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On a variant of Čebyšev’s inequality of the Mercer type
Journal of Inequalities and Applications volume 2020, Article number: 242 (2020)
Abstract
We consider the discrete Jensen–Mercer inequality and Čebyšev’s inequality of the Mercer type. We establish bounds for Čebyšev’s functional of the Mercer type and bounds for the Jensen–Mercer functional in terms of the discrete Ostrowski inequality. Consequentially, we obtain new refinements of the considered inequalities.
1 Introduction
Let \(n\geq 2\) and let \(\boldsymbol{w}= ( w_{1},\ldots ,w_{n} ) \) be a real n-tuple such that
In [5] the following Čebyšev’s inequality of the Mercer type:
was proved for any real n-tuples \(\boldsymbol{x}= ( x_{1}, \ldots ,x_{n} ) \) and \(\boldsymbol{y}= ( y_{1},\ldots ,y_{n} ) \) monotone in the same direction and real numbers a, b, c, d such that
If x and y are monotonic in the opposite directions, inequality (2) is reversed.
Here, to be more precise, we cite that result with the slightly different notation.
In the same paper, the authors considered Čebyšev’s functional (or Čebyšev’s difference) of the Mercer type defined as the difference of the right- and left-hand sides of inequality (2). They established bounds in terms of the discrete Ostrowski inequality. Here we give more accurate bounds, which also provide refinements of inequality (2). In addition, using these results, we establish Ostrowski-like bounds for the Jensen–Mercer functional and, consequentially, a refinement of the Jensen–Mercer inequality.
2 Bounds for the Čebyšev’s functional of the Mercer type
Let \(m\geq 2\) and let \(\boldsymbol{p}= ( p_{1},\ldots ,p_{m} ) \) be a real m-tuple such that
Then \(\overline{P}_{k}=\sum_{i=k}^{m}p_{i}\geq 0\), \(k=1, \ldots ,m \). Furthermore, from the summation by parts (sometimes called the Abel transformation) it follows that the identity
holds for any two real m-tuples \(\boldsymbol{\xi }= ( \xi _{1},\ldots ,\xi _{m} ) \) and \(\boldsymbol{\zeta }= ( \zeta _{1},\ldots ,\zeta _{m} ) \), where \(\Delta \xi _{i}=\xi _{i+1}-\xi _{i}\), \(\Delta \zeta _{i}=\zeta _{i+1}-\zeta _{i}\), \(i=1,\ldots ,m-1\) (see [7, 8]).
Here, and in the rest of the paper, we assume \(\sum_{j=k}^{l}x_{j}=0\) when \(k>l\).
Lemma 1
Let \(n\geq 2\) and let w be a real n-tuple such that (1) is fulfilled. Then for any real n-tuples x, y and real numbers a, b, c, d satisfying (3), the identity
holds, where \(\Delta x_{i}=x_{i+1}-x_{i}\), \(\Delta y_{i}=y_{i+1}-y_{i}\), \(i=1,\ldots ,n-1\).
Proof
For \(m=n+2\), we define m-tuples p, ξ, and ζ as
Since w satisfies (1) it follows that
Hence, we can apply identity (5). Its left-hand side is
It can be easily seen that
hence, on the right-hand side of (5) we have
Calculating separately summands for \(i=1\) and \(i=m-1\), we obtain
Therefore,
which is equal to the right-hand side of (6). □
Using identity (6) and imposing stricter conditions than (3), we obtain refinements of inequality (2) which are more accurate than those previously established in [5].
Theorem 1
Let \(n\geq 2\) and let w be a real n-tuple such that (1) is fulfilled. Let x, y be real n-tuples monotonic in the same direction. Suppose that real numbers a, b, c, d and nonnegative real numbers r, s satisfy
Then
If x and y are monotonic in the opposite directions, then the inequalities in (10) are reversed and the term rs appears with the negative sign.
Proof
Under the given assumptions, using identity (6), we obtain
Since
we obtain the first inequality in (10). Since r, s are nonnegative real numbers and obviously
the second inequality in (10) immediately follows. □
Using identity (6) and the triangle inequality, we can establish bounds for the Čebyšev’s functional (or Čebyšev’s difference) of the Mercer type in terms of the discrete Ostrowski inequality.
Throughout the rest of the paper, let \([ a,b ] \) and \([ c,d ] \) be intervals in \(\mathbb{R}\), where \(a< b\), \(c< d\).
Theorem 2
Let \(n\geq 2\) and let w be a real n-tuple such that conditions (1) are fulfilled. Then for any real n-tuples \(\boldsymbol{x}\in [ a,b ] ^{n}\), \(\boldsymbol{y}\in [ c,d ] ^{n}\) the following inequalities hold:
Proof
Using identity (6) and the triangle inequality, we have
because \(W_{i}\) and \(\overline{W}_{i}\) are nonnegative for all \(i=1,\ldots ,n \). Since \(\vert y_{1}-c \vert \), \(\vert d-y_{n} \vert \), \(\vert \Delta y_{i} \vert \) for all \(i=1,\ldots ,n\), are less or equal to \(d-c\), and \(\vert x_{1}-a \vert \), \(\vert b-x_{n} \vert \), \(\vert \Delta x_{i} \vert \) for all \(i=1,\ldots ,n\), are less or equal to \(b-a\), we obtain inequalities (11). □
Remark 1
If in Theorem 2 we add assumption that R, S are nonnegative real numbers such that
then we obtain refinements of the two inequalities proved in [5] under the same assumption. Namely, we have inequalities
and, as a special case when \(w_{i}=1\) (\(i=1,\ldots ,n \)), we have inequalities
3 Bounds for the Jensen–Mercer functional
Jensen–Mercer inequality
for a convex function \(f: ( \alpha ,\beta ) \rightarrow \mathbb{R} \), real n-tuple \(\boldsymbol{x}\in [ a,b ] ^{n}\), and positive real n-tuple w, where \(-\infty \leq \alpha < a< b<\beta \leq \infty \), was proved in [6]. In [1], it was proved that it remains valid when x is monotonic and w satisfies conditions (1).
Using our results from the previous section, we establish Ostrowski-like bounds for the Jensen–Mercer functional, i.e., the difference of the right- and left-hand sides of inequality (15).
Theorem 3
Let \(f: ( \alpha ,\beta ) \rightarrow \mathbb{R} \) be a differentiable function and suppose that γ, δ are real numbers such that \(\gamma \leq f^{\prime } ( x ) \leq \delta \), for all \(x\in ( \alpha ,\beta ) \). Let \(n\geq 2\) and suppose that n-tuple \(\boldsymbol{x}\in [ a,b ] ^{n}\), where \(-\infty \leq \alpha < a< b<\beta \leq \infty \), satisfies conditions (12). Let w be a real n-tuple such that conditions (1) are fulfilled and \(a+b-\frac{1}{W_{n}}\sum_{i=1}^{n}w_{i}x_{i}\in [ a,b ] \). Then
Proof
By the mean-value theorem, for any \(\zeta ,\eta \in ( \alpha ,\beta ) \), there exists some ξ between them such that \(f ( \zeta ) -f ( \eta ) =f^{\prime } ( \xi ) ( \zeta -\eta ) \). Hence, choosing \(\zeta =x_{i}\) and \(\eta =a+b-\frac{1}{W_{n}}\sum_{i=1}^{n}w_{i}x_{i}\), we obtain
Multiplying (17) by \(-\frac{w_{i}}{W_{n}}\), and then summing over i, we have
Choosing \(\zeta =a\), \(\zeta =b\), respectively, and \(\eta =a+b-\frac{1}{W_{n}}\sum_{i=1}^{n}w_{i}x_{i}\), we have
Summing the above three equalities, we obtain
Since \(\gamma \leq f^{\prime } ( x ) \leq \delta \), for all \(x\in ( \alpha ,\beta ) \), it holds
and inequalities (16) immediately follow from Theorem 2 and Remark 1. □
Remark 2
An integral variant of identity (5) can be found in [9] and there is a way to obtain integral variants in terms of Riemann–Stieltjes integral of the Jensen–Mercer inequality from the Jensen–Steffensen inequality (see, for example, [2–4]). Hence, our discrete results can be extended to the continuous case.
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Acknowledgements
We would like to thank the reviewers for their effort to read the paper thoroughly and give us very useful suggestions how to improve it.
Funding
This publication was supported by the University of Split, Faculty of Science and by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008).
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Dedicated to Professor Shoshana Abramovich on the occasion of her 80th birthday.
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Matković, A., Pečarić, J. On a variant of Čebyšev’s inequality of the Mercer type. J Inequal Appl 2020, 242 (2020). https://doi.org/10.1186/s13660-020-02509-3
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DOI: https://doi.org/10.1186/s13660-020-02509-3