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Discrete majorization type inequalities for convex functions on rectangles

  • Muhammad Adil Khan
  • Shan-He Wu
  • Hidayat Ullah
  • Yu-Ming ChuEmail author
Open Access
Research

Abstract

In this paper, we present several discrete majorization type inequalities for the convex functions defined on rectangles.

Keywords

Majorization inequality Favard’s inequality Convex function Coordinate convex function Support line inequality 

MSC

26D15 26B25 

1 Introduction and preliminaries

We start this section by giving some brief introduction about convex function and related results.

Definition 1.1

Let \(\textbf{I}\subseteq \mathbb{R}^{n}\) be a convex set. Then the function \(\phi :\textbf{I}\mapsto \mathbb{R}\) is said to be convex if the inequality
$$ \phi \bigl(\zeta \textbf{x}+(1-\zeta )\textbf{y}\bigr)\leq \zeta \phi ( \textbf{x})+(1-\zeta )\phi (\textbf{y}) $$
holds for all \(\textbf{x},\textbf{y}\in \textbf{I}\) and \(\zeta \in [0,1]\).
It is well-known that a convex function may not be differentiable. If the function ϕ is convex, then the support line inequality
$$ \phi (\textbf{x})-\phi (\textbf{y})\geq \nabla _{+}\phi (\textbf{y}) ( \textbf{x}-\textbf{y}) $$
holds for all \(\textbf{x}, \textbf{y}\in \textbf{I}\), where
$$ \nabla _{+}\phi (\textbf{y}) (\textbf{x}-\textbf{y})= \biggl\langle \frac{ \partial \phi _{+}(\textbf{y})}{\partial \textbf{y}}, (\textbf{x}- \textbf{y}) \biggr\rangle ,\qquad \frac{\partial \phi _{+}(\textbf{y})}{ \partial \textbf{y}}= \biggl(\frac{\partial \phi _{+}(\textbf{y})}{ \partial y_{1}},\frac{\partial \phi _{+}(\textbf{y})}{\partial y_{2}}, \ldots , \frac{\partial \phi _{+}(\textbf{y})}{\partial y_{n}} \biggr) $$
for \(\textbf{x}=(x_{1},x_{2},\dots ,x_{n})\), \(\textbf{y}=(y_{1},y_{2}, \dots ,y_{n})\in \textbf{I}\) and \(\langle \cdot ,\cdot \rangle \) is the ordinary inner product in \(\mathbb{R}^{n}\).

Convex functions have many important applications in mathematics, physics, statistics and engineering [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Currently, many refinements, variants, generalizations and extensions for the convexity can be found in the literature [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50].

In [51], Dragomir introduced the definition of the coordinate convex functions as follows:

Definition 1.2

(See [51])

Let \([a_{1},a_{2}]\) and \([b_{1},b_{2}]\) be two intervals in \(\mathbb{R}\) and \(S=[a_{1},a_{2}]\times [b_{1},b_{2}]\). A function \(\phi :S\mapsto \mathbb{R}\) is said to be coordinate convex on S if the partial functions \(\phi _{y}: [a_{1},a_{2}]\mapsto \mathbb{R}\) and \(\phi _{x}:[b_{1},b_{2}]\mapsto \mathbb{R}\) defined by
$$ \phi _{y}(u)=\phi (u,y), \qquad \phi _{x}(v)=\phi (x,v) $$
are convex.

Lemma 1.3

(See [51])

Every convex function defined on a rectangle is coordinate convex, but the converse is not true, in general.

In the remaining part of this section, we give a comprehensive introduction about majorization theory.

Let \(n\geq 2\), \(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\) and \(\textbf{b}=(b_{1},b_{2},\dots ,b_{n})\) be two n-tuples of real numbers, and
$$ a_{[1]}\geq a_{[2]}\geq \cdots \geq a_{[n]}, \qquad b_{[1]}\geq b_{[2]} \geq \cdots \geq b_{[n]} $$
be their ordered arrangement.

Definition 1.4

The n-tuple b is said to be majorized by the n-tuple a, or a majorizes b, in symbols \(\textbf{a}\succ \textbf{b}\), if
$$\begin{aligned}& \sum_{i=1}^{k}a_{[i]}\geq \sum _{i=1}^{k}b_{[i]} \quad (k=1,2,\dots ,n-1), \\& \sum_{i=1}^{n}b_{i}=\sum _{i=1}^{n}a_{i}. \end{aligned}$$

Let \(\textbf{a}, \textbf{b}\in \mathbb{R}^{n}\) be two vectors such that a majorizes b. Then from the basic knowledge of linear algebra we clearly see that there exist a set of probabilities \((q_{1},q_{2},\dots ,q_{n})\) with \(\sum_{i=1}^{n}q_{i}=1\) and a set of permutations \((P_{1},P_{2},\dots ,P_{n})\) such that \(\textbf{a}=\sum_{i=1}^{n}P_{i}q_{i}\textbf{b}\). Alternatively, it can be shown that there exists a doubly stochastic matrix D such that \(\textbf{a}=D \textbf{b}\). In fact, the latter characterization of majorization relation implies that the set of vectors a that satisfy \(\textbf{a}\succ \textbf{b}\) is the convex hull spanned by the n! points formed from the permutations of the elements of b.

Let S and T be two Hermitian operators. Then we say that the Hermitian operator S majorizes the Hermitian operator T if the set of eigenvalues of S majorizes the set of eigenvalues values of T.

Majorization is a partial order relation between the vectors, which precisely defines the vague notion that the components of one vector are “less spread out” or “more nearly equal” than the components of another vector. And the functions that preserve the majorization order are called Schur convex functions. Many problems arising in signal processing and communications involve comparing vector-valued strategies or solving optimization problems with vector- or matrix-valued variables. Majorization theory is a key tool that allows us to solve or simplify these problems.

The following Theorem 1.5 is well-known in the literature as the majorization theorem and for its proof we refer to Marshall and Olkin [52]. This result is due to Hardy, Littlewood and Pólya [53] and it can also be found in [54].

Theorem 1.5

LetIbe an interval in\(\mathbb{R}\), and\(\textbf{a}=(a _{1},a_{2},\dots ,a_{n})\)and\(\textbf{b}=(b_{1},b_{2},\dots ,b_{n})\)be twon-tuples such that\(a_{i},b_{i}\in \mathbf{{I}}\) (\(i=1,2, \dots ,n\)). Then the inequality
$$ \sum_{i=1}^{n}\phi (a_{i})\geq \sum_{i=1}^{n}\phi (b_{i}) $$
holds for every continuous convex function\(\phi :\mathbf {I}\mapsto \mathbb{R}\)if and only if\(\textbf{a}\succ \textbf{b}\).

The following Theorem 1.6 is a weighted version of Theorem 1.5 and is given by Fuchs [55].

Theorem 1.6

Let\(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\)and\(\textbf{b}=(b_{1},b _{2},\dots , b_{n})\)be two decreasingn-tuples such that\(a_{i}, b_{i}\in \mathbf{{I}}\) (\(i=1,2,\dots ,n\)), and\(\textbf{p}=(p _{1},p_{2},\dots ,p_{n})\)be a realn-tuple with
$$\begin{aligned}& \sum_{i=1}^{k}p_{i}a_{i} \geq \sum_{i=1}^{k}p_{i}b_{i} \quad (k=1,2, \dots ,n-1), \\& \sum_{i=1}^{n}p_{i}a_{i}= \sum_{i=1}^{n}p_{i}b_{i}. \end{aligned}$$
Then the inequality
$$ \sum_{i=1}^{n}p_{i}\phi (a_{i})\geq \sum_{i=1}^{n}p_{i} \phi (b_{i}) $$
(1.1)
holds for each continuous convex function\(\phi :\mathbf {I}\mapsto \mathbb{R}\).

Another result similar to that above with some relaxed conditions on a, b and stricter condition on function ϕ was obtained by Bullen, Vasić and Stanković [56].

Theorem 1.7

Let\(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\)and\(\textbf{b}=(b_{1},b _{2},\dots ,b_{n})\)be two decreasingn-tuples, and\(\textbf{p}=(p _{1},p_{2},\dots , p_{n})\)be a realn-tuple. If the inequality
$$ \sum_{i=1}^{k}p_{i}a_{i} \geq \sum_{i=1}^{k}p_{i}b_{i} $$
(1.2)
holds for\(k=1,2,\dots ,n\), then inequality (1.1) holds for each continuous increasing convex function\(\phi :\mathbf {I}\mapsto \mathbb{R}\). Ifaandbare increasingn-tuples and the reverse inequality in (1.2) holds for\(k=1,2,\dots ,n\), then inequality (1.1) holds for each continuous decreasing convex function\(\phi :\mathbf {I}\mapsto \mathbb{R}\).

Dragomir [57] presented another majorization result, which has been obtained by using support line and Chebyshev’s inequalities.

Theorem 1.8

LetIbe an interval in\(\mathbb{R}\), \(\phi :\mathbf {I}\mapsto \mathbb{R}\)be a convex function, \(\textbf{a}=(a_{1}, a_{2},\dots , a _{n})\)and\(\textbf{b}=(b_{1}, b_{2},\dots , b_{n})\)be two realn-tuples such that\(a_{i}, b_{i}\in \mathbf{{I}}\) (\(i=1,2,\dots ,n\)), and\(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\)be a non-negative realn-tuple with\(P_{n}=\sum_{i=1}^{n}p_{i}>0\). Ifband\(\textbf{a}-\textbf{b}\)are monotonic in the same sense with\(\sum_{i=1}^{n}p_{i}a_{i}=\sum_{i=1}^{n}p_{i}ab_{i}\), then one has
$$ \sum_{i=1}^{n}p_{i} \phi (a_{i})\geq \sum_{i=1}^{n}p_{i} \phi (b_{i}). $$
(1.3)
Ifϕis strictly convex onIand\(p_{i}>0\) (\(i=1,2, \dots ,n\)), then equality holds in (1.3) if and only if\(a_{i}=b_{i}\)for all\(i=1,2,\dots ,n\).

In this paper, our focus is on the majorization type results for the convex functions defined on rectangles. We shall extend classical majorization inequality for majorized tuples and establish weighted versions of majorization inequalities for certain tuples, for example, monotonic tuples in the same sense, monotonic tuples in mean, etc. For obtaining these results, we use Chebyshev’s inequality, Abel transformation, support line inequality of convex function and the fact that every convex function defined on rectangles is coordinate convex. At the end of the paper, we provide Favard’s type inequalities by using the generalized majorization results.

2 Main results

We start by giving a majorization inequality for the convex functions defined on rectangles by using majorized tuples.

Theorem 2.1

Let\(\mathbf {I}_{1}\)and\(\mathbf {I}_{2}\)be any two intervals in\(\mathbb{R}\), \(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\)and\(\textbf{b}=(b _{1},b_{2},\dots ,b_{n})\)be twon-tuples such that\(a_{i},b_{i} \in \mathbf {I}_{1}\) (\(i=1,2,\dots ,n\)), and\(\textbf{c}=(c_{1},c_{2}, \dots ,c_{m})\)and\(\textbf{d}=(d_{1},d_{2},\dots ,d_{m})\)be twom-tuples such that\(c_{j},d_{j}\in \mathbf {I}_{2}\) (\(j=1,2,\dots ,m\)). If\(\textbf{a}\succ \textbf{b}\)and\(\textbf{c}\succ \textbf{d}\), then the inequality
$$ \sum_{i=1}^{n}\sum _{j=1}^{m}\phi (a_{i},c_{j})\geq \sum_{i=1}^{n} \sum _{j=1}^{m}\phi (b_{i},d_{j}) $$
(2.1)
holds for each convex function\(\phi :\textbf{I}_{1}\times \textbf{I} _{2}\mapsto \mathbb{R}\).

Proof

Without loss of generality, we assume that the tuples a, b, c and d are in decreasing order and \(a_{i}\neq b_{i}\), \(c_{j}\neq d_{j}\) for all \(i\in \{1,2,\dots ,n\}\) and \(j\in \{1,2,\dots ,m\}\). Let
$$\begin{aligned}& A_{k}=\sum_{i=1}^{k}a_{i},\qquad B_{k}=\sum_{i=1}^{k}b_{i} \quad (k=1,2, \dots ,n), \\& C_{l}=\sum_{j=1}^{l}c_{j},\qquad D_{l}=\sum_{j=1}^{l}d_{j} \quad (l=1,2, \dots ,m), \\& A_{0}=B_{0}=C_{0}=D_{0}=0. \end{aligned}$$
Then it follows from the definition of majorization that
$$ A_{n}=B_{n},\qquad C_{m}=D_{m}. $$
Let \(t_{i,j}\) and \(s_{i,j}\) be defined by
$$\begin{aligned}& t_{i,j}:=\nabla \phi (a_{i},b_{i};c_{j})= \frac{\phi (a_{i},c_{j})- \phi (b_{i},c_{j})}{a_{i}-b_{i}}, \\& s_{i,j}:=\nabla \phi (b_{i};c_{j},d_{j})= \frac{\phi (b_{i},c_{j})- \phi (b_{i},d_{j})}{c_{j}-d_{j}}. \end{aligned}$$
Then we clearly see that
$$\begin{aligned} \phi (a_{i},c_{j})-\phi (b_{i},d_{j}) =& \phi (a_{i},c_{j})-\phi (b_{i},c _{j})+ \phi (b_{i},c_{j})-\phi (b_{i},d_{j}) \\ =&\frac{\phi (a_{i},c_{j})-\phi (b_{i},c_{j})}{a_{i}-b_{i}}(a_{i}-b _{i})+\frac{\phi (b_{i},c_{j})-\phi (b_{i},d_{j})}{c_{j}-d_{j}}(c_{j}-d _{j}) \\ =&t_{i,j} (A_{i}-A_{i-1}-B_{i}+B_{i-1} )+s_{i,j} (C_{j}-C _{j-1}-D_{j}+D_{j-1} ). \end{aligned}$$
Summing over all i and j gives
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}\phi (a_{i},c_{j})-\sum _{i=1}^{n}\sum_{j=1} ^{m}\phi (b_{i},d_{j}) \\& \quad =\sum_{i=1}^{n}\sum _{j=1}^{m}t_{i,j} (A_{i}-A_{i-1}-B_{i}+B_{i-1} ) \\& \quad \quad {}+\sum_{i=1}^{n}\sum _{j=1}^{m}s_{i,j} (C_{j}-C_{j-1}-D_{j}+D_{j-1} ) \\& \quad =\sum_{j=1}^{m} \Biggl[\sum _{i=1}^{n}t_{i,j} (A_{i}-B_{i} )- \sum_{i=1}^{n}t_{i,j} (A_{i-1}-B_{i-1} ) \Biggr] \\& \qquad {}+\sum_{i=1}^{n} \Biggl[\sum _{j=1}^{m}s_{i,j} (C_{j}-D_{j} )- \sum_{j=1}^{m}s_{i,j} (C_{j-1}-D_{j-1} ) \Biggr] \\& \quad =\sum_{j=1}^{m} \Biggl[\sum _{i=1}^{n-1}t_{i,j} (A_{i}-B_{i} )- \sum_{i=2}^{n}t_{i,j} (A_{i-1}-B_{i-1} ) \Biggr] \\& \qquad {}+\sum_{i=1}^{n} \Biggl[\sum _{j=1}^{m-1}s_{i,j} (C_{j}-D_{j} )- \sum_{j=2}^{m}s_{i,j} (C_{j-1}-D_{j-1} ) \Biggr] \\& \quad =\sum_{j=1}^{m} \Biggl[\sum _{i=1}^{n-1}t_{i,j} (A_{i}-B_{i} )- \sum_{i=1}^{n-1}t_{i+1,j} (A_{i}-B_{i} ) \Biggr] \\& \qquad {}+\sum_{i=1}^{n} \Biggl[\sum _{j=1}^{m-1}s_{i,j} (C_{j}-D_{j} )- \sum_{j=1}^{m-1}s_{i,j+1} (C_{j}-D_{j} ) \Biggr] \\& \quad =\sum_{j=1}^{m} \Biggl[\sum _{i=1}^{n-1} (t_{i,j}-t_{i+1,j} ) (A_{i}-B_{i} ) \Biggr] \\& \qquad {}+\sum_{i=1}^{n} \Biggl[\sum _{j=1}^{m-1} (s_{i,j}-s_{i,j+1} ) (C_{j}-D_{j} ) \Biggr]. \end{aligned}$$
(2.2)
Since ϕ is a convex function on \(\textbf{I}_{1}\times \textbf{I}_{2}\), ϕ is a coordinate convex function on \(\textbf{I}_{1}\times \textbf{I}_{2}\). Thus, \(t_{i,j}\) is decreasing with respect to i for each fixed j and \(s_{i,j}\) is decreasing with respect to j for each fixed i. Hence \(t_{i,j}-t_{i+1,j}\geq 0\) for all \(i\in \{1,2,\dots ,n-1\}\) and \(s_{i,j}-s_{i,j+1}\geq 0\) for all \(j\in \{1,2,\dots ,m-1\}\). From the definition of majorization we get \(A_{i}-B_{i}\geq 0\) for all \(i\in \{1,2,\dots ,n-1\}\) and \(C_{j}-D _{j}\geq 0\) for all \(j\in \{1,2,\dots ,m-1\}\). Therefore, the right-hand side of (2.2) is non-negative, and hence we have
$$ \sum_{i=1}^{n}\sum _{j=1}^{m}\phi (a_{i},c_{j})-\sum _{i=1}^{n}\sum_{j=1} ^{m}\phi (b_{i},d_{j})\geq 0, $$
which is equivalent to (2.1). □

In the following Theorem 2.2, we prove a general inequality for the convex functions defined on rectangles, which implies majorization inequality for certain tuples.

Theorem 2.2

Let\(\mathbf {I}_{1}\)and\(\mathbf {I}_{2}\)be any two intervals in\(\mathbb{R}\), \(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\)and\(\textbf{b}=(b _{1},b_{2},\dots ,b_{n})\)be twon-tuples such that\(a_{i}, b_{i} \in \mathbf {I}_{1}\) (\(i=1,2,\dots ,n\)), \(\textbf{c}=(c_{1},c_{2},\dots ,c _{m})\)and\(\textbf{d}=(d_{1},d_{2},\dots ,d_{m})\)be twom-tuples such that\(c_{j},d_{j}\in \mathbf {I}_{2}\) (\(j=1,2,\dots ,m\)) and\(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\)and\(\textbf{w}=(w_{1},w_{2}, \dots ,w_{m})\)be any positive realn- andm-tuples, respectively. If\(\phi :\mathbf {I}_{1}\times \mathbf {I}_{2}\mapsto \mathbb{R}\)is a convex function, then one has
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (a_{i},c_{j})-\sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (b_{i},d_{j}) \\& \quad \geq \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}t_{i}(a_{i}-b_{i})+ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j}(c_{j}-d_{j}), \end{aligned}$$
(2.3)
where\(t_{i}\)is the positive partial derivative ofϕwith respect to the first variable at\(b_{i}\) (\(i=1,2,\dots ,n\)) and\(s_{j}\)is the positive partial derivative ofϕwith respect to the second variable at\(d_{j}\) (\(j=1,2,\dots ,m\)).

Proof

It follows from the convexity of the function \(\phi :\mathbf {I}_{1} \times \mathbf {I}_{2}\mapsto \mathbb{R}\) that
$$ \phi (x,y)-\phi (w,z)\geq \bigl\langle \nabla \phi (w,z),(x-w,y-z)\bigr\rangle $$
for all \((x,y),(w,z)\in \mathbf {I}_{1}\times \mathbf {I}_{2}\). That is,
$$ \phi (x,y)-\phi (w,z)\geq \frac{\partial \phi }{\partial w}(w,z) (x-w)+\frac{ \partial \phi }{\partial z}(w,z) (y-z). $$
(2.4)
Now, applying (2.4) and by choosing \(x\rightarrow a_{i}\), \(y\rightarrow c_{i}\), \(w\rightarrow b_{i}\) and \(z\rightarrow d_{j}\), we get
$$ \phi (a_{i},c_{j})-\phi (b_{i},d_{j}) \geq t_{i}(a_{i}-b_{i})+s_{j}(c _{j}-d_{j}). $$
(2.5)
Multiplying both sides of (2.5) by \(p_{i}w_{j}\) and summing over the indices, we obtain
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (a_{i},c_{j})-\sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (b_{i},d_{j}) \\& \quad \geq \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}t_{i}(a_{i}-b_{i})+ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j}(c_{j}-d_{j}). \end{aligned}$$
 □

If \(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\) and \(\textbf{w}=(w_{1},w _{2},\dots ,w_{m})\) are two tuples, then throughout this paper \(P_{k}\) and \(W_{j}\) are defined by \(P_{k}=\sum_{i=1}^{k}p_{i}\) and \(W_{j}=\sum_{i=1}^{j}w_{i}\), \(k=1,2,\dots ,n\) and \(j=1,2,\dots ,m\).

Some majorization type results, which are obtained from the above Theorem 2.2, are given in the form of the following Propositions 2.3 and 2.5.

Proposition 2.3

Assume that all the hypotheses of Theorem 2.2hold. Additionally, band\(\textbf{a}-\textbf{b}\)are monotonic in the same sense, andcand\(\textbf{c}-\textbf{d}\)are monotonic in the same sense. If
$$ \sum_{i=1}^{n}a_{i}p_{i}= \sum_{i=1}^{n}b_{i}p_{i} $$
(2.6)
and
$$ \sum_{j=1}^{m}c_{j}w_{j}= \sum_{j=1}^{m}d_{j}w_{j}, $$
(2.7)
then
$$ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (a_{i},c_{j})\geq \sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (b_{i},d_{j}). $$
(2.8)

Proof

Since ϕ is a convex function on \(\textbf{I}_{1}\times \textbf{I}_{2}\), ϕ is a coordinate convex function on \(\textbf{I}_{1}\times \textbf{I}_{2}\). If b is an increasing n-tuple, then \((t_{1},t_{2},\dots ,t_{n})\) is an increasing n-tuple, where \(t_{i}\) is the positive partial derivative of ϕ with respect to the first variable at \(b_{i}\) (\(i=1,2,\dots ,n\)). If b and \(\textbf{a}-\textbf{b}\) are increasing n-tuples, then, applying Chebyshev’s inequality to the first term on right-hand side of (2.3) and using (2.6), we have
$$\begin{aligned} \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}t_{i}(a_{i}-b_{i}) =& \sum_{j=1} ^{m}w_{j} \Biggl[\sum _{i=1}^{n}p_{i}t_{i}(a_{i}-b_{i}) \Biggr] \\ \geq& \sum_{j=1}^{m}w_{j} \Biggl[ \frac{1}{P_{n}}\sum_{i=1}^{n}p_{i}t _{i}\sum_{i=1}^{n}p_{i}(a_{i}-b_{i}) \Biggr]=0. \end{aligned}$$
(2.9)
Similarly, since ϕ is a coordinate convex function on \(\textbf{I}_{1}\times \textbf{I}_{2}\), if d is an increasing m-tuple, then \((s_{1},s_{2},\dots ,s_{m})\) is an increasing m-tuple, where \(s_{j}\) is the positive partial derivative of ϕ with respect to the second variable at \(d_{j}\) (\(j=1,2,\dots ,m\)). If d and \(\textbf{c}-\textbf{d}\) are increasing m-tuples, then, applying Chebyshev’s inequality to the second term on right-hand side of (2.3) and using (2.7), we have
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j}(c_{j}-d_{j}) \\& \quad = \sum_{i=1} ^{n}p_{i} \Biggl[\sum _{j=1}^{m}w_{j}s_{j}(c_{j}-d_{j}) \Biggr] \\& \quad \geq \sum_{i=1}^{n}p_{i} \Biggl[ \frac{1}{W_{m}}\sum_{j=1}^{m}w_{j}s _{j}\sum_{j=1}^{m}w_{j}(c_{j}-d_{j}) \Biggr]=0. \end{aligned}$$
(2.10)
Using (2.9) and (2.10) in (2.3), we get
$$ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j}(a_{i}-b_{i})- \sum_{i=1} ^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j}(c_{j}-d_{j}) \geq 0, $$
which is equivalent to (2.8).

Similarly, we can prove inequality (2.8) in the remaining cases. □

Remark 2.4

In what follows, a convex function is said to be monotonic increasing if it is monotonic increasing with respect to each of its variables.

Proposition 2.5

Let all the assumptions of Theorem 2.2hold. If\(\phi :\mathbf {I}_{1} \times \mathbf {I}_{2}\mapsto \mathbb{R}\)is an increasing convex function, band\(\textbf{a}-\textbf{b}\)are monotonic in the same sense, dand\(\textbf{c}-\textbf{d}\)are monotonic in the same sense, and
$$ \sum_{i=1}^{n}a_{i}p_{i} \geq \sum_{i=1}^{n}b_{i}p_{i} $$
(2.11)
and
$$ \sum_{j=1}^{m}c_{j}w_{j} \geq \sum_{j=1}^{m}d_{j}w_{j}, $$
then inequality (2.8) holds.

Proof

Since ϕ is an increasing function on \(\textbf{I}_{1}\times \textbf{I}_{2}\), we have \(t_{i}\geq 0 \) (\(i=1,2,\dots ,n\)), where \(t_{i}\) is the positive partial derivative of ϕ with respect to the first variable at \(b_{i}\) (\(i=1,2,\dots ,n\)), thus
$$ \sum_{i=1}^{n}p_{i}t_{i} \geq 0. $$
(2.12)
Using (2.11) and (2.12) in the right-hand side of (2.9), we have
$$\begin{aligned} \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}t_{i}(a_{i}-b_{i}) \geq 0. \end{aligned}$$
(2.13)
Similarly, we have
$$ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j}(c_{j}-d_{j}) \geq 0. $$
(2.14)
Using (2.13) and (2.14) in (2.3), we get (2.8).

Similarly, we can prove inequality (2.8) in the remaining cases. □

The following Theorem 2.6 is another weighted discrete version of majorization theorem.

Theorem 2.6

Let\(\textbf{I}_{1}\), \(\textbf{I}_{2}\)be two intervals in\(\mathbb{R}\), \(\phi :\mathbf {I}_{1}\times \mathbf {I}_{2}\mapsto \mathbb{R}\)be a convex function, \(\textbf{a}=(a_{1},a_{2},\dots , a_{n})\)and\(\textbf{b}=(b_{1},b_{2},\dots , b_{n})\)be twon-tuples such that\(a_{i},b_{i}\in \mathbf {I}_{1}\) (\(i=1,2,\dots ,n\)), \(\textbf{c}=(c_{1},c _{2},\dots ,c_{m})\)and\(\textbf{d}=(d_{1},d_{2},\dots ,d_{m})\)be twom-tuples such that\(c_{j},d_{j}\in \mathbf {I}_{2}\) (\(j=1,2,\dots ,m\)), \(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\)and\(\textbf{w}=(w_{1},w_{2}, \dots ,w_{n})\)be two any positive realn- andm-tuples, respectively, and
$$\begin{aligned}& \sum_{i=1}^{k}b_{i}p_{i} \leq \sum_{i=1}^{k}a_{i}p_{i} \quad (k=1,2, \dots ,n-1), \end{aligned}$$
(2.15)
$$\begin{aligned}& \sum_{i=1}^{k}d_{j}w_{j} \leq \sum_{i=1}^{k}c_{j}w_{j} \quad (k=1,2, \dots ,m-1), \end{aligned}$$
(2.16)
$$\begin{aligned}& \sum_{i=1}^{n}b_{i}p_{i}= \sum_{i=1}^{n}a_{i}p_{i}, \end{aligned}$$
(2.17)
$$\begin{aligned}& \sum_{i=1}^{m}d_{j}w_{j}= \sum_{i=1}^{m}c_{j}w_{j}. \end{aligned}$$
(2.18)
Then the following statements are true:
  1. (i)
    Ifbanddare decreasingn- andm-tuples, respectively, then
    $$ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (b_{i},d_{j})\leq \sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (a_{i},c_{j}). $$
    (2.19)
     
  2. (ii)
    Ifaandcare increasingn- andm-tuples, respectively, then
    $$ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (a_{i},c_{j})\leq \sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (b_{i},d_{j}). $$
    (2.20)
     

Proof

For part (i), we use Abel’s transformation to prove part (i). Let
$$ A_{0}=B_{0}=0,\qquad A_{k}=\sum _{i=1}^{k}p_{i}a_{i},\qquad B_{k}= \sum_{i=1}^{k}p_{i}b_{i} \quad (k=1,2,\dots ,n) $$
and
$$ C_{0}=D_{0}=0, \qquad C_{k}=\sum _{j=1}^{k}w_{j}c_{j},\qquad D_{k}= \sum_{j=1}^{k}w_{j}d_{j} \quad (k=1,2,\dots ,m). $$
Then from (2.17) and (2.18) we have
$$ A_{n}=B_{n},\qquad C_{m}=D_{m}. $$
Since ϕ is a convex function on \(\textbf{I}_{1}\times \textbf{I}_{2}\), ϕ is a coordinate convex function on \(\textbf{I}_{1}\times \textbf{I}_{2}\). If b and d are decreasing n- and m-tuples, respectively, then \((t_{1},t_{2}, \dots ,t_{n})\) and \((s_{1},s_{2},\dots ,s_{m})\) are decreasing n- and m-tuples, respectively, where \(t_{i}\) is the positive partial derivative of ϕ with respect to the first variable at \(b_{i}\) (\(i=1,2,\dots ,n\)) and \(s_{j}\) is the positive partial derivative of ϕ with respect to the second variable at \(d_{j} \) (\(j=1,2, \dots ,m\)). It follows from (2.3) that
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (a_{i},c_{j})-\sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (b_{i},d_{j}) \\& \quad \geq \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}t_{i}(a_{i}-b_{i})+ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j}(c_{j}-d_{j}) \\& \quad =\sum_{j=1}^{m}w_{j} \Biggl[\sum _{i=1}^{n}t_{i}(p_{i}a_{i}-p_{i}b_{i}) \Biggr]+ \sum_{i=1}^{n}p_{i} \Biggl[\sum_{j=1}^{m}s_{j}(w_{j}c_{j}-w_{j}d_{j}) \Biggr] \\& \quad =\sum_{j=1}^{m}w_{j} \Biggl[\sum _{i=1}^{n}t_{i}(A_{i}-A_{i-1}-B_{i}+B _{i-1}) \Biggr] \\& \qquad {}+\sum_{i=1}^{n}p_{i} \Biggl[\sum _{j=1}^{m}s_{j}(C_{i}-C_{i-1}-D_{i}+D _{i-1}) \Biggr] \\& \quad =\sum_{j=1}^{m}w_{j} \Biggl[\sum _{i=1}^{n}t_{i}(A_{i}-B_{i})- \sum_{i=1} ^{n}t_{i}(A_{i-1}-B_{i-1}) \Biggr] \\& \qquad {}+\sum_{i=1}^{n}p_{i} \Biggl[\sum _{j=1}^{m}s_{j}(C_{i}-D_{i})- \sum_{j=1} ^{m}s_{j}(C_{i-1}-D_{i-1}) \Biggr] \\& \quad =\sum_{j=1}^{m}w_{j} \Biggl[\sum _{i=1}^{n-1}(t_{i}-t_{i+1}) (A_{i}-B _{i}) \Biggr] +\sum_{i=1}^{n}p_{i} \Biggl[\sum _{j=1}^{m-1}(s_{j}-s_{j+1}) (C_{i}-D _{i}) \Biggr]. \end{aligned}$$
(2.21)
Since \((t_{1},t_{2},\dots ,t_{n})\) and \((s_{1},s_{2},\dots ,s_{m})\) are decreasing n- and m-tuples, respectively, \(t_{i}- t_{i+1}\geq 0\) (\(i=1,2,\dots ,n-1\)) and \(s_{j}-s_{j+1}\geq 0\) (\(j=1,2,\dots ,m-1\)). Also from the assumptions (2.15) and (2.16) we know that \(A_{i}-B_{i} \geq 0\) (\(i=1,2,\dots ,n-1\)) and \(C_{j}-D_{j}\geq 0\) (\(j=1,2,\dots , m-1\)). Thus
$$ \sum_{j=1}^{m}w_{j} \Biggl[\sum _{i=1}^{n-1}(t_{i}-t_{i+1}) (A_{i}-B_{i}) \Biggr]+ \sum_{i=1}^{n}p_{i} \Biggl[\sum_{j=1}^{m-1}(s_{j}-s_{j+1}) (C_{i}-D_{i}) \Biggr] \geq 0. $$
(2.22)
Using (2.22) in (2.21), we get
$$ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (a_{i},c_{j})-\sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (b_{i},d_{j})\geq 0, $$
which is equivalent to (2.19).

Similarly, we can prove inequality (2.20) for the remaining cases. □

Definition 2.7

Let \(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\) be a positive real n-tuple. Then the real n-tuple \(\textbf{a}=(a_{1},a_{2},\dots ,a _{n})\) is said to be monotonic increasing in mean relative to \(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\) if
$$ \frac{1}{P_{k}}\sum_{i=1}^{k}p_{i}a_{i} \leq \frac{1}{P_{k+1}}\sum_{i=1} ^{k+1}a_{i}p_{i} \quad (k=1,2,\dots ,n-1), $$
and decreasing in mean relative to \(\textbf{p}=(p_{1},p_{2},\dots ,p _{n})\) if
$$ \frac{1}{P_{k}}\sum_{i=1}^{k}p_{i}a_{i} \geq \frac{1}{P_{k+1}}\sum_{i=1} ^{k+1}a_{i}p_{i} \quad (k=1,2,\dots ,n-1). $$

The following Lemma 2.8 is due to Biernacki [58] (for a generalization, see Burkill and Mirsky [59]).

Lemma 2.8

Let\(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\)and\(\textbf{b}=(b_{1},b _{2},\dots ,b_{n})\)be any two realn-tuples, which are monotonic in mean relative to positive realn-tuple\(\textbf{p}=(p_{1},p_{2}, \dots ,p_{n})\)in the same sense, that is,
$$ \frac{1}{P_{k}}\sum_{i=1}^{k}p_{i}a_{i} \lesseqqgtr \frac{1}{P_{k+1}} \sum_{i=1}^{k+1}a_{i}p_{i} \quad (k=1,2,\dots ,n-1) $$
and
$$ \frac{1}{P_{k}}\sum_{i=1}^{k}p_{i}b_{i} \lesseqqgtr \frac{1}{P_{k+1}} \sum_{i=1}^{k+1}b_{i}p_{i} \quad (k=1,2,\dots ,n-1). $$
Then
$$ \frac{1}{P_{n}}\sum_{i=1}^{n}p_{i}a_{i}b_{i} \geq \frac{1}{P_{n}}\sum_{i=1}^{n}a_{i}p_{i} \frac{1}{P_{n}}\sum_{i=1}^{n}b_{i}p_{i}. $$
(2.23)

If one tuple is decreasing in mean and the other one is increasing in mean, then the reverse inequality holds in (2.23).

Now, we state another result for convex functions and for arbitrary monotonic tuples in mean.

Theorem 2.9

Let all the assumptions of Theorem 2.2hold. Additionally, ifband\(\textbf{a}-\textbf{b}\)are monotonicn-tuples in mean relative to\(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\)in the same sense, andcand\(\textbf{c}-\textbf{d}\)are monotonicm-tuples in mean relative to\(\textbf{w}=(w_{1},w_{2},\dots ,w_{m})\)in the same sense, then
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (a_{i},c_{j})-\sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (b_{i},d_{j}) \\& \quad \geq \frac{1}{P_{n}}\sum_{j=1}^{m}\sum _{i=1}^{n}w_{j}p_{i}t_{i} \sum_{i=1}^{n}p_{i}(a_{i}-b_{i}) \\& \qquad {}+\frac{1}{W_{m}}\sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j} \sum_{j=1} ^{m}w_{j}(c_{j}-d_{j}), \end{aligned}$$
(2.24)
where\(t_{i}\)is the positive partial derivative ofϕwith respect to the first variable at\(b_{i}\) (\(i=1,2,\dots ,n\)) and\(s_{j}\)is the partial positive derivative ofϕwith respect to the second variable at\(d_{j}\) (\(j=1,2,\dots ,m\)).

Proof

It follows from the proof of Proposition 2.3 that \((t_{1},t_{2}, \dots ,t_{n})\) is an increasing n-tuple. Now if b and \(\textbf{a}-\textbf{b}\) are monotonic increasing in mean relative to p, then, applying Chebyshev’s inequality to first term on the right-hand side of (2.3), we have
$$\begin{aligned} \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}t_{i}(a_{i}-b_{i}) =& \sum_{j=1} ^{m}w_{j} \Biggl[\sum _{i=1}^{n}p_{i}t_{i}(a_{i}-b_{i}) \Biggr] \\ \geq& \sum_{j=1}^{m}w_{j} \Biggl[ \frac{1}{P_{n}}\sum_{i=1}^{n}p_{i}t _{i}\sum_{i=1}^{n}p_{i}(a_{i}-b_{i}) \Biggr] \\ =&\frac{1}{P_{n}}\sum_{j=1}^{m}\sum _{i=1}^{n}w_{j}p_{i}t_{i} \sum_{i=1} ^{n}p_{i}(a_{i}-b_{i}). \end{aligned}$$
(2.25)
Similarly, we have
$$ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j}(c_{j}-d_{j}) \geq \frac{1}{W _{m}}\sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}s_{j} \sum_{j=1}^{m}w_{j}(c _{j}-d_{j}). $$
(2.26)
Using (2.25) and (2.26) in (2.3), we get (2.24).

Similarly, we can prove inequality (2.24) in the remaining cases. □

Corollary 2.10

Assume that all the hypotheses of Theorem 2.9hold. Additionally, if
$$ \sum_{i=1}^{n}a_{i}p_{i}= \sum_{i=1}^{n}b_{i}p_{i} $$
(2.27)
and
$$ \sum_{j=1}^{m}c_{j}w_{j}= \sum_{j=1}^{m}d_{j}w_{j}, $$
(2.28)
then
$$ \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi (a_{i},c_{j})\geq \sum_{i=1} ^{n}\sum_{j=1}^{m}p_{i}w_{j} \phi (b_{i},d_{j}). $$
(2.29)

Proof

Using (2.27) and (2.28) on the right-hand side of (2.24), we get (2.29). □

In the following Corollary 2.11, we obtain a majorization inequality by using an increasing convex function.

Corollary 2.11

Let all the assumptions of Theorem 2.9hold. If\(\phi :\mathbf {I}_{1} \times \mathbf {I}_{2}\mapsto \mathbb{R}\)is an increasing convex function, and
$$ \sum_{i=1}^{n}a_{i}p_{i} \geq \sum_{i=1}^{n}b_{i}p_{i} $$
(2.30)
and
$$ \sum_{j=1}^{m}c_{j}w_{j} \geq \sum_{j=1}^{m}d_{j}w_{j}, $$
(2.31)
then inequality (2.29) holds.

Proof

Since ϕ is an increasing function on \(\textbf{I}_{1}\times \textbf{I}_{2}\), we get that \(t_{i}\geq 0\), \(s_{j}\geq 0\) (\(i=1,2,\dots ,n\), \(j=1,2,\dots ,m\)), where \(t_{i}\) is the positive partial derivative of ϕ with respect to the first variable at \(b_{i}\) (\(i=1,2,\dots ,n\)) and \(s_{j}\) is the partial positive derivative of ϕ with respect to the second variable at \(d_{j}\) (\(j=1,2,\dots ,m\)), thus
$$\begin{aligned}& \sum_{i=1}^{n}p_{i}t_{i} \geq 0, \end{aligned}$$
(2.32)
$$\begin{aligned}& \sum_{j=1}^{m}w_{j}s_{j} \geq 0. \end{aligned}$$
(2.33)

Hence using (2.30), (2.31), (2.32) and (2.33) on the right-hand side of (2.24), we obtain inequality (2.29). □

The following Lemma 2.12 can be found in the literature [60].

Lemma 2.12

Let\(\textbf{v}=(v_{1}, v_{2}, \dots , v_{n})\)be a positive realn-tuple and\(\textbf{x}=(x_{1}, x_{2}, \dots , x_{n})\)be an increasing realn-tuple. Then the inequality
$$ \sum_{i=1}^{k}x_{i}v_{i} \sum_{i=1}^{n}v_{i}\leq \sum _{i=1}^{n}x_{i}v _{i} \sum_{i=1}^{k}v_{i} $$
(2.34)
holds for\(k=1,2,\dots ,n\). Ifxis a decreasing realn-tuple, then the reverse inequality holds in (2.32).

If \(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\) and \(\textbf{b}=(b_{1},b _{2},\dots ,b_{n})\) with \(b_{i}\neq 0\) for \(i=1, 2, \dots , n\), then \(\frac{\textbf{a}}{\textbf{b}}=(\frac{a_{1}}{b_{1}},\frac{a_{2}}{b _{2}},\dots ,\frac{a_{n}}{b_{n}})\).

Theorem 2.13

Let\(\mathbf {I}_{1}\)and\(\mathbf {I}_{2}\)be any two intervals in\(\mathbb{R}\), \(\phi :\mathbf {I}_{1}\times \mathbf {I}_{2}\mapsto \mathbb{R}\)be a convex function, \(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\)and\(\textbf{b}=(b_{1},b_{2},\dots ,b_{n})\)be two positiven-tuples such that\(a_{i},b_{i}\in \mathbf {I}_{1}\) (\(i=1,2,\dots ,n\)), \(\textbf{c}=(c _{1},c_{2},\dots ,c_{m})\)and\(\textbf{d}=(d_{1},d_{2},\dots ,d_{m})\)be two positivem-tuples such that\(c_{j},d_{j}\in \mathbf {I}_{2}\) (\(j=1,2,\dots ,m\)), \(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\)and\(\textbf{w}=(w_{1},w_{2},\dots ,w_{m})\)be any positive realn- andm-tuples, respectively, and\(\textbf{a}/\textbf{b}\)and\(\textbf{c}/ \textbf{d}\)are decreasingn- andm-tuples, respectively. Then the following statements are true:
  1. (i)
    Ifais an increasingn-tuple andcis an increasingm-tuple, then
    $$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{a_{i}}{\sum_{i=1}^{n}p_{i}a_{i}},\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr) \\& \quad \leq \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{b_{i}}{ \sum_{i=1}^{n}b_{i}p_{i}},\frac{d_{j}}{\sum_{j=1}^{m}w_{j}d_{j}} \biggr). \end{aligned}$$
    (2.35)
     
  2. (ii)

    Ifbis a decreasingn-tuple anddis a decreasingm-tuple, then the reverse inequality holds in (2.33).

    If\(\textbf{a}/\textbf{b}\)and\(\textbf{c}/\textbf{d}\)are increasingnandm-tuples, respectively, then we have the following statements:

     
  3. (iii)
    Ifbis an increasingn-tuple anddis an increasingm-tuple, then
    $$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{b_{i}}{\sum_{i=1}^{n}b_{i}p_{i}},\frac{d_{j}}{\sum_{j=1}^{m}w_{j}d_{j}} \biggr) \\& \quad \leq \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{a_{i}}{ \sum_{i=1}^{n}p_{i}a_{i}},\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr). \end{aligned}$$
    (2.36)
     
  4. (iv)

    Ifais a decreasingn-tuple andcis a decreasingm-tuple, then the reverse inequality holds in (2.34).

     

Proof

(i) Let \(\textbf{a}/\textbf{b}\) and \(\textbf{c}/\textbf{d}\) are decreasing n- and m-tuples, respectively. Then using Lemma 2.12 with \(\textbf{x}=\textbf{a}/\textbf{b}\) and \(\textbf{v}=\textbf{p} \textbf{b}\), we obtain
$$ \sum_{i=1}^{n}p_{i}a_{i} \sum_{i=1}^{k}p_{i}b_{i} \leq \sum_{i=1}^{k}a _{i}p_{i} \sum_{i=1}^{n}p_{i}b_{i} \quad (k=1,2,\dots ,n). $$
That is,
$$ \sum_{i=1}^{k}p_{i} \biggl( \frac{b_{i}}{\sum_{i=1}^{n}p_{i}b_{i}} \biggr) \leq \sum_{i=1}^{k}p_{i} \biggl(\frac{a_{i}}{\sum_{i=1}^{n}p_{i}a_{i}} \biggr) \quad (k=1,2,\dots ,n). $$
(2.37)
Similarly, using Lemma 2.12 with \(\textbf{x}=\textbf{c}/\textbf{d}\) and \(\textbf{v}=\textbf{d}\textbf{w}\), we get
$$ \sum_{i=1}^{n}w_{j}c_{j} \sum_{i=1}^{k}w_{j}d_{j} \leq \sum_{i=1}^{k}c _{j}w_{j} \sum_{i=1}^{n}d_{j}w_{j} \quad (k=1,2,\dots ,m). $$
That is,
$$ \sum_{j=1}^{k}w_{j} \biggl( \frac{d_{j}}{\sum_{j=1}^{m}w_{j}d_{j}} \biggr) \leq \sum_{j=1}^{k}w_{j} \biggl(\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr)\quad (k=1,2,\dots ,m). $$
(2.38)
Also, it is obvious that
$$ \sum_{i=1}^{n}p_{i} \biggl( \frac{b_{i}}{\sum_{i=1}^{n}p_{i}b_{i}} \biggr)= \sum_{i=1}^{n}p_{i} \biggl(\frac{a_{i}}{\sum_{i=1}^{n}p_{i}a_{i}} \biggr) $$
(2.39)
and
$$ \sum_{j=1}^{n}w_{j} \biggl( \frac{d_{j}}{\sum_{j=1}^{m}w_{j}d_{j}} \biggr)= \sum_{j=1}^{n}w_{j} \biggl(\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr). $$
(2.40)
If a and c are increasing, then using Theorem 2.6(ii) and (2.35)–(2.38), we have
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{a_{i}}{\sum_{i=1}^{n}p_{i}a_{i}},\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr) \\& \quad \leq \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{b_{i}}{ \sum_{i=1}^{n}b_{i}p_{i}},\frac{d_{j}}{\sum_{j=1}^{m}w_{j}d_{j}} \biggr). \end{aligned}$$
Similarly, we can prove the remaining cases. □

Definition 2.14

(See [61])

An n-tuple \(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\) is said to be concave if
$$ 2a_{k}\geq a_{k+1}+a_{k-1} $$
for all \(k=2, 3, \dots ,n-1\).

Definition 2.15

(See [61])

An n-tuple \(\textbf{a}=(a_{1},a_{2},\dots ,a_{n})\) is said to be convex if
$$ 2a_{k}\leq a_{k+1}+a_{k-1} $$
for all \(k=2, 3, \dots ,n-1\).

Corollary 2.16

Let\(\phi :[0,\infty )\times [0,\infty )\longrightarrow \mathbb{R}\)be a convex function, \(\textbf{a}=(a_{1},a_{2},\dots , a_{n})\)be a positiven-tuple, \(\textbf{c}=(c_{1},c_{2},\dots ,c_{m})\)be a positivem-tuple, and\(\textbf{p}=(p_{1},p_{2},\dots ,p_{n})\)and\(\textbf{w}=(w_{1},w_{2},\dots ,w_{m})\)be any positive realn- andm-tuples, respectively. Then the following statements are true:
  1. (i)
    Ifais an increasing concaven-tuple andcis an increasing concavem-tuple, then
    $$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{i-1}{\sum_{i=1} ^{n}p_{i}({i-1})},\frac{j-1}{\sum_{j=1}^{m}w_{j}({j-1)}} \biggr) \\& \quad \geq \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{a_{i}}{ \sum_{i=1}^{n}p_{i}a_{i}},\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr). \end{aligned}$$
    (2.41)
     
  2. (ii)

    Ifais an increasing convexn-tuple with\(a_{1}=0\)andcis an increasing convexm-tuple with\(c_{1}=0\), then the reverse inequality holds in (2.39).

     
  3. (iii)
    Ifais a decreasing concaven-tuple andcis a decreasing concavem-tuple, then
    $$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{n-i}{\sum_{i=1} ^{n}p_{i}({n-i})},\frac{m-j}{\sum_{j=1}^{m}w_{j}({m-j)}} \biggr) \\& \quad \geq \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{a_{i}}{ \sum_{i=1}^{n}p_{i}a_{i}},\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr). \end{aligned}$$
    (2.42)
     
  4. (iv)

    Ifais a decreasing convexn-tuple with\(a_{n}=0\)andcis a decreasing convexm-tuple with\(c_{m}=0\), then the reverse inequality holds in (2.40).

     

Proof

(i) Let \(\textbf{b}=(b_{1}, b_{2}, \dots , b_{n})\) and \(\textbf{d}=(d _{1}, d_{2}, \dots , d_{m})\) be respectively the n- and m-tuples such that \(b_{1}=\epsilon < a_{1}/a_{2}\), \(d_{1}=\delta < c_{1}/c_{2}\), \(b_{i}=i-1\) for \(i=2, 3, \dots , n\), and \(d_{j}=j-1\) for \(j=2, 3, \dots , m\). Then \(\textbf{a}/\textbf{b}\) and \(\textbf{c}/\textbf{d}\) are decreasing n- and m-tuples, respectively. It follows from Theorem 2.13 that
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{a_{i}}{\sum_{i=1}^{n}p_{i}a_{i}},\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr) \\& \quad \leq p_{1}w_{1}\phi \biggl(\frac{\epsilon }{\epsilon p_{1}+\sum_{i=2} ^{n}(i-1)p_{i}}, \frac{\delta }{w_{1}\delta +\sum_{j=2}^{m}(j-1)w_{j}} \biggr) \\& \qquad {}+\sum_{i=2}^{n}\sum _{j=2}^{m}p_{i}w_{j}\phi \biggl( \frac{i-1}{\epsilon p_{1}+\sum_{i=2}^{n}(i-1)p_{i}},\frac{j-1}{w_{1}\delta +\sum_{j=2} ^{m}(j-1)w_{j}} \biggr). \end{aligned}$$
Taking \(\epsilon \rightarrow 0\) and \(\delta \rightarrow 0\), we obtain
$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{a_{i}}{\sum_{i=1}^{n}p_{i}a_{i}},\frac{c_{j}}{\sum_{j=1}^{m}w_{j}c_{j}} \biggr) \\& \quad \leq p_{1}w_{1}\phi (0,0)+\sum _{i=2}^{n} \sum_{j=2}^{m}p_{i}w_{j} \phi \biggl(\frac{i-1}{\epsilon p_{1}+\sum_{i=2}^{n}(i-1)p_{i}},\frac{j-1}{w _{1}\delta +\sum_{j=2}^{m}(j-1)w_{j}} \biggr) \\& \quad =\sum_{i=1}^{n}\sum _{j=1}^{m}p_{i}w_{j}\phi \biggl( \frac{i-1}{\epsilon p_{1}+\sum_{i=2}^{n}(i-1)p_{i}},\frac{j-1}{w_{1}\delta +\sum_{j=2} ^{m}(j-1)w_{j}} \biggr). \end{aligned}$$

This proves (2.39).

Similarly, we can use other parts of Theorem 2.13 to prove the required results for the remaining cases. □

Remark 2.17

For some related integral majorization inequalities for the convex functions defined on rectangles, see [62]; and for some other recent results related to majorization, see [63, 64, 65].

3 Results and discussion

In the article, we have generalized the classical majorization inequality for majorized tuples, established several weighted version of majorization inequalities for certain tuples and provided Favard’s type inequalities by the use of Chebyshev’s inequality, Abel transformation and support line inequality.

4 Conclusions

In this paper, we have extended some discrete majorization type inequalities of convex functions from intervals to rectangles. The given results are generalizations of the previously known results. Our approach may have further applications in the theory of majorization.

Notes

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Funding

This work was supported by the Natural Science Foundation of China (Grant Nos. 11701176, 11626101, 11601485) and the Natural Science Foundation of Fujian Province (Grant No. 2016J01023).

Competing interests

The authors declare that they have no competing interests.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of ScienceHunan City UniversityYiyangChina
  2. 2.Department of MathematicsUniversity of PeshawarPeshawarPakistan
  3. 3.Department of MathematicsLongyan UniversityLongyanChina
  4. 4.Department of MathematicsHuzhou UniversityHuzhouChina

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