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 x = (x 1 , x 2 , . . . , x n ) and y = (y 1 , y 2 , . . . , y n ), x is said to be majorized by y (denoted x ≺ y) if m i=1 x [i] ≤ m i=1 y [i] for m = 1, 2, . . . , n -1 and n i=1 x i = n i=1 y i , where x [1] ≥ x [2] ≥ · · · ≥ x [n] and y [1] ≥ y [2] ≥ · · · ≥ y [n] 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 f : I → R be a continuous convex function on the interval I, and let x = (x 1 , x 2 , . . . , x n ) and y = (y 1 , y 2 , . . . , y n ) be two n-tuples such that x i , y i ∈ I (i = 1, 2, . . . , n). If x is majorized by y, then (1.1) 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. (

then for every continuous convex function
then for every continuous increasing convex function In 1995, Maligranda, Pečarić, and Persson [27] established the following analogue of the Fuchs inequality.
In 1933, Favard [18] proved the following results. (1.6) As a consequence of Theorem 1.4, the following inequality was also established in [18].
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.
In this paper, convex functions considered are supposed to be twice differentiable. It is well know that if the function φ is convex, then . , x n ), y = (y 1 , y 2 , . . . , y n ) ∈ , and ·, · is the usual inner product in R n . In the literature, there are many generalizations of convex functions in different directions. One of them is coordinate convex functions introduced by Dragomir [16].  h is an increasing function on [a, b], 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.
Put x = f (t), y = h(s), w = g(t), z = k(s) in the last inequality and assume that .
Then we have and Then, from the assumptions in Theorem 3.1 we conclude that

k(s) w(t)p(s) dt ds
Also, k and g are decreasing, so that k (s) ≤ 0 and g (t) ≤ 0. Thus it follows that  Additionally, it is easy to observe that

k(s)u(s) ds w(t)u(s) dt ds,
which is the desired inequality (3.8).
(ii) If we perform an interchange f − → g (g − → f ) and k − → l (l − → k), 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. in inequality (3.8) and choosing g(t) = ta and k(s) = sc, we get the required inequality (4.1). Applying the above substitution to the reverse inequality of (3.9) and choosing g(t) = bt and k(s) = ds, we derive inequality (4.2).
We deduce that Q (t) > 0 for t ∈ [a, b), since f is positive concave function on [a, b]. Thus, Q(t) = f (t)/(bt) is an increasing function on [a, b). In the same way, we can prove that l(s)/(ds) is an increasing function on [c, d). 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 w(t) = 1, u(s) = 1, f = , and ψ(x, y) = ϒ(x) 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 ϒ(x) = x q (q > 1). 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.