Abstract
In the article, we establish several Ostrowski type inequalities involving the conformable fractional integrals. As applications, we find new inequalities for the arithmetic and generalized logarithmic means.
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1 Introduction
Let \(I\subseteq\mathbb{R}\) be an interval and \(I^{\circ}\) the interior of I. Then the classical Ostrowski inequality [1] states that a real-valued function \(f: I\rightarrow\mathbb{R}\) satisfies the inequality
with the best possible constant \(1/4\) if \(a_{1}, a_{2}\in I^{\circ}\) with \(a_{1}< a_{2}\) and \(|f^{\prime}(x)|\leq M\) for all \(x\in[a_{1}, a_{2}]\).
Recently, the Ostrowski inequality has attracted the attention of many researchers, many remarkable generalizations, extensions, variants and applications can be found in the literature [2–24].
Let \(0<\alpha\leq1\) and g be a real-valued function defined on \([0, \infty)\). Then the (conformable) fractional derivative \(D_{\alpha }(g)(t)\) [23] of order α of g at \(t>0\) is defined by
g is said to be α-differentiable if the conformable fractional derivative of order α of g exists. In what follows, we write \(g^{\alpha}(t)\) or \(\frac{d_{\alpha}}{d_{\alpha}t}(g)\) for \(D_{\alpha }(g)(t)\) to denote the conformable fractional derivative of order α of g. The conformable fractional derivative at 0 is defined as \(g^{\alpha}(0)=\lim_{t\rightarrow0^{+}}g^{\alpha}(t)\).
Let \(\alpha\in(0, 1]\) and \(0\leq a< b\). Then the function \(h: [a, b]\rightarrow\mathbb{R}\) is said to be α-fractional integrable on \([a, b]\) if the integral
exists and is finite. All α-fractional integrable functions on \([a, b]\) are denoted by \(L_{\alpha}^{1}([a, b])\).
Remark 1.1
Note that the relation between the Riemann integral and the conformable fractional integral is given by
Let \(\alpha\in(0, 1]\) and f, g be α-differentiable at \(t>0\). Then it is well known that
for all \(n\in\mathbb{R}\);
for all constant \(c\in\mathbb{R}\);
for all \(a, b\in\mathbb{R}\);
if f is differentiable at \(g(t)\).
The main purpose of the article is to find the Ostrowski type inequalities involving the conformable fractional integrals and give their applications in certain bivariate means.
2 Main results
Lemma 2.1
Let \(0<\alpha\leq1\), \(0\leq a_{1}< a_{2}\) and \(h: [a_{1}, a_{2}]\rightarrow\mathbb{R}\) be an α-fractional differentiable function. Then the identity
holds if \(D_{\alpha}(h)\in L_{\alpha}^{1}([a_{1}, a_{2}])\).
Proof
It follows from integration by parts that
□
Theorem 2.2
Let \(0<\alpha\leq1\), \(0\leq a_{1}< a_{2}\), \(h: [a_{1}, a_{2}]\rightarrow\mathbb{R}\) be an α-fractional differentiable function and \(D_{\alpha}(h)\in L_{\alpha}^{1}([a_{1}, a_{2}])\). Then the inequality
holds if \(|h^{\prime}(x)|\) is convex, where
Proof
Let \(y>0\), \(\varphi_{1}(y)=y^{\alpha-1}\) and \(\varphi_{2}(y)=-y^{\alpha }\). Then we clearly see that the functions \(\varphi_{1}\) and \(\varphi _{2}\) both are convex. It follows from Lemma 2.1 and the convexity of \(\varphi_{1}\), \(\varphi_{2}\) and \(|h'|\) that
□
Corollary 2.3
Let \(x=(a_{1}+a_{2})/2\). Then Theorem 2.2 leads to
Remark 2.4
If \(\alpha=1\), then Corollary 2.3 becomes
where the second inequality is obtained by using the convexity of \(|h'|\).
Theorem 2.5
Let \(q>1\), \(M>0\), \(0<\alpha\leq1\), \(0\leq a_{1}< a_{2}\), \(h: [a_{1}, a_{2}]\rightarrow\mathbb{R}\) be an α-fractional differentiable function and \(D_{\alpha}(h)\in L_{\alpha}^{1}([a_{1}, a_{2}])\). Then the inequality
holds if \(|h^{\prime}|^{q}\) is convex on \([a_{1}, a_{2}]\) and \(|h^{\prime}(x)|^{q}\leq M\), where
Proof
From Lemma 2.1, power-mean inequality and the convexity of \(|h^{\prime }|^{q}\) together with the identities
and
we clearly see that
Therefore, Theorem 2.5 follows easily from (2.1)–(2.5). □
Remark 2.6
Let \(\alpha=1\). Then Theorem 2.5 leads to
where
Theorem 2.7
Let \(q>1\), \(M>0\), \(0<\alpha\leq1\), \(0\leq a_{1}< a_{2}\), \(h: [a_{1}, a_{2}]\rightarrow\mathbb{R}\) be an α-fractional differentiable function and \(D_{\alpha}(h)\in L_{\alpha}^{1}([a_{1}, a_{2}])\). Then the inequality
holds if \(|h^{\prime}|^{q}\) is convex on \([a_{1}, a_{2}]\) and \(|h^{\prime}(x)|^{q}\leq M\), where
Proof
It follows from the proof of Theorem 2.2 that
From the power-mean inequality and convexity of \(|h^{\prime}|^{q}\) together with the identities
and
we get
Therefore, Theorem 2.7 follows easily from (2.6)–(2.10). □
Theorem 2.8
Let \(q>1\), \(0<\alpha\leq1\), \(0\leq a_{1}< a_{2}\), \(h: [a_{1}, a_{2}]\rightarrow\mathbb{R}\) be an α-fractional differentiable function and \(D_{\alpha}(h)\in L_{\alpha}^{1}([a_{1}, a_{2}])\). Then the inequality
holds if \(|h^{\prime}|^{q}\) is concave on \([a_{1}, a_{2}]\), where
Proof
It is well known that \(|h^{\prime}|\) is concave due to \(|h^{\prime }|^{q}\) being concave. It follows from Lemma 2.1 that
Making use of Jensen’s integral inequality, we have
where we have used the identities
□
Remark 2.9
If \(\alpha=1\), then Theorem 2.8 becomes
Theorem 2.10
Let \(q>1\), \(0<\alpha\leq1\), \(0\leq a_{1}< a_{2}\), \(h: [a_{1}, a_{2}]\rightarrow\mathbb{R}\) be an α-fractional differentiable function and \(D_{\alpha}(h)\in L_{\alpha}^{1}([a_{1}, a_{2}])\). Then the inequality
holds if \(|h^{\prime}|^{q}\) is concave on \([a_{1}, a_{2}]\), where
Proof
From the concavity of \(|h^{\prime}|^{q}\) we know that \(|h^{\prime}|\) is also concave, then from Lemma 2.1 we have
It follows from the Jensen integral inequality that
□
Remark 2.11
If \(\alpha=1\), then Theorem 2.10 leads to
Remark 2.12
If \(\alpha=1\) and \(x=(a_{1}+a_{2})/2\), then Theorem 2.10 becomes
3 Applications to means
Let \(a, b>0\) with \(a\neq b\). Then the arithmetic mean \(A(a, b)\), logarithmic mean \(L(a,b)\) and generalized logarithmic mean \(L_{(\alpha, r)}(a,b)\) of a and b are defined by
respectively.
Recently, the bivariate means have been the subject of intensive research, many remarkable inequalities for the bivariate means can be found in the literature [25–60].
Let \(h(x)=x^{r}\) and \(h(x)=1/x\). Then Corollary 2.3 immediately leads to Theorems 3.1 and 3.2.
Theorem 3.1
Let \(r>1\) and \(\alpha\in(0, 1]\). Then the inequality
holds for all \(a_{1}, a_{2}>0\).
Theorem 3.2
Let \(r>1\) and \(\alpha\in(0, 1]\). Then the inequality
holds for all \(a_{1}, a_{2}>0\).
4 Results and discussion
There are many results devoted to the well-known Ostrowski inequality. This inequality has many applications in the area of numerical analysis. In this paper, we give results for Ostrowski inequality containing conformable fractional integrals and their applications for means. First, we prove an identity associated with the Ostrowski inequality for conformable fractional integrals. By using this identity and convexity of different classes of functions and some well-known inequalities, we obtain several results for the inequality. The inequalities derived here are also pointed out to correspond to some known results, being special cases. At the end, we also present applications for means. The presented idea may stimulate further research in the theory of conformable fractional integrals.
5 Conclusion
In this paper, we prove an identity associated with the Ostrowski inequality for conformable fractional integral, present several Ostrowski type inequalities involving the conformable fractional integrals, and provide the applications in bivariate means theory. The idea and results presented are novel and interesting.
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Acknowledgements
The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 61374086, 11371125, 11401191), the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101) and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant No. Y201635325).
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Adil Khan, M., Begum, S., Khurshid, Y. et al. Ostrowski type inequalities involving conformable fractional integrals. J Inequal Appl 2018, 70 (2018). https://doi.org/10.1186/s13660-018-1664-4
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DOI: https://doi.org/10.1186/s13660-018-1664-4