2.1

A function defined on a convex subset Ω of is said to be convex if for all and .

In this paper, convex functions considered are supposed to be twice differentiable. It is well know that if the function ϕ is convex, then where , , and is the usual inner product in .

In the literature, there are many generalizations of convex functions in different directions. One of them is coordinate convex functions introduced by Dragomir [16].

Definition 2.2

([16])

Let us consider a bidimensional interval . A function is said to be coordinate convex if the partial functions defined as and defined as are convex where defined for all and .

Lemma 2.3

([16])

Every convex function is coordinate convex.

Lemma 2.4

([27])

Let v be a weight function on .

If h is a decreasing function on , then

If h is an increasing function on , then

Majorization inequalities for functions defined on rectangles

In this section, we establish some majorization integral inequalities for functions defined on rectangles. The following theorem is a generalization of Theorem 1.3 mentioned in the Introduction.

Theorem 3.1

Let w and p be positive continuous functions on and respectively, and let f, g and h, k be positive differentiable functions on and respectively. Suppose that is a convex function and that and

If g and k are decreasing functions on and respectively, then

If f and h are increasing functions on and respectively, then

Proof

□

Since is a convex function, we have

Put , , , in the last inequality and assume that Then we have Set and Then, from the assumptions in Theorem 3.1 we conclude that Multiplying both sides of inequality (3.3) by , we get

Integrating both sides of inequality (3.4) gives

By Fubini’s theorem we have

Using integration by parts, we obtain which yields

Since ϕ is convex on , by Lemma 2.3 we conclude that ϕ is coordinate convex on , and thus , . Also, k and g are decreasing, so that and . Thus it follows that and

Combining (3.5), (3.6), and (3.7) yields which implies the desired inequality (3.1).

Inequality (3.2) can be proved in the same way as inequality (3.1). Theorem 3.1 is proved.

Theorem 3.2

Let w and u be positive continuous functions on and , respectively, and let f, g and k, l be positive differentiable functions on and , respectively. Suppose that is a convex function. If f and l are decreasing functions on and , respectively, then the reverse inequality of (3.9) holds.

Let and be decreasing functions on and , respectively. If f and l are increasing functions on and , respectively, then If g and k are decreasing functions on and , respectively, then the reverse inequality of (3.8) holds.

Let and be increasing functions on and , respectively. If g and k are increasing functions on and , respectively, then

□

Using Lemma 2.4 with substitution and in (2.3), we obtain

Also, putting and in (2.3) gives

From (3.10) and (3.11) we deduce that

Additionally, it is easy to observe that

By relations (3.12), (3.13), (3.14), and (3.15), using inequality (3.2) given in Theorem 3.1, we obtain which is the desired inequality (3.8).

Using the majorization relations (3.12), (3.13), (3.14), and (3.15) and inequality (3.1), we get the reversed inequality of (3.8).

If we perform an interchange and , then inequality (3.9) follows immediately from (3.8). The reversed inequality of (3.9) can be deduced from the reversed inequality of (3.8) by using the same interchange. This completes the proof of Theorem 3.2.

Applications to the generalization of Favard’s inequality

As an application of Theorem 3.2, we establish some Favard-type inequalities for functions defined on rectangles, which generalize the results of Theorem 1.4 described in the Introduction.

Corollary 4.1

Let w and u be positive continuous functions on and , respectively, and let f and l be positive differentiable functions on and , respectively. Suppose that is a convex function.

Let and be decreasing functions on and , respectively. If f and l are increasing functions on and , respectively, then

Let and be increasing function on and , respectively. If f and l are decreasing functions on and , respectively, then

Note the simple fact that if () is a convex function, then (, ) is also a convex function. Using Theorem 3.2 with substitution in inequality (3.8) and choosing and , we get the required inequality (4.1). Applying the above substitution to the reverse inequality of (3.9) and choosing and , we derive inequality (4.2). □

Corollary 4.2

Let w and u be positive continuous functions on and , respectively, and let f and l be positive differentiable functions on and , respectively. Suppose that is a convex function.

Let and be decreasing functions on and , respectively. If f and l are increasing functions on and , respectively, then where and .

Let and be increasing functions on and , respectively. If f and l are decreasing functions on and , respectively, then where and .

Substituting and into the right-hand sides of (4.1), we get which, together with inequality (4.1), leads to the desired inequality (4.3).

Similarly, we can deduce inequality (4.4) by substituting and into the right-hand sides of (4.2). □

Corollary 4.3

Let w and u be positive continuous functions on and , respectively. Suppose that is a convex function. where and .

If f and l are positive increasing concave functions on and , respectively, then

where and .

If f and l are positive decreasing concave functions on and , respectively, then

□

By the first part of Corollary 4.2, to prove inequality (4.5), it suffices to prove that and are decreasing functions on and , respectively.

Consider the function

Differentiation of with respect to t gives

It follows that for since f is a positive concave function on . Thus we conclude that is a decreasing function on . In the same way, we can prove that is a decreasing function on . This proves inequality (4.5).

Similarly, we have We deduce that for , since f is positive concave function on . Thus, is an increasing function on . In the same way, we can prove that is an increasing function on . Therefore inequality (4.6) follows from the second part of Corollary 4.2.

Final remarks

Obviously, the results of Corollary 4.3 are generalizations of those given in Theorem 1.6 relating to Favard’s inequality. Indeed, if we put , , , and in (4.5) and (4.6), respectively, then we obtain the Favard inequality (1.6). Further, we can deduce the Favard inequality (1.7) by taking (). It is worth noting that the majorization inequalities asserted in Theorem 3.1 play a key role in proving Theorem 3.2. Further, with the help of Theorem 3.2, we obtain some significant results in Corollaries 4.1, 4.2, and 4.3.