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Bellman–Steffensen type inequalities
- Julije Jakšetić^{1},
- Josip Pečarić^{2} and
- Ksenija Smoljak Kalamir^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-018-1882-9
© The Author(s) 2018
- Received: 19 July 2018
- Accepted: 14 October 2018
- Published: 22 October 2018
Abstract
In this paper some Bellman–Steffensen type inequalities are generalized for positive measures. Using sublinearity of a class of convex functions and Jensen’s inequality, nonnormalized versions of Steffensen’s inequality are obtained. Further, linear functionals, from obtained Bellman–Steffensen type inequalities, are produced and their action on families of exponentially convex functions is studied.
Keywords
- Steffensen’s inequality
- Bellman–Steffensen type inequality
- Measure theory
- Exponential convexity
MSC
- 26D15
- 26D20
1 Introduction
Since its appearance in 1918 Steffensen’s inequality [1] has been a subject of investigation by many mathematicians because it plays an important role not only in the theory of inequalities but also in statistics, functional equations, time scales, special functions, etc. A comprehensive survey on generalizations and applications of Steffensen’s inequality can be found in [2].
In 1959 Bellman gave an \(L^{p}\) generalization of Steffensen’s inequality (see [3]) for which Godunova, Levin and Čebaevskaya noted that it is incorrect as stated (see [4]). Further, in [5] Pečarić showed that the Bellman generalization of Steffensen’s inequality is true with very simple modifications of conditions. Using some substitutions in his result from [5], Pečarić also proved the following modification of Steffensen’s inequality in [6].
Theorem 1.1
In [7] Mitrinović and Pečarić gave necessary and sufficient conditions for inequality (1.2). The purpose of this paper is to generalize the aforementioned result for positive measures, using the approach from [8] and [9], and to give some applications related to exponential convexity.
Let \(\mathcal{B}([a,b])\) be the Borel σ-algebra generated on \([a,b]\). In [10] the authors proved the following measure theoretic generalization of Steffensen’s inequality.
Theorem 1.2
Remark 1.1
2 Main results
Motivated by Theorem 1.1 and necessary and sufficient conditions given in [7], we prove some generalizations of Bellman–Steffensen type inequalities for positive measures.
Theorem 2.1
For a nondecreasing, right-continuous function \(f:[a,b]\rightarrow \mathbb{R}\), inequality (2.1) is reversed.
Proof
In the following lemma we recall the property of sublinearity of the class of convex functions.
Lemma 2.1
Theorem 2.2
Proof
Remark 2.1
In Theorems 2.1 and 2.2 we proved similar results to those obtained by Liu in [11] but we only need μ to be finite and positive instead of finite continuous and strictly increasing as in [11].
We continue with some more general Bellman–Steffensen type inequalities related to the function \(f/k\).
Theorem 2.3
For a nondecreasing, right-continuous function \(f/k:[a,b]\rightarrow \mathbb{R}\) inequality (2.5) is reversed.
Proof
Theorem 2.4
Proof
3 Applications
In this section we use classes of log-convex, exponentially convex and n-exponentially convex functions. Definitions and properties of these classes of functions can be found, e.g., in Pečarić, Proschan and Tong [12], Bernstein [13], Pečarić and Perić [14], and Jakšetić, Pečarić [15].
The following example will be useful in our applications.
Example 3.1
- (i)
\(f(x)=e^{\alpha x}\) is exponentially convex on \(\mathbb{R}\), for any \(\alpha\in \mathbb{R}\).
- (ii)
\(g(x)=x^{-\alpha}\) is exponentially convex on \((0,\infty )\), for any \(\alpha>0\).
The following families of functions given in the next two lemmas will be useful in constructing exponentially convex functions.
Lemma 3.1
Proof
Since \(\frac{d}{dx} (\frac{\varphi_{p}(x)}{k(x)} )=x^{p-1}>0\) on \((0,\infty)\) for each \(p\in \mathbb{R}\) we have that \(x\mapsto(\varphi_{p}/k)(x)\) is increasing on \((0,\infty)\). From Example 3.1 the mappings \(p\mapsto e^{p\log x}\) and \(p\mapsto\frac{1}{p}\) are exponentially convex, and since \(p\mapsto\frac{x^{p}}{p}=e^{p\log x}\cdot\frac{1}{p}\), the second conclusion follows. □
Similarly we obtain the following lemma.
Lemma 3.2
Theorem 3.1
- (i)
The function Φ is continuous on \(\mathbb{R}\).
- (ii)If \(n\in \mathbb{N}\) and \(p_{1},\ldots,p_{n}\in \mathbb{R}\) are arbitrary, then the matrixis positive semidefinite.$$\biggl[\Phi \biggl(\frac{p_{j}+p_{k}}{2} \biggr) \biggr]_{j,k=1}^{n} $$
- (iii)
The function Φ is exponentially convex on \(\mathbb{R}\).
- (iv)
The function Φ is log-convex on \(\mathbb{R}\).
- (v)If \(p,q,r\in \mathbb{R}\) are such that \(p< q< r\), then$$ \Phi(q)^{r-p}\leq \Phi(p)^{r-q} \Phi(r)^{q-p}. $$
Proof
(i) Continuity of the function \(p\mapsto\Phi(p)\) is obvious for \(p\in \mathbb{R}\setminus\{0\}\). For \(p=0\) it is directly checked using Heine characterization.
Claims (iii), (iv), (v) are simple consequences of (i) and (ii). □
Theorem 3.2
- (i)
The function F is continuous on \((0,\infty)\).
- (ii)If \(n\in \mathbb{N}\) and \(p_{1},\ldots,p_{n}\in(0,\infty)\) are arbitrary, then the matrixis positive semidefinite.$$\biggl[F \biggl(\frac{p_{j}+p_{k}}{2} \biggr) \biggr]_{j,k=1}^{n} $$
- (iii)
The function F is exponentially convex on \((0,\infty)\).
- (iv)
The function F is log-convex on \((0,\infty)\).
- (v)If \(p,q,r\in(0,\infty)\) are such that \(p< q< r\), then$$ F(q)^{r-p}\leq F(p)^{r-q} F(r)^{q-p}. $$
Proof
(i) Continuity of the function \(p\mapsto F(p)\) is obvious.
Claims (iii), (iv), (v) are simple consequences of (i) and (ii). □
In the following theorem we give the Lagrange-type mean value theorem.
Theorem 3.3
Proof
Using the standard Cauchy type mean value theorem, we obtain the following corollary.
Corollary 3.1
Remark 3.1
Theorem 3.4
- (i)
The function H is continuous on \((1,\infty)\).
- (ii)If \(n\in \mathbb{N}\) and \(p_{1},\ldots,p_{n}\in(1,\infty)\) are arbitrary, then the matrixis positive semidefinite.$$\biggl[H \biggl(\frac{p_{j}+p_{k}}{2} \biggr) \biggr]_{j,k=1}^{n} $$
- (iii)
The function H is exponentially convex on \((1,\infty)\).
- (iv)
The function H is log-convex on \((1,\infty)\).
- (v)If \(p,q,r\in(1,\infty)\) are such that \(p< q< r\), then$$ H(q)^{r-p}\leq H(p)^{r-q} H(r)^{q-p}. $$
Proof
(i) Continuity of the function \(p\mapsto H(p)\) is obvious.
Claims (iii), (iv), (v) are simple consequences of (i) and (ii). □
Similar to Corollary 3.1 we also have the following corollary.
Corollary 3.2
Remark 3.2
Theorem 3.5
- (i)
S is n-exponentially convex in the Jensen sense on J.
- (ii)If S is continuous on J, then it is n-exponentially convex on J and for \(p, q, r \in J\) such that \(p < q < r\), we have$$ S(q)^{r-p}\leq S(p)^{r-q}S(r)^{q-p}. $$
- (iii)If S is positive and differentiable on J, then for every \(p, q, u,v \in J\) such that \(p\leq u, q \leq v\), we havewhere \(\widetilde{M}(p,q)\) is defined by$$ \widetilde{M}(p,q)\leq\widetilde{M}(u,v), $$$$ \widetilde{M}(p,q) = \textstyle\begin{cases} (\frac{S(p)}{S(q)} )^{\frac{1}{p-q}},& p\neq q; \\ \mathrm{exp} (\frac{\frac{d }{dp} (S(p) )}{S(p)} ), & p=q. \end{cases} $$
Proof
(ii) Since S is continuous on J, then it is n-exponentially convex.
(iii) This is a consequence of the characterization of convexity by the monotonicity of the first order divided differences (see [12, p. 4]). □
Theorem 3.6
- (i)If \(n\in \mathbb{N}\) and \(p_{1},\ldots,p_{n}\in \mathbb{R}\) are arbitrary, then the matrixis positive semidefinite.$$\biggl[H \biggl(\frac{p_{k}+p_{m}}{2} \biggr) \biggr]_{k,m=1}^{n} $$
- (ii)
If the function H is continuous on J, then H is exponentially convex on J.
- (iii)If H is positive and differentiable on J, then for every \(p, q, u,v \in J\) such that \(p\leq u, q \leq v\), we havewhere \(\widehat{M}(p,q)\) is defined by$$ \widehat{M}(p,q)\leq\widehat{M}(u,v), $$$$ \widehat{M}(p,q) = \textstyle\begin{cases} (\frac{H(p)}{H(q)} )^{\frac{1}{p-q}},& p\neq q; \\ \mathrm{exp} (\frac{\frac{d }{dp} (H(p) )}{H(p)} ), & p=q. \end{cases} $$
Proof
(ii) This follows from part (i).
(iii) This is a consequence of the characterization of convexity by the monotonicity of the first order divided differences (see [12, p. 4]). □
Declarations
Acknowledgements
The authors would like to thank an anonymous referee for his valuable remarks and suggestions that improved an earlier version of the manuscript.
Availability of data and materials
Not applicable.
Funding
No funding was received.
Authors’ contributions
The authors jointly worked on the results and they read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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’ Affiliations
References
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