1 Introduction and preliminaries
For any given partial ordering ⪯ of a set χ, real-valued functions ϕ defined on χ, which satisfy \(\phi(x)\leq\phi(y)\) whenever \(x\preceq y\), are variously referred to as ‘monotonic’, ‘isotonic’, or ‘order-preserving’. We consider a partial ordering of majorization.
For two vectors \(\mathbf{x}, \mathbf{y}\in\mathbb{R}^{m}\) we say that x
majorizes
y or y is majorized by x and write \(\mathbf{y}\prec\mathbf {x}\) if Here \(x_{[i]}\) and \(y_{[i]}\) denote the elements of x and y sorted in decreasing order. Majorization on vectors determines the degree of similarity between the vector elements.
$${ \sum_{i=1}^{k}} y_{[i]} \leq { \sum_{i=1}^{k}} x_{[i]}\quad\text{for }k=1,\ldots,m-1\quad\text{and}\quad { \sum _{i=1}^{m}} y_{i}={ \sum _{i=1}^{m}} x_{i}. $$
Anzeige
For the concept of majorization, the order-preserving functions were first systematically studied by Schur (see [1, 2], p.79). In his honor, such functions are said to be ‘convex in the sense of Schur’, ‘Schur convex’, or ‘S-convex.’
Many of the inequalities that arise from a majorization can be obtained simply by identifying an appropriate order-preserving function. Historically, such inequalities have often been proved by direct methods without an awareness that a majorization underlies the validity of the inequality. The classical example of this is the Hadamard determinant inequality, where the underlying majorization was discovered by Schur.
Instead of discussing a partial order on vectors in \(\mathbb{R}^{m}\), it is natural to consider ordering matrices. Majorization in its usual sense applies to vectors with fixed element totals. For \(m\times n\) matrices, several avenues of generalization are open. Two matrices can be considered as being ordered if one is obtainable from the other by postmultiplication by a doubly stochastic matrix. This relates \(m\times n\) matrices with fixed row totals. There are, however, several variations on this approach that merit attention.
An important tool in the study of majorization is the next theorem, due to Hardy et al. [3], which gives connections with matrix theory, more specifically with doubly stochastic matrices, i.e. nonnegative square matrices with all rows and columns sums being equal to one.
Anzeige
Theorem 1
Let
\(\mathbf{x}=(x_{1},\ldots,x_{m})\), \(\mathbf{y}=(y_{1},\ldots,y_{m})\in \mathbb{R}^{m}\). Then the following statements are equivalent:
(i)
\(\mathbf{y}\prec\mathbf{x}\);
(ii)
there is a doubly stochastic matrix
A
such that
\(\mathbf{y}=\mathbf{xA}\);
(iii)
the inequality
\(\sum_{i=1}^{m}\phi(y_{i}) \leq\sum_{i=1}^{m}\phi(x_{i})\)
holds for each convex continuous function
\(\phi :\mathbb{R}\rightarrow\mathbb{R}\).
For many purposes, the condition \(\mathbf{y}=\mathbf{xA}\) is more convenient than the partial sums conditions defining majorization.
Sherman [4] considered a weighted relation of majorization, for nonnegative weights \(a_{j}\) and \(b_{i}\) and proved a more general result which includes the row stochastic \(k\times l\) matrix, i.e. matrix \(\mathbf{A}=(a_{ij})\in\mathcal{M}_{kl}(\mathbb{R})\) such that By \(\mathbf{A}^{T}=(a_{ji})\in\mathcal{M}_{lk}(\mathbb{R})\) we denote the transpose of A.
$${ \sum_{i=1}^{k}} b_{i}y_{i} \leq { \sum_{j=1}^{l}} a_{j}x_{j}$$
$$\begin{aligned} &a_{ij} \geq0\quad\text{for all }i=1,\ldots,k,j=1,\ldots,l, \\ & \sum_{j=1}^{l} a_{ij} =1\quad\text{for all }i=1,\ldots,k. \end{aligned}$$
Sherman’s result can be formulated as the following theorem (see [4]).
Theorem 2
Let
\(\mathbf{x}\in[\alpha,\beta]^{l}\), \(\mathbf{y}\in[\alpha,\beta]^{k}\), \(\mathbf{a}\in[0,\infty)^{l}\), \(\mathbf{b}\in[0,\infty)^{k}\)
and
for some row stochastic matrix
\(\mathbf{A}\in\mathcal{M}_{kl}(\mathbb{R})\). Then for every convex function
\(\phi:[\alpha,\beta]\rightarrow\mathbb {R}\)
we have
If
ϕ
is concave, then the reverse inequality in (1.2) holds.
$$ \mathbf{y}=\mathbf{x}\mathbf{A}^{T}\quad\textit{and}\quad\mathbf {a}=\mathbf{bA} $$
(1.1)
$$ \sum_{i=1}^{k}b_{i} \phi(y_{i})\leq\sum_{j=1}^{l}a_{j} \phi(x_{j}). $$
(1.2)
Notice that if we set \(k=l\) and \(a_{j}=b_{i}\) for all \(i,j=1,\ldots,k\), the condition \(\mathbf{a}=\mathbf{bA}\) ensures the stochasticity on columns, so in that case we deal with doubly stochastic matrices. Then, as a special case of Sherman’s inequality, we get a weighted version of the majorization inequality:
$${ \sum_{i=1}^{k}} a_{i}\phi(y_{i})\leq { \sum_{i=1}^{k}} a_{i} \phi(x_{i}). $$
Denoting \(A_{k}=\sum_{i=1}^{k}a_{i}\) and putting \(y_{1}=y_{2}=\cdots=y_{k}=\frac{1}{A_{k}}\sum_{i=1}^{k}a_{i}x_{i}\), we obtain Jensen’s inequality in the form
$$\phi \Biggl( \frac{1}{A_{k}}{ \sum _{i=1}^{k}} a_{i}x_{i} \Biggr) \leq\frac{1}{A_{k}}{ \sum _{i=1}^{k}} a_{i} \phi(x_{i}). $$
Particularly, for \(A_{k}=1\), we have
$$ \phi \Biggl( { \sum_{i=1}^{k}} a_{i}x_{i} \Biggr) \leq { \sum_{i=1}^{k}} a_{i} \phi(x_{i}). $$
(1.3)
On the other hand, the proof of Sherman’s inequality (1.2) is based on Jensen’s inequality (1.3). Since the matrix \(\mathbf{A}\in\mathcal{M}_{kl}(\mathbb{R})\) is row stochastic and (1.1) holds, for every convex function \(\phi:[\alpha,\beta]\rightarrow\mathbb{R}\) we have
$$\begin{aligned} { \sum_{i=1}^{k}} b_{i}\phi(y_{i}) & ={ \sum_{i=1}^{k}} b_{i} \phi \Biggl( { \sum_{j=1}^{l}} x_{j}a_{ij} \Biggr) \leq { \sum_{i=1}^{k}} b_{i}{ \sum_{j=1}^{l}} a_{ij} \phi ( x_{j} ) \\ & ={ \sum_{j=1}^{l}} \Biggl( { \sum_{i=1}^{k}} b_{i}a_{ij} \Biggr) \phi ( x_{j} ) ={ \sum_{j=1}^{l}} a_{j}\phi(x_{j}). \end{aligned}$$
In this paper, we consider a difference of Sherman’s inequality and establish generalizations of Sherman’s inequality (1.2) which hold for n-convex functions which are in special cases convex in the usual sense. Moreover, we obtain an extension to real, not necessarily nonnegative entries of the vectors a, b, and the matrix A. Some related results can be found in [5, 6].
$$\sum_{j=1}^{l}a_{j} \phi(x_{j})-\sum_{i=1}^{k}b_{i} \phi(y_{i}) $$
The notion of n-convexity was defined in terms of divided differences by Popoviciu. Divided differences are a very important notion by dealing with the functions when discussing the degree of smoothness. A function \(\phi :[\alpha ,\beta]\rightarrow\mathbb{R}\) is n-convex, \(n\geq0\), if its nth order divided differences \([x_{0},\ldots,x_{n};\phi]\) are nonnegative for all choices of \((n+1)\) distinct points \(x_{i}\in[\alpha,\beta]\), \(i=0,\ldots,n\). Thus, a 0-convex function is nonnegative, a 1-convex function is nondecreasing, and a 2-convex function is convex in the usual sense. If ϕ is n-convex, then without loss of generality we can assume that ϕ is n-times differentiable and \(\phi^{(n)}\geq0\) (see [7]).
The techniques that we use in the paper are based on classical real analysis and the application of the Abel-Gontscharoff interpolation. The Abel-Gontscharoff interpolation problem in the real case was introduced in 1935 by Whittaker [8] and subsequently by Gontscharoff [9] and Davis [10]. The next theorem presents the Abel-Gontscharoff interpolating polynomial for two points with integral remainder (see [11]).
Theorem 3
Let
\(n,m\in\mathbb{N}\), \(n\geq2\), \(0\leq m\leq n-1\), and
\(\phi\in C^{n}([\alpha,\beta])\). Then
where
\(Q_{n-1}\)
is the Abel-Gontscharoff interpolation for two points of degree
\(n-1\), i.e.
and the remainder is given by
where
\(G_{mn}(u,t)\)
is Green’s function defined by
$$\phi(t)=Q_{n-1} ( \alpha,\beta,\phi,t ) +R ( \phi,u ) , $$
$$\begin{aligned} Q_{n-1} ( \alpha,\beta,\phi,t ) ={}&\sum_{s=0}^{m}\frac{ ( t-\alpha ) ^{s}}{s!}\phi^{(s)}(\alpha) \\ &{} +\sum_{r=0}^{n-m-2} \Biggl[ \sum _{s=0}^{r}\frac{ ( t-\alpha ) ^{m+1+s} ( \alpha-\beta ) ^{r-s}}{ ( m+1+s ) ! ( r-s ) !} \Biggr] \phi^{(m+1+r)}(\beta) \end{aligned}$$
$$R ( \phi,u ) = \int_{\alpha}^{\beta}G_{mn}(u,t) \phi^{(n)}(t)\,dt, $$
$$ G_{mn}(u,t)=\frac{1}{(n-1)!}\textstyle\begin{cases}{ \sum_{s=0}^{m}} \binom{n-1}{s} ( u-\alpha ) ^{s} ( \alpha-t ) ^{n-s-1} , & \alpha\leq t\leq u;\\ -{ \sum_{s=m+1}^{n-1}} \binom{n-1}{s} ( u-\alpha ) ^{s} ( \alpha-t ) ^{n-s-1}, & u\leq t\leq\beta. \end{cases} $$
(1.4)
Remark 1
Further, for \(\alpha\leq t\), \(u\leq\beta\) the following inequalities hold:
$$\begin{aligned} &(-1)^{n-m-1}\frac{\partial^{s}G_{mn}(u,t)}{\partial u^{s}} \geq 0,\quad 0\leq s\leq m, \\ &(-1)^{n-s}\frac{\partial^{s}G_{mn}(u,t)}{\partial u^{s}} \geq 0,\quad m+1\leq s\leq n-1. \end{aligned}$$
2 Generalizations of Sherman’s theorem
Applying interpolation by the Abel-Gontscharoff polynomial we derive an identity related to generalized Sherman’s inequality.
Theorem 4
Let
\(\mathbf{x}\in[\alpha,\beta]^{l}\), \(\mathbf{y}\in[\alpha,\beta]^{k}\), \(\mathbf{a}\in\mathbb{R}^{l}\)
and
\(\mathbf{b}\in\mathbb{R}^{k}\)
be such that (1.1) holds for some matrix
\(\mathbf{A}\in\mathcal{M}_{kl}(\mathbb{R})\)
whose entries satisfy the condition
\({ \sum_{j=1}^{l}} a_{ij}=1\), \(i=1,\ldots,k\). Let
\(n,m\in\mathbb{N}\), \(n\geq2\), \(0\leq m\leq n-1\), \(\phi\in C^{n}([\alpha,\beta])\), and
\(G_{mn}\)
be defined by (1.4). Then
$$\begin{aligned} & \sum_{j=1}^{l}a_{j} \phi(x_{j})-\sum_{i=1}^{k}b_{i} \phi (y_{i}) \\ &\quad =\sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl( \sum_{j=1}^{l}a_{j}(x_{j}- \alpha)^{s}-\sum_{i=1}^{k}b_{i}(y_{i}- \alpha )^{s} \Biggr) \\ &\qquad{} +\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{(-1)^{r-s}(\beta-\alpha)^{r-s} \phi^{(m+1+r)}(\beta)}{(m+1+s)!(r-s)!} \Biggl( { \sum_{j=1}^{l}} a_{j}(x_{j}-\alpha)^{m+1+s}-{ \sum_{i=1}^{k}} b_{i}(y_{i}- \alpha)^{m+1+s} \Biggr) \\ &\qquad{} + \int_{\alpha}^{\beta} \Biggl( \sum _{j=1}^{l}a_{j}G_{mn}(x_{j},t)- \sum_{i=1}^{k}b_{i}G_{mn}(y_{i},t) \Biggr) \phi ^{(n)}(t)\,dt. \end{aligned}$$
(2.1)
Proof
Using Theorem 3 we can represent every function \(\phi\in C^{n}([\alpha,\beta])\) in the form
$$\begin{aligned} \phi(u) =&\sum_{s=0}^{m}\frac{(u-\alpha)^{s}}{s!} \phi^{(s)}(\alpha) \\ &{} +\sum_{r=0}^{n-m-2} \Biggl[ \sum _{s=0}^{r}\frac{(u-\alpha)^{m+1+s}(-1)^{r-s}(\beta-\alpha)^{r-s}}{(m+1+s)!(r-s)!} \Biggr] \phi^{(m+1+r)}(\beta) \\ &{} + \int_{\alpha}^{\beta}G_{mn}(u,t) \phi^{(n)}(t)\,dt. \end{aligned}$$
(2.2)
By an easy calculation, applying (2.2) in \(\sum_{j=1}^{l}a_{j}\phi(x_{j})-\sum_{i=1}^{k}b_{i}\phi(y_{i})\), we get Since (1.1) holds, Therefore, (2.1) follows. □
$$\begin{aligned} & \sum_{j=1}^{l}a_{j} \phi(x_{j})-\sum_{i=1}^{k}b_{i} \phi (y_{i}) \\ &\quad =\sum_{s=0}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl( \sum_{j=1}^{l}a_{j}(x_{j}- \alpha)^{s}-\sum_{i=1}^{k}b_{i}(y_{i}- \alpha )^{s} \Biggr) \\ & \qquad{}+\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{(-1)^{r-s}(\beta-\alpha)^{r-s} \phi^{(m+1+r)}(\beta)}{(m+1+s)!(r-s)!} \Biggl( { \sum_{j=1}^{l}} a_{j}(x_{j}-\alpha)^{m+1+s}-{ \sum_{i=1}^{k}} b_{i}(y_{i}- \alpha)^{m+1+s} \Biggr) \\ &\qquad{} + \int_{\alpha}^{\beta} \Biggl( \sum _{j=1}^{l}a_{j}G_{mn}(x_{j},t)- \sum_{i=1}^{k}b_{i}G_{mn}(y_{i},t) \Biggr) \phi^{(n)}(t)\,dt. \end{aligned}$$
$$\frac{\phi^{(s)}(\alpha)}{s!} \Biggl( \sum_{j=1}^{l}a_{j}(x_{j}- \alpha)^{s}-\sum_{i=1}^{k}b_{i}(y_{i}- \alpha)^{s} \Biggr) =0\quad\text{for }s=0,1. $$
The following theorem extends Sherman’s result to convex functions of higher order and to real, not necessarily nonnegative entries of the vectors a, b, and the matrix A.
Theorem 5
Suppose that all the assumptions of Theorem
4
hold. Additionally, let
ϕ
be
n-convex on
\([\alpha,\beta]\)
and
Then
If the reverse inequality in (2.3) holds, then the reverse inequality in (2.4) holds.
$$ \sum_{i=1}^{k}b_{i}G_{mn} ( y_{i},t ) \leq\sum_{j=1}^{l}a_{j}G_{mn} ( x_{j},t ) ,\quad t\in[\alpha,\beta]. $$
(2.3)
$$\begin{aligned} &\sum_{j=1}^{l}a_{j}\phi ( x_{j} ) -\sum_{i=1}^{k}b_{i} \phi ( y_{i} ) \\ &\quad \geq\sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl( \sum_{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{s} \Biggr) \\ &\qquad{} +\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{ ( -1 ) ^{r-s} ( \beta-\alpha ) ^{r-s}\phi^{(m+1+r)}(\beta)}{(m+1+s)!(r-s)!} \\ &\qquad{}\times\Biggl( \sum _{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{m+1+s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{m+1+s} \Biggr) . \end{aligned}$$
(2.4)
Proof
Notice that when we take into account Sherman’s condition of nonnegativity of the vectors a, b, and the matrix A, the assumption (2.3) is equivalent to the requirement that \(G_{mn} ( \cdot,t ) \), \(t\in[\alpha,\beta]\), must be convex on \([\alpha,\beta]\). So, for \(n=2\) and \(0\leq m\leq1\), the assumption (2.3) is immediately satisfied and then the inequality (2.4) holds. Moreover, in that case, the right-hand side of (2.4) is equal to zero, so we have
i.e. we get Sherman’s inequality as a direct consequence. For an arbitrary \(n\geq3\) and \(0\leq m\leq1\), we use Remark 1, i.e. we consider the following inequality: Hence, the convexity of \(G_{mn}(\cdot,t)\) depends on the parity of n. If n is even, then \(\frac{\partial^{2}G_{mn}(u,t)}{\partial u^{2}}\geq0\), i.e.
\(G_{mn}(\cdot,t)\) is convex and assumption (2.3) is satisfied. Moreover, the inequality (2.4) holds. For odd n we get the reverse inequality. For all other choices, the following generalization holds.
$$\sum_{j=1}^{l}a_{j} \phi(x_{j})-\sum_{i=1}^{k}b_{i} \phi (y_{i})\geq0, $$
$$(-1)^{n-2}\frac{\partial^{2}G_{mn}(u,t)}{\partial u^{2}}\geq0. $$
Theorem 6
Proof
(i) By Remark 1, the following inequality holds: In case \(n-m\) is odd (\(n-m-1\) is even), we have
i.e.
\(G_{mn} ( \cdot,t ) \), \(t\in[\alpha,\beta ]\), is convex on \([\alpha,\beta]\). Then by Sherman’s theorem we have
i.e. the assumption (2.3) is satisfied. Hence, applying Theorem 5 we get (2.4).
$$(-1)^{n-m-1}\frac{\partial^{2}G_{mn}(u,t)}{\partial u^{2}}\geq0,\quad \alpha\leq u,t\leq\beta. $$
$$\frac{\partial^{2}G_{mn}(u,t)}{\partial u^{2}}\geq0, $$
$$\sum_{i=1}^{k}b_{i}G_{mn} ( y_{i},s ) \leq\sum_{j=1}^{l}a_{j}G_{mn} ( x_{j},s ) , $$
(ii) Similarly we can prove this part. □
Theorem 7
Suppose that all assumptions of Theorem
2
hold. Additionally, let
\(n,m\in\mathbb{N}\), \(n\geq2\), \(0\leq m\leq n-1\), \(\phi\in C^{n}([\alpha,\beta])\)
be
n-convex and
\(F:[\alpha,\beta ]\rightarrow\mathbb{R}\)
be defined by
$$\begin{aligned} F(t) ={}&\sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} ( t-\alpha ) ^{s} \\ &{} +\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{ ( -1 ) ^{r-s} ( \beta-\alpha ) ^{r-s}}{ ( m+1+s ) ! ( r-s ) !}\phi^{(m+1+r)}(\beta) ( t-\alpha ) ^{m+1+s}. \end{aligned}$$
(2.5)
Proof
(i) Let (2.4) holds. If F is convex, then by Sherman’s theorem we have which, changing the order of summation, can be written in the form Therefore, the right-hand side of (2.4) is nonnegative and the inequality (1.2) immediately follows.
$$\sum_{j=1}^{l}a_{j}F ( x_{j} ) -\sum_{i=1}^{k}b_{i}F ( y_{i} ) \geq0, $$
$$\begin{aligned} &\sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl( \sum_{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{s} \Biggr) \\ &\qquad{} +\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{ ( -1 ) ^{r-s} ( \beta-\alpha ) ^{r-s}\phi^{(m+1+r)}(\beta)}{(m+1+s)!(r-s)!}\\ &\qquad{}\times \Biggl( \sum _{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{m+1+s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{m+1+s} \Biggr) \\ &\quad \geq0. \end{aligned}$$
(ii) Similarly we can prove this part. □
Remark 2
Note that the function \(t\mapsto(t-\alpha)^{p}\) is convex on \([\alpha,\beta]\) for each \(p=2,\ldots,n-1\), i.e.
\(\sum_{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{p}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{p}\geq0\), for each \(p=2,\ldots,n-1\).
(i)
If (2.4) holds and in addition \(\phi^{(s)}(\alpha )\geq0\) for \(s=0,\ldots,m\) and \(\phi^{(m+1+s)}(\beta)\geq0\) if \(r-s\) is even and \(\phi^{(m+1+s)}(\beta)\leq0\) if \(r-s\) is odd for \(s=0,\ldots,r\) and \(r=0,\ldots,n-m-2\), then the right-hand side of (2.4) is nonnegative, i.e. the inequality (1.2) holds.
(ii)
If the reverse of (2.4) holds and in addition \(\phi^{(s)}(\alpha)\leq0\) for \(s=0,\ldots,m\) and \(\phi^{(m+1+s)}(\beta)\leq 0\) if \(r-s\) is even, and \(\phi^{(m+1+s)}(\beta)\geq0\) if \(r-s\) is odd for \(s=0,\ldots,r\) and \(r=0,\ldots,n-m-2\), then the right-hand side of (2.4) is negative, i.e. the reverse inequality in (1.2) holds.
3 Upper bound for generalized Sherman’s inequality
In the previous section, under special conditions which are imposed in Theorem 7 and Remark 2, we obtained the following estimations for Sherman’s difference:
$$\begin{aligned} &\sum_{j=1}^{l}a_{j}\phi ( x_{j} ) -\sum_{i=1}^{k}b_{i} \phi ( y_{i} ) \\ &\quad \geq\sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl( \sum_{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{s} \Biggr) \\ &\qquad{} +\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{ ( -1 ) ^{r-s} ( \beta-\alpha ) ^{r-s}\phi^{(m+1+r)}(\beta)}{(m+1+s)!(r-s)!} \\ &\qquad{}\times \Biggl( \sum _{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{m+1+s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{m+1+s} \Biggr) \\ &\quad \geq0. \end{aligned}$$
(3.1)
In this section we present upper bounds for obtained generalization. In the proofs of some estimations we use recent results related to the Čebyšev functional, which for two Lebesgue integrable functions \(f,g:[a,b]\rightarrow \mathbb{R}\) is defined by
$$T(f,g)=\frac{1}{b-a} \int_{a}^{b}f(t)g(t)\,dt-\frac{1}{b-a} \int_{a}^{b}f(t)\,dt\cdot \frac{1}{b-a} \int_{a}^{b}g(t)\,dt. $$
With \(\Vert \cdot \Vert _{p}\), \(1\leq p\leq\infty\), we denote the usual Lebesgue norms on space \(L_{p}[a,b]\).
Theorem 8
([12], Theorem 1)
Let
\(f:[a,b]\rightarrow\mathbb{R}\)
be a Lebesgue integrable function and
\(g:[a,b]\rightarrow\mathbb{R}\)
be an absolutely continuous function with
\((\cdot-a)(b-\cdot)[g^{\prime }]^{2}\in L_{1}[a,b]\). Then
The constant
\(\frac{1}{\sqrt{2}}\)
in (3.2) is the best possible.
$$ \bigl|T(f,g)\bigr|\leq\frac{1}{\sqrt{2}} \bigl[ T(f,f) \bigr] ^{\frac{1}{2}} \frac {1}{\sqrt{b-a}} \biggl( \int_{a}^{b}(x-a) (b-x) \bigl[ g^{\prime }(x) \bigr] ^{2}\,dx \biggr) ^{\frac{1}{2}}. $$
(3.2)
Theorem 9
([12], Theorem 2)
Assume that
\(g:[a,b]\rightarrow \mathbb{R}\)
is monotonic nondecreasing on
\([a,b]\)
and
\(f:[a,b]\rightarrow\mathbb {R}\)
is absolutely continuous with
\(f^{\prime}\in L_{\infty}[a,b]\). Then
The constant
\(\frac{1}{2}\)
in (3.3) is the best possible.
$$ \bigl|T(f,g)\bigr|\leq\frac{1}{2(b-a)}\bigl\Vert f^{\prime}\bigr\Vert _{\infty }\int _{a}^{b}(x-a) (b-x)\,dg(x). $$
(3.3)
To avoid many notations, we define the function \(\mathcal{F}:[\alpha ,\beta]\rightarrow\mathbb{R}\) by under assumptions of Theorem 4. We also consider the Čebyšev functional
$$ \mathcal{F}(t)=\sum_{j=1}^{l}a_{j}G_{mn} ( x_{j},t ) -\sum_{i=1}^{k}b_{i}G_{mn} ( y_{i},t ) , $$
(3.4)
$$T(\mathcal{F},\mathcal{F})=\frac{1}{\beta-\alpha} \int_{\alpha}^{\beta }\mathcal{F}^{2}(t)\,dt- \biggl( \frac{1}{\beta-\alpha} \int_{\alpha}^{\beta }\mathcal{F}(t)\,dt \biggr) ^{2}. $$
Theorem 10
Suppose that all the assumptions of Theorem
4
hold. Additionally, let
\((\cdot-\alpha)(\beta-\cdot)(\phi^{(n+1)})^{2}\in L_{1}[\alpha,\beta]\)
and
\(\mathcal{F}\)
be defined as in (3.4). Then the following identity holds:
where the remainder
\(R_{n}(\alpha,\beta;\phi)\)
satisfies the estimation
$$\begin{aligned} &\sum_{j=1}^{l}a_{j}\phi ( x_{j} ) -\sum_{i=1}^{k}b_{i} \phi ( y_{i} ) \\ &\quad =\sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl[ \sum_{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{s} \Biggr] \\ &\qquad{} +\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{ ( -1 ) ^{r-s} ( \beta-\alpha ) ^{r-s}\phi^{(m+1+r)}(\beta)}{(s+1+m)!(r-s)!} \Biggl[ \sum _{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{m+1+s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{m+1+s} \Biggr] \\ & \qquad{}+\frac{\phi^{(n-1)}(\beta)-\phi^{(n-1)}(\alpha)}{\beta-\alpha} \int _{\alpha }^{\beta}\mathcal{F}(t)\,dt+R_{n}( \alpha,\beta;\phi), \end{aligned}$$
(3.5)
$$\bigl\vert R_{n}(\alpha,\beta;\phi)\bigr\vert \leq\sqrt{ \frac{\beta -\alpha }{2}} \bigl[ T(\mathcal{F},\mathcal{F}) \bigr] ^{\frac{1}{2}}\biggl\vert \int_{\alpha}^{\beta}(t-\alpha) (\beta-t) \bigl[ \phi^{(n+1)}(t) \bigr] ^{2}\,dt\biggr\vert ^{\frac{1}{2}}. $$
Proof
By applying Theorem 8 for \(f\rightarrow\mathcal{F}\) and \(g\rightarrow\phi^{(n)}\) we get Therefore we have where the remainder \(R_{n}(\alpha,\beta;\phi)\) satisfies the estimation. Now from the identity (2.1) we obtain (3.5). □
$$\begin{aligned} &\biggl\vert \frac{1}{\beta-\alpha} \int_{\alpha}^{\beta}\mathcal {F}(t)\phi^{(n)}(t) \,dt-\frac{1}{\beta-\alpha} \int_{\alpha}^{\beta }\mathcal {F}(t)\,dt\cdot \frac{1}{\beta-\alpha} \int_{\alpha}^{\beta}\phi^{(n)}(t)\,dt \biggr\vert \\ &\quad \leq \frac{1}{\sqrt{2}} \bigl[ T(\mathcal{F},\mathcal{F}) \bigr] ^{\frac{1}{2}}\frac{1}{\sqrt{\beta-\alpha}}\biggl\vert \int_{\alpha }^{\beta }(t-\alpha) (\beta-t) \bigl[ \phi^{(n+1)}(t) \bigr] ^{2}\,dt\biggr\vert ^{\frac{1}{2}}. \end{aligned}$$
$$\int_{\alpha}^{\beta}\mathcal{F}(t)\phi^{(n)}(t) \,dt=\frac{\phi^{(n-1)}(\beta)-\phi^{(n-1)}(\alpha)}{\beta-\alpha} \int_{\alpha}^{\beta}\mathcal{F}(t) \,dt+R_{n}(\alpha,\beta;\phi), $$
The following Grüss type inequality also holds.
Theorem 11
Suppose that all the assumptions of Theorem
4
hold. Additionally, let
\(\phi^{(n+1)}\geq0\)
on
\([\alpha,\beta]\)
and
\(\mathcal {F}\)
be defined as in (3.4). Then the identity (3.5) holds and the remainder
\(R(\phi;a,b)\)
satisfies the bound
$$ \bigl\vert R_{n}(\alpha,\beta;\phi)\bigr\vert \leq\bigl\Vert \mathcal {F}^{\prime}\bigr\Vert _{\infty} \biggl\{ \frac{\phi^{(n-1)}(\beta)+\phi ^{(n-1)}(\alpha)}{2}-\frac{\phi^{(n-2)}(\beta)-\phi^{(n-2)}(\alpha)}{\beta-\alpha} \biggr\} . $$
(3.6)
Proof
Applying Theorem 9 for \(f\rightarrow\mathcal{F}\) and \(g\rightarrow \phi^{(n)}\) we obtain Since using the identities (2.1) and (3.7) we deduce (3.6). □
$$\begin{aligned} &\biggl\vert \frac{1}{\beta-\alpha} \int_{\alpha}^{\beta}\mathcal {F}(t)\phi^{(n)}(t) \,dt-\frac{1}{\beta-\alpha} \int_{\alpha}^{\beta }\mathcal {F}(t)\,dt\cdot\frac{1}{\beta-\alpha} \int_{\alpha}^{\beta}\phi ^{(n)}(t)\,dt\biggr\vert \\ & \quad \leq\frac{1}{2(\beta-\alpha)}\bigl\Vert \mathcal{F}^{\prime}\bigr\Vert _{\infty} \int_{\alpha}^{\beta}(t-\alpha) (\beta-t) \phi^{(n+1)}(t)\,dt. \end{aligned}$$
(3.7)
$$\begin{aligned} &\int_{\alpha}^{\beta}(t-\alpha) (\beta-t) \phi^{(n+1)}(t)\,dt\\ &\quad=\int _{\alpha}^{\beta} \bigl[ 2t-(\alpha+\beta) \bigr] \phi^{(n)}(t)\,dt \\ & \quad=(\beta-\alpha) \bigl[ \phi^{(n-1)}(\beta)+\phi^{(n-1)}(\alpha) \bigr] -2 \bigl( \phi^{(n-2)}(\beta)-\phi^{(n-2)}(\alpha) \bigr) , \end{aligned}$$
We present an upper bound for generalized Sherman’s inequality which is an Ostrowski type inequality.
Theorem 12
Suppose that all the assumptions of Theorem
4
hold. Let
\((p,q)\)
be a pair of conjugate exponents, that is, \(1\leq p\), \(q\leq\infty\), \(\frac{1}{p}+\frac{1}{q}=1\). Then
The constant on the right-hand side of (3.8) is sharp for
\(1< p\leq\infty\)
and the best possible for
\(p=1\).
$$\begin{aligned} & \Biggl|\sum_{j=1}^{l}a_{j}\phi ( x_{j} ) -\sum_{i=1}^{k}b_{i} \phi ( y_{i} ) \\ & \qquad{}- \sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl[ \sum_{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{s} \Biggr] \\ &\qquad{} -\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{ ( -1 ) ^{r-s} ( \beta-\alpha ) ^{r-s}\phi^{(m+1+r)}(\beta)}{(m+1+s)!(r-s)!} \\ &\qquad{}\times \Biggl( \sum _{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{m+1+s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{m+1+s} \Biggr) \Biggr| \\ &\quad \leq\bigl\Vert \phi^{(n)}\bigr\Vert _{p} \Biggl( \int_{\alpha}^{\beta }\Biggl\vert \sum _{j=1}^{l}a_{j}G_{mn} ( x_{j},t ) -\sum_{i=1}^{k}b_{i}G_{mn} ( y_{i},t ) \Biggr\vert ^{q}\,dt \Biggr) ^{\frac {1}{q}}. \end{aligned}$$
(3.8)
Proof
Applying the well-known Hölder inequality to the identity (2.1) we have where \(\mathcal{F}(t)\) is defined as in (3.4).
$$\begin{aligned} & \Biggl|\sum_{j=1}^{l}a_{j} \phi(x_{j})-\sum_{i=1}^{k}b_{i} \phi(y_{i}) \\ &\qquad{} -\sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl( \sum_{j=1}^{l}a_{j}(x_{j}- \alpha)^{s}-\sum_{i=1}^{k}b_{i}(y_{i}- \alpha )^{s} \Biggr) \\ &\qquad{} -\sum_{r=0}^{n-m-2}\sum _{s=0}^{r}\frac{(-1)^{r-s}(\beta -\alpha)^{r-s}\phi^{(m+1+r)}(\beta)}{(m+1+s)!(r-s)!} \\ &\qquad{}\times \Biggl( { \sum_{j=1}^{l}} a_{j}(x_{j}-\alpha)^{m+1+s}-{ \sum_{i=1}^{k}} b_{i}(y_{i}- \alpha)^{m+1+s} \Biggr) \Biggr| \\ &\quad =\Biggl\vert \int_{\alpha}^{\beta} \Biggl( \sum _{j=1}^{l}a_{j}G_{mn}(x_{j},t)- \sum_{i=1}^{k}b_{i}G_{mn}(y_{i},t) \Biggr) \phi^{(n)}(t)\,dt\Biggr\vert \\ &\quad \leq\bigl\Vert \phi^{ ( n ) }\bigr\Vert _{p} \biggl( \int_{\alpha}^{\beta}\bigl\vert \mathcal{F}(t)\bigr\vert ^{q}\,dt \biggr) ^{\frac{1}{q}}, \end{aligned}$$
(3.9)
For the proof of the sharpness of the constant \(( \int_{\alpha}^{\beta} \vert \mathcal{F}(t)\vert ^{q}\,dt ) ^{\frac{1}{q}}\) let us find a function ϕ for which the equality in (3.9) is obtained.
For \(1< p<\infty\) take ϕ to be such that For \(p=\infty\) take For \(p=1\) we prove that is the best possible inequality.
$$\phi^{(n)}(t)=\operatorname{sgn}\mathcal{F}(t)\bigl\vert \mathcal{F}(t)\bigr\vert ^{\frac {1}{p-1}}. $$
$$\phi^{(n)}(t)=\operatorname{sgn}\mathcal{F}(t). $$
$$ \biggl\vert \int_{\alpha}^{\beta}\mathcal{F}(t)\phi^{(n)}(t)\,dt \biggr\vert \leq\max_{t\in[\alpha,\beta]}\bigl\vert \mathcal{F}(t)\bigr\vert \biggl( \int_{\alpha}^{\beta}\bigl\vert \phi^{(n)}(t)\bigr\vert \,dt \biggr) $$
(3.10)
Suppose that \(\vert \mathcal{F}(t)\vert \) attains its maximum at \(t_{0}\in[ \alpha,\beta]\).
First we assume that \(\mathcal{F}(t_{0})>0\). For ϵ small enough we define \(\phi_{\epsilon}(t)\) by Then for ϵ small enough Now from the inequality (3.10) we have Since the statement follows.
$$\phi_{\epsilon}(t) := \textstyle\begin{cases} 0, & \alpha\leq t\leq t_{0},\\ \frac{1}{\epsilon n!} ( t-t_{0} ) ^{n}, & t_{0}\leq t\leq t_{0}+\epsilon,\\ \frac{1}{n!} ( t-t_{0} ) ^{n-1}, & t_{0}+\epsilon\leq t\leq \beta. \end{cases} $$
$$\biggl\vert \int_{\alpha}^{\beta}\mathcal{F}(t)\phi^{(n)}(t) \biggr\vert =\biggl\vert \int_{t_{0}}^{t_{0}+\epsilon}\mathcal{F}(t)\frac{1}{\epsilon }\,dt \biggr\vert =\frac{1}{\epsilon} \int_{t_{0}}^{t_{0}+\epsilon}\mathcal {F}(t)\,dt. $$
$$\frac{1}{\epsilon} \int_{t_{0}}^{t_{0}+\epsilon}\mathcal{F}(t)\,dt\leq \mathcal{F}(t_{0}) \int_{t_{0}}^{t_{0}+\epsilon}\frac{1}{\epsilon }\,dt= \mathcal{F}(t_{0}). $$
$$\lim_{\epsilon\rightarrow0}\frac{1}{\epsilon} \int _{t_{0}}^{t_{0}+\epsilon }\mathcal{F}(t)\,dt= \mathcal{F}(t_{0}) $$
In the case \(\mathcal{F}(t_{0})<0\), we define \(\phi_{\epsilon}(t)\) by and the rest of the proof is the same as above. □
$$\phi_{\epsilon}(t) := \textstyle\begin{cases} \frac{1}{n!} ( t-t_{0}-\epsilon ) ^{n-1}, & \alpha\leq t\leq t_{0},\\ -\frac{1}{\epsilon n!} ( t-t_{0}-\epsilon ) ^{n}, & t_{0}\leq t\leq t_{0}+\epsilon,\\ 0, & t_{0}+\epsilon\leq t\leq\beta, \end{cases} $$
In the sequel we consider a particular case of Green’s function \(G_{mn}(u,t)\) defined by (1.4). For \(n=2\), \(m=1\), we have and
$$ G_{12}(u,t)=\textstyle\begin{cases} u-t, & \alpha\leq t\leq u,\\ 0, & u\leq t\leq\beta, \end{cases} $$
(3.11)
$$\begin{aligned} & \sum_{j=1}^{l}a_{j}G_{12}(x_{j},t)- \sum_{i=1}^{k}b_{i}G_{12}(y_{i},t) \\ & \quad =\sum_{\mathbf{j}}a_{\mathbf{j}}(x_{\mathbf{j}}-t)- \sum_{\mathbf{i}}b_{\mathbf{i}}(y_{\mathbf{i}}-t),\quad \mathbf{j}\in\{j;x_{j}\geq t\}, \mathbf{i}\in \{i;y_{i}\geq t\}. \end{aligned}$$
(3.12)
Corollary 1
Let
\(\phi\in C^{2}([\alpha,\beta])\), \(\mathbf{x}\in [\alpha,\beta]^{l}\), \(\mathbf{y}\in[\alpha,\beta]^{k}\), \(\mathbf{a}\in\mathbb{R}^{l}\), and
\(\mathbf{b}\in\mathbb{R}^{k}\)
be such that (1.1) holds for some matrix
\(\mathbf{A}\in\mathcal{M}_{kl}(\mathbb{R})\)
whose entries satisfy the condition
\({ \sum_{j=1}^{l}} a_{ij}=1\), \(i=1,\ldots,k\). Let
\((p,q)\)
be a pair of conjugate exponents, that is, \(1\leq p\), \(q\leq\infty\), \(\frac{1}{p}+\frac{1}{q}=1\). Then
The constant on the right-hand side of (3.13) is sharp for
\(1< p\leq\infty\)
and the best possible for
\(p=1\).
$$\begin{aligned} & \Biggl\vert \sum_{j=1}^{l}a_{j} \phi ( x_{j} ) -\sum_{i=1}^{k}b_{i} \phi ( y_{i} ) \Biggr\vert \\ &\quad \leq\bigl\Vert \phi^{\prime\prime}\bigr\Vert _{p} \Biggl( \int_{\alpha }^{\beta}\Biggl\vert \sum _{j=1}^{l}a_{j}G_{12} ( x_{j},t ) -\sum_{i=1}^{k}b_{i}G_{12} ( y_{i},t ) \Biggr\vert ^{q}\,dt \Biggr) ^{\frac{1}{q}}. \end{aligned}$$
(3.13)
Remark 3
If additionally suppose that vectors a, b, and matrix A are nonnegative and ϕ is convex, then the difference \(\sum_{j=1}^{l}a_{j}\phi ( x_{j} ) -\sum_{i=1}^{k}b_{i}\phi ( y_{i} ) \) is nonnegative and we have In the sequel we consider some particular cases of this result.
$$\begin{aligned} 0 & \leq\sum_{j=1}^{l}a_{j}\phi ( x_{j} ) -\sum_{i=1}^{k}b_{i} \phi ( y_{i} ) \\ & \leq\bigl\Vert \phi^{\prime\prime}\bigr\Vert _{p} \Biggl( \int_{\alpha }^{\beta}\Biggl\vert \sum _{j=1}^{l}a_{j}G_{12} ( x_{j},t ) -\sum_{i=1}^{k}b_{i}G_{12} ( y_{i},t ) \Biggr\vert ^{q}\,dt \Biggr) ^{\frac{1}{q}}. \end{aligned}$$
(3.14)
(i)
If \(k=l\) and all weights \(b_{i}\) and \(a_{j}\) are equal, we get the following estimate for the weighted majorization inequality:
$$\begin{aligned} 0 & \leq\sum_{i=1}^{k}a_{i}\phi ( x_{i} ) -\sum_{i=1}^{k}a_{i} \phi ( y_{i} ) \\ & \leq\bigl\Vert \phi^{\prime\prime}\bigr\Vert _{p} \Biggl( \int _{\alpha }^{\beta}\Biggl\vert \sum _{i=1}^{k}a_{i}G_{12} ( x_{i},t ) -\sum_{i=1}^{k}a_{i}G_{12} ( y_{i},t ) \Biggr\vert ^{q}\,dt \Biggr) ^{\frac{1}{q}}. \end{aligned}$$
(3.15)
(ii)
If we denote \(A_{k}=\sum_{i=1}^{k}a_{i}\) and put \(y_{1}=y_{2}=\cdots=y_{k}=\frac{1}{A_{k}}\sum_{i=1}^{k}a_{i}x_{i}=\bar{x}\), from (3.15) as an easy consequence we get estimate for Jensen’s inequality: Specially, setting \(A_{k}=1\), we have \(\bar{x}=\sum_{i=1}^{k}a_{i}x_{i}\) and
$$\begin{aligned} 0 & \leq\sum_{i=1}^{k}a_{i}\phi ( x_{i} ) -A_{k}\phi ( \bar{x} ) \\ & \leq\bigl\Vert \phi^{\prime\prime}\bigr\Vert _{p} \Biggl( \int _{\alpha }^{\beta}\Biggl\vert \sum _{i=1}^{k}a_{i}G_{12} ( x_{i},t ) -A_{k}G_{12} ( \bar{x},t ) \Biggr\vert ^{q}\,dt \Biggr) ^{\frac{1}{q}}. \end{aligned}$$
$$\begin{aligned} 0 & \leq\sum_{i=1}^{k}a_{i}\phi ( x_{i} ) -\phi ( \bar {x} ) \\ & \leq\bigl\Vert \phi^{\prime\prime}\bigr\Vert _{p} \Biggl( \int _{\alpha }^{\beta}\Biggl\vert \sum _{i=1}^{k}a_{i}G_{12} ( x_{i},t ) -G_{12} ( \bar{x},t ) \Biggr\vert ^{q}\,dt \Biggr) ^{\frac{1}{q}}. \end{aligned}$$
(iii)
For \(k=2\), \(a_{1}=a_{2}=1\), \(x_{1}=\alpha\), and \(x_{2}=\beta\), we have where Specially, for \(p=1\), \(q=\infty\), we obtain where the constant \(\frac{1}{4}\) is the best possible in the sense that it cannot be replaced by a smaller constant.
$$\begin{aligned} 0 & \leq\phi ( \alpha ) +\phi ( \beta ) -2\phi \biggl( \frac{\alpha+\beta}{2} \biggr) \\ & \leq\bigl\Vert \phi^{\prime\prime}\bigr\Vert _{p} \biggl( \int _{\alpha }^{\beta} \biggl( G_{12} ( \alpha,t ) +G_{12} ( \beta,t ) -2G_{12} \biggl( \frac{\alpha+\beta}{2},t \biggr) \biggr) ^{q}\,dt \biggr) ^{\frac{1}{q}}, \end{aligned}$$
$$G_{12} ( \alpha,t ) +G_{12} ( \beta,t ) -2G_{12} \biggl( \frac{\alpha+\beta}{2},t \biggr) =\textstyle\begin{cases} t-\alpha, & \alpha\leq t\leq\frac{\alpha+\beta}{2},\\ \beta-t, & \frac{\alpha+\beta}{2}\leq t\leq\beta. \end{cases} $$
$$0\leq\frac{\phi ( \alpha ) +\phi ( \beta ) }{2}-\phi \biggl( \frac{\alpha+\beta}{2} \biggr) \leq \frac{1}{4} \bigl( \phi^{\prime}(\beta)-\phi^{\prime}(\alpha) \bigr) (\beta-\alpha ), $$
4 Applications
In this section we discuss the application of Corollary 1, i.e. the inequality (3.14), to the lower and upper bounds estimations of some relationships between well-known means.
Let \(0<\alpha<\beta\), \(\mathbf{x}\in[\alpha,\beta]^{l}\), and \(\mathbf{a}\in[0,\infty)^{l}\) whose entries satisfy the condition \(\sum_{j=1}^{l}a_{j}=1\). The weighted power mean of order \(s\in\mathbb {R}\) is defined by
$$M_{s}(\mathbf{a};\mathbf{x})= \textstyle\begin{cases} ( { \sum _{j=1}^{l}} a_{j}x_{j}^{s} ) ^{\frac{1}{s}}, & s\neq0, \\ { \prod_{j=1}^{l}} x_{j}^{a_{j}}, & s=0.\end{cases} $$
The classical weighted means are defined as
$$\begin{aligned} &M_{0}(\mathbf{a};\mathbf{x}) =G(\mathbf{a}; \mathbf{x})=\prod_{j=1}^{l}x_{j}^{a_{j}},\quad \text{the geometric mean,} \\ &M_{1}(\mathbf{a};\mathbf{x)} =A(\mathbf{a};\mathbf{x})= \sum_{j=1}^{l}a_{j}x_{j},\quad \text{the arithmetic mean,} \\ &M_{-1}(\mathbf{a};\mathbf{x)} =H(\mathbf{a};\mathbf{x})= \frac {1}{\sum_{j=1}^{l}\frac{a_{j}}{x_{j}}},\quad\text{the harmonic mean.} \end{aligned}$$
In an analogous way, for \(\mathbf{y}\in[\alpha,\beta]^{k}\), \(\mathbf{b}\in[0,\infty)^{k}\) with \(\sum_{i=1}^{k}b_{i}=1\), we define and then accordingly the classical weighted means \(G(\mathbf{b};\mathbf{y})\), \(A(\mathbf{b};\mathbf{y})\), and \(H(\mathbf{b};\mathbf{y})\).
$$M_{s}(\mathbf{b};\mathbf{y})= \textstyle\begin{cases} ( { \sum _{i=1}^{k}} b_{i}y_{i}^{s} ) ^{\frac{1}{s}}, & s\neq0, \\ { \prod_{i=1}^{k}} y_{i}^{b_{i}}, & s=0,\end{cases} $$
Corollary 2
Let
\(s\geq2\)
and
\(0<\alpha<\beta\). Let
\(\mathbf{x}\in[\alpha,\beta ]^{l}\), \(\mathbf{y}\in[\alpha,\beta]^{k}\), \(\mathbf{a}\in[ 0,\infty )^{l}\), and
\(\mathbf{b}\in[0,\infty)^{k}\)
be such that (1.1) holds for some row stochastic matrix
\(\mathbf{A} \in\mathcal{M}_{kl}(\mathbb{R})\)
and the entries of
a
and
b
satisfy the condition
\(\sum_{j=1}^{l}a_{j}=\sum_{i=1}^{k}b_{i}=1\). Then
where
\(\mathcal{G}(t)\)
denotes the difference (3.12).
$$\begin{aligned} &0 \leq M_{s}^{s}(\mathbf{a};\mathbf{x})-M_{s}^{s}( \mathbf{b};\mathbf {y})\leq s(s-1) \biggl( \int_{\alpha}^{\beta}t^{p(s-2)}\,dt \biggr) ^{\frac {1}{p}}\Vert \mathcal{G}\Vert _{q}, \\ &1 \leq\frac{G(\mathbf{b};\mathbf{y})}{G(\mathbf{a};\mathbf{x})}\leq \exp \biggl[ \biggl( \int_{\alpha}^{\beta} \biggl( \frac{1}{t^{2}} \biggr) ^{p}\,dt \biggr) ^{\frac{1}{p}}\Vert \mathcal{G}\Vert _{q} \biggr] , \\ &0 \leq\frac{H(\mathbf{b};\mathbf{y})-H(\mathbf{b};\mathbf{y})}{H(\mathbf{a};\mathbf{x})H(\mathbf{b};\mathbf{y})}\leq \biggl( \int _{\alpha }^{\beta} \biggl( \frac{2}{t^{3}} \biggr) ^{p}\,dt \biggr) ^{\frac{1}{p}}\Vert \mathcal{G}\Vert _{q}, \end{aligned}$$
The constants on the right-hand side of inequalities are sharp for
\(1< p\leq \infty\)
and the best possible for
\(p=1\).
Proof
Applying (3.14) to the function \(\phi(x)=x^{s}\), \(\phi (x)=\frac{x^{s}}{s(s-1)}\), \(\phi(x)=-\ln x\), and \(\phi(x)=\frac{1}{x}\), respectively. □
Remark 4
Particular cases of the previous results for \(p=1\), \(q=\infty\): Replacing \(x_{j}\rightarrow\frac{1}{x_{j}}\), \(y_{i}\rightarrow\frac {1}{y_{i}}\), we have \(A(\mathbf{a};\mathbf{x})\rightarrow\frac{1}{H(\mathbf{a};\mathbf{x})}\), \(A(\mathbf{b};\mathbf{y})\rightarrow\frac{1}{H(\mathbf {b};\mathbf{y})}\), and the last inequality becomes where \(\tilde{\mathcal{G}}(t)=\sum_{\mathbf{j}}a_{\mathbf{j}}(\frac {1}{x_{\mathbf{j}}}-t)-\sum_{\mathbf{i}}b_{\mathbf{i}}(\frac {1}{y_{\mathbf{i}}}-t)\), \(\mathbf{j}\in\{j;\frac{1}{x_{j}}\geq t\}\), \(\mathbf{i}\in\{i;\frac{1}{y_{i}}\geq t\}\).
$$\begin{aligned} &0 \leq M_{s}^{s}(\mathbf{a};\mathbf{x})-M_{s}^{s}( \mathbf{b};\mathbf{y})\leq s\bigl(\beta^{s-1}- \alpha^{s-1}\bigr)\Vert \mathcal{G}\Vert _{\infty}, \\ &1 \leq\frac{G(\mathbf{b};\mathbf{y})}{G(\mathbf{a};\mathbf{x})}\leq \exp \biggl( \frac{\beta-\alpha}{\alpha\beta} \Vert \mathcal{G}\Vert _{\infty} \biggr) , \\ &0 \leq\frac{H(\mathbf{b};\mathbf{y})-H(\mathbf{a};\mathbf{x})}{H(\mathbf{a};\mathbf{x})H(\mathbf{b};\mathbf{y})}\leq \biggl( \frac {\beta^{2}-\alpha^{2}}{\alpha^{2}\beta^{2}} \biggr) \Vert \mathcal {G}\Vert _{\infty}. \end{aligned}$$
$$0\leq A(\mathbf{a};\mathbf{x})-A(\mathbf{b};\mathbf{y})\leq \bigl( \beta ^{2}-\alpha^{2} \bigr) \Vert \tilde{\mathcal{G}}\Vert _{\infty}, $$
Corollary 3
Let
\(0<\alpha<\beta\)
and
\(\mathbf{x}\in[\alpha,\beta]^{l}\), \(\mathbf{y}\in[\alpha,\beta]^{k}\), \(\mathbf{a}\in[0,\infty)^{l}\), and
\(\mathbf{b}\in[0,\infty)^{k}\)
be such that (1.1) holds for some row stochastic matrix
\(\mathbf{A}\in\mathcal{M}_{kl}(\mathbb{R})\)
and the entries of
a
and
b
satisfy the condition
\(\sum_{j=1}^{l}a_{j}=\sum_{i=1}^{k}b_{i}=1\). Then
where
\(\mathcal{G}(t)\)
denotes the difference (3.12).
$$\begin{aligned} &1 \leq\frac{\prod_{j=1}^{l}x_{j}^{a_{j}x_{j}}}{\prod_{i=1}^{k}y_{i}^{b_{i}y_{i}}}\leq\exp \biggl[ \biggl( \int_{\alpha}^{\beta} \biggl( \frac{1}{p} \biggr) ^{p}\,dt \biggr) ^{\frac{1}{p}}\Vert \mathcal{G}\Vert _{q} \biggr] , \\ &0 \leq\sum_{j=1}^{l}a_{j}e^{x_{j}}- \sum_{i=1}^{k}b_{i}e^{y_{i}} \leq \biggl( \int_{\alpha}^{\beta}e^{tp}\,dt \biggr) ^{\frac{1}{p}}\Vert \mathcal {G}\Vert _{q}, \\ &1 \leq\frac{{ \prod_{j=1}^{l}} (1+e^{x_{j}})^{a_{j}}}{{ \prod_{i=1}^{k}} (1+e^{y_{i}})^{b_{i}}}\leq\exp \biggl( \biggl( \int_{\alpha}^{\beta} \biggl( \frac{e^{t}}{(1+e^{t})^{2}} \biggr) ^{p}\,dt \biggr) ^{\frac{1}{p}}\Vert \mathcal{G}\Vert _{q} \biggr) , \end{aligned}$$
The constants on the right-hand side of inequalities are sharp for
\(1< p\leq \infty\)
and the best possible for
\(p=1\).
Proof
Apply (3.14) to the functions \(\phi(x)=x\ln x\), \(\phi(x)=e^{x}\), and \(\phi(x)=\ln(1+e^{x})\), respectively. □
Remark 5
Particular cases of the previous results for \(p=1\), \(q=\infty\):
$$\begin{aligned} &1 \leq\frac{\prod_{j=1}^{l}x_{j}^{a_{j}x_{j}}}{\prod_{i=1}^{k}y_{i}^{b_{i}y_{i}}}\leq \biggl( \frac{\beta}{\alpha} \biggr) ^{\Vert \mathcal{G}\Vert _{\infty}}, \\ &0 \leq\sum_{j=1}^{l}a_{j}e^{x_{j}}- \sum_{i=1}^{k}b_{i}e^{y_{i}}\leq\bigl(e^{\beta}-e^{\alpha}\bigr)\Vert \mathcal{G}\Vert _{\infty}, \\ &1 \leq\frac{{ \prod_{j=1}^{l}} (1+e^{x_{j}})^{a_{j}}}{{ \prod_{i=1}^{k}} (1+e^{y_{i}})^{b_{i}}}\leq\exp \biggl[ \frac{e^{\beta}-e^{\alpha}}{ ( 1+e^{\beta} ) (1+e^{\alpha})}\Vert \mathcal{G} \Vert _{\infty } \biggr] . \end{aligned}$$
If we consider a particular case of our result given in Remark 3(ii) and apply it to convex functions from the previous two corollaries, we could obtain some new upper bounds for Jensen’s inequality. Related estimates are given in [13‐15].
Finally, we indicate one more interesting application.
Using (2.4), under the assumptions of Theorem 5, we define the linear functional \(A:C^{n}([\alpha,\beta])\rightarrow\mathbb{R}\) by It is obvious that if \(\phi\in C^{n}([\alpha,\beta])\) is n-convex, then \(A(\phi)\geq0\). Using the linearity and positivity of this functional we may derive corresponding mean-value theorems. Moreover, we may produce new classes of exponentially convex functions and as a result we get new means of the Cauchy type applying the same method as given in [5, 16].
$$\begin{aligned} A(\phi) ={}&\sum_{j=1}^{l}a_{j} \phi ( x_{j} ) -\sum_{i=1}^{k}b_{i} \phi ( y_{i} ) \\ & {}-\sum_{s=2}^{m}\frac{\phi^{(s)}(\alpha)}{s!} \Biggl[ \sum_{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{s} \Biggr] \\ &{} -\sum_{r=0}^{n-m-2}\sum _{m=0}^{r}\frac{ ( -1 ) ^{r-s} ( \beta-\alpha ) ^{r-s}\phi^{(m+1+r)}(\beta)}{(m+1+s)!(r-s)!}\\ &{}\times \Biggl[ \sum _{j=1}^{l}a_{j} ( x_{j}-\alpha ) ^{m+1+s}-\sum_{i=1}^{k}b_{i} ( y_{i}-\alpha ) ^{m+1+s} \Biggr] . \end{aligned}$$
5 Conclusions
In this paper we give generalizations of Sherman’s inequality from which the classical majorization inequality, as well as Jensen’s inequality, follows as a special case. The obtained results hold for convex functions of higher order, which are in a special case convex in the usual sense. Moreover, the obtained generalizations represent an extension to real, not necessarily nonnegative entries of the vectors a, b, and the matrix A. The methods used are based on classical real analysis and the application of the Abel-Gontscharoff formula and Green’s function and can be extended to the investigation of other inequalities.
Acknowledgements
The research of the first and third author has been supported by the Croatian Science Foundation under the project 5435. The authors would like to thank the referees for their valuable comments and suggestions which essentially improved the quality of this paper.
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to deriving all the results of this article, and they read and approved the final manuscript.