- Research
- Open Access
- Published:
Inequalities for α-fractional differentiable functions
Journal of Inequalities and Applications volume 2017, Article number: 93 (2017)
Abstract
In this article, we present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals. As applications, we establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.
1 Introduction
A real-valued function \(\psi: \mathrm{I}\subseteq \mathbb{R}\rightarrow\mathbb{R}\) is said to be convex on I if the inequality
holds for all \(\xi, \zeta\in\mathrm{I}\) and \(\theta\in[0, 1]\). ψ is said to be concave on I if inequality (1.1) is reversed.
Let \(\psi: \mathrm{I}\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a convex function on the interval I, and \(c_{1}, c_{2} \in\mathrm{I}\) with \(c_{1}< c_{2}\). Then the double inequality
is known in the literature as the Hermite-Hadamard inequality for convex functions [1–3]. Both inequalities hold in the reversed direction if ψ is concave on the interval I. In particular, many classical inequalities for means can be derived from (1.2) for appropriate particular selections of the function ψ.
Recently, the improvements, generalizations, refinements and applications for the Hermite-Hadamard inequality have attracted the attention of many researchers [4–22].
Dragomir and Agarwal [23] proved the following results connected with the right hand part of (1.2).
Theorem 1.1
See [23], Lemma 2.1
Let \(\psi:\mathrm{I}^{\circ}\subseteq\mathbb{R}\rightarrow \mathbb{R}\) be a differentiable mapping on \(\mathrm{I}^{\circ}\). Then the identity
holds for all \(c_{1}, c_{2}\in\mathrm{I}^{\circ}\) with \(c_{1}< c_{2}\) if \(\psi^{\prime}\in\mathrm{L}[c_{1}, c_{2}]\), where \(\mathrm{I}^{\circ}\) denotes the interior of I.
Theorem 1.2
See [23], Theorem 2.2
Let \(\psi:\mathrm{I}^{\circ}\subseteq\mathbb{R}\rightarrow \mathbb{R}\) be a differentiable mapping on \(\mathrm{I}^{\circ}\). Then the inequality
holds for \(c_{1}, c_{2}\in\mathrm{I}^{\circ}\) with \(c_{1}< c_{2}\) if \(\vert \psi^{\prime} \vert \) is convex on \([c_{1}, c_{2}]\).
Making use of Theorem 1.1, Pearce and Pečarić [24] established Theorem 1.3 as follows.
Theorem 1.3
See [24], Theorem 1
Let \(c_{1}, c_{2}\in\mathrm{I}\subseteq\mathbb{R}\) with \(c_{1}< c_{2}\), \(\psi:\mathrm{I}^{\circ}\subseteq \mathbb{R}\rightarrow\mathbb{R}\) be a differentiable mapping on \(\mathrm{I}^{\circ}\) and \(q\geq1\). Then the inequality
is valid if the mapping \(\vert \psi^{\prime} \vert ^{q}\) is convex on the interval \([c_{1}, c_{2}]\).
Next, we recall several elementary definitions and important results in the theory of conformable fractional calculus, which will be used throughout the article, we refer the interested reader to [25–32].
The conformable fractional derivative of order \(0<\alpha\leq1\) for a function \(\psi: (0, \infty)\rightarrow\mathbb{R}\) at \(\xi>0\) is defined by
and the fractional derivative at 0 is defined as \(\mathrm{D}_{\alpha}(\psi)(0)=\lim_{\xi\rightarrow 0^{+}}\mathrm{D}_{\alpha}(\psi)(\xi)\).
The (left) fractional derivative starting from \(c_{1}\) of a function \(\psi: [c_{1}, \infty)\rightarrow\mathbb{R}\) of order \(0<\alpha \leq1\) is defined by
and we write \(\mathrm{D}_{\alpha}^{c_{1}}(\psi)=\mathrm{D}_{\alpha}^{0}(\psi)=\mathrm {D}_{\alpha}(\psi)\) if \(c_{1}=0\). For more details see [26].
Let \(\alpha\in(0, 1]\) and \(\psi, \phi\) be α-differentiable at \(\xi>0\). Then we have
where ψ is differentiable at \(\phi(\xi)\) in equation (1.3). In particular,
if ψ is differentiable.
Let \(\alpha\in(0, 1] \) and \(0\leq c_{1} < c_{2}\). A function \(\psi : [c_{1}, c_{2}] \rightarrow\mathbb{R} \) is said to be α-fractional integrable on \([c_{1}, c_{2}]\) if the integral
exists and is finite. All the α-fractional integrable functions on \([c_{1}, c_{2}]\) are denoted by \(\mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\).
It is well known that
if \(\psi, \phi: [c_{1}, c_{2}] \rightarrow\mathbb{R}\) are two functions such that ψϕ is differentiable.
Very recently, Anderson [33] established a Hermite-Hadamard type inequality for fractional differentiable functions as follows.
Theorem 1.4
Let \(\alpha\in(0, 1]\) and \(\psi: [c_{1}, c_{2}]\rightarrow\mathbb{R}\) be an α-fractional differentiable function. Then the inequality
holds if \(\mathrm{D}_{\alpha}(\psi)\) is increasing on \([c_{1}, c_{2}]\). Moreover, if the function ψ is decreasing on \([c_{1}, c_{2}]\), then one has
Remark 1.5
We clearly see that inequalities (1.4) and (1.5) reduce to inequality (1.2) if \(\alpha=1\).
The main purpose of the article is to present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals, establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.
2 Main results
In order to prove our main results we need a lemma, which we present in this section.
Lemma 2.1
Let \(\alpha\in(0, 1]\), \(c_{1}, c_{2} \in\mathbb{R}\) with \(0\leq c_{1}< c_{2}\) and \(\psi:[c_{1}, c_{2}]\rightarrow\mathbb {R}\) be an α-fractional differentiable function on \((c_{1}, c_{2})\). Then the identity
holds if \(\mathrm{D}_{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\).
Proof
Let \(\xi=\theta c_{1}+(1-\theta)c_{2}\). Then making use of integration by parts, we get
Therefore, Lemma 2.1 follows easily from (2.1). □
Remark 2.2
We clearly see that the identity given in Lemma 2.1 reduces to the identity given in Theorem 1.1 if \(\alpha=1\).
Theorem 2.3
Let \(\alpha\in(0, 1]\), \(c_{1}, c_{2}\in\mathbb{R}\) with \(0\leq c_{1} < c_{2}\) and \(\psi:[c_{1}, c_{2}] \rightarrow\mathbb{R}\) be an α-differentiable function. Then the inequality
holds if \(\mathrm{D} _{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\) and \(\vert \psi^{\prime } \vert \) is convex on \([c_{1}, c_{2}]\).
Proof
It follows from Lemma 2.1 and the convexities of the functions \(\xi\rightarrow\xi^{\alpha-1}\) and \(\xi\rightarrow-\xi^{\alpha}\) on \((0, \infty)\) together with the convexity of \(\vert \psi^{\prime } \vert \) on \([c_{1}, c_{2}]\) that
□
Remark 2.4
Let \(\alpha=1\). Then inequality (2.2) becomes
Theorem 2.5
Let \(\alpha\in(0, 1]\), \(q>1\), \(c_{1}, c_{2}\in\mathbb{R}\) with \(0\leq c_{1} < c_{2}\) and \(\psi:[c_{1}, c_{2}] \rightarrow \mathbb{R}\) be an α-differentiable function on \((c_{1}, c_{2})\). Then the inequality
is valid if \(\mathrm{D}_{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\) and \(\vert \psi^{\prime } \vert ^{q}\) is convex on \([c_{1}, c_{2}]\), where
Proof
From Lemma 2.1 and the well-known Hölder mean inequality together with the convexity of \(\vert \psi^{\prime} \vert ^{q}\) on the interval \([c_{1}, c_{2}]\) we clearly see that
Therefore, inequality (2.3) follows easily from (2.4)-(2.8). □
Remark 2.6
Let \(\alpha=1\). Then inequality (2.3) becomes
with
Theorem 2.7
Let \(\alpha\in(0, 1]\), \(q>1\), \(c_{1}, c_{2}\in\mathbb{R} \) with \(0\leq c_{1} < c_{2}\) and \(\psi:[c_{1}, c_{2}] \rightarrow \mathbb{R}\) be an α-differentiable function on \((c_{1}, c_{2})\). Then the inequality
holds if \(\mathrm{D}_{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\) and \(\vert \psi^{\prime } \vert ^{q}\) is concave on \([c_{1}, c_{2}]\), where \(\mathrm{A}_{1}(\alpha)\) and \(\mathrm{B}_{1}(\alpha)\) are defined as in Theorem 2.5, and \(\mathrm{C}_{1}(\alpha)\) and \(\mathrm{C}_{2}(\alpha)\) are defined by
Proof
It follows from the concavity of \(\vert \psi' \vert ^{q}\) and the Hölder mean inequality that
which implies that \(\vert \psi' \vert \) is also concave. Making use of Lemma 2.1 and the Jensen integral inequality, we have
Therefore, inequality (2.9) follows easily from (2.10)-(2.12). □
Remark 2.8
Let \(\alpha=1\). Then inequality (2.9) leads to
3 Applications to special means of real numbers
Let \(\alpha\in(0,1]\), \(r\in\mathbb{R}\), \(r\neq0, -\alpha\) and \(a, b>0\) with \(a\neq b\). Then the arithmetic mean \(\mathrm{A}(a, b)\), logarithmic mean \(\mathrm{L}(a, b)\) and \((\alpha, r)\)th generalized logarithmic mean \(\mathrm{L}_{(\alpha, r)}(a,b)\) of a and b are defined by
respectively. Then from Theorems 2.3 and 2.5 together with the convexities of the functions \(\xi\rightarrow\xi^{r}\) and \(\xi\rightarrow1/\xi\) on the interval \((0, \infty)\) we get several new inequalities for the arithmetic, logarithmic and generalized logarithmic means as follows.
Theorem 3.1
Let \(c_{1},c_{2}\in\mathbb{R}\) with \(0< c_{1}< c_{2}\), \(r>1\), \(q>1\) and \(\alpha\in(0, 1]\). Then we have
where \(\mathrm{A}_{1}(\alpha)\), \(\mathrm{A}_{2}(\alpha)\), \(\mathrm{A}_{3}(\alpha)\), \(\mathrm{B}_{1}(\alpha)\), \(\mathrm{B}_{2}(\alpha)\) and \(\mathrm{B}_{3}(\alpha)\) are defined as in Theorem 2.5.
4 Applications to the trapezoidal formula
Let Δ be a division \(c_{1}=\xi_{0}<\xi_{1}<\cdots<\xi_{n-1}<\xi_{n}=c_{2}\) of the interval \([c_{1}, c_{2}]\) and consider the quadrature formula
where
is the trapezoidal version and \(\mathrm{E}_{\alpha}(\psi, \Delta)\) denotes the associated approximation error. In this section, we are going to derive several new error estimations for the trapezoidal formula.
Theorem 4.1
Let \(\alpha\in(0, 1]\), \(c_{1}, c_{2}\in\mathbb{R}\) with \(0\leq c_{1} < c_{2}\), \(\psi:[c_{1}, c_{2}]\rightarrow\mathbb{R}\) be an α-differentiable function on \((c_{1}, c_{2})\) and Δ be a division \(c_{1}=\xi_{0}<\xi_{1}<\cdots<\xi_{n-1}<\xi_{n}=c_{2}\) of the interval \([c_{1}, c_{2}]\). Then the inequality
holds if \(\mathrm{D}_{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1},c_{2}])\) and \(\vert \psi^{\prime } \vert \) is convex on \([c_{1},c_{2}]\).
Proof
Applying Theorem 2.3 on the subinterval \([\xi_{i},\xi _{i+1}]\) \((i= 0, 1,\ldots, n - 1)\) of the division Δ, we have
It follows from (4.1) and the convexity of \(\vert \psi^{\prime}(\xi ) \vert \) on the interval \([c_{1}, c_{2}]\) that
□
Making use of arguments analogous to the proof of Theorem 4.1, we get Theorem 4.2 immediately.
Theorem 4.2
Let \(\alpha\in(0, 1]\), \(q>1\), \(c_{1}, c_{2}\in\mathbb{R}\) with \(0\leq c_{1} < c_{2}\), \(\psi:[c_{1}, c_{2}]\rightarrow\mathbb{R}\) be an α-differentiable function on \((c_{1}, c_{2})\) and Δ be a division \(c_{1}=\xi_{0}<\xi_{1}<\cdots<\xi_{n-1}<\xi_{n}=c_{2}\) of the interval \([c_{1}, c_{2}]\). Then the inequality
holds if \(\mathrm{D}_{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1},c_{2}])\) and \(\vert \psi^{\prime } \vert ^{q}\) is convex on \([c_{1},c_{2}]\), where \(\mathrm{A}_{1}(\alpha)\), \(\mathrm{A}_{2}(\alpha)\), \(\mathrm{A}_{3}(\alpha)\), \(\mathrm{B}_{1}(\alpha)\), \(\mathrm{B}_{2}(\alpha)\) and \(\mathrm{B}_{3}(\alpha)\) are defined as in Theorem 2.5.
5 Conclusion
In this work, we find an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals, present some new inequalities for the arithmetic, logarithmic and generalized logarithmic means of two positive real numbers and provide the error estimations for the trapezoidal formula.
References
Hermite, C: Sur deux limites d’une intégrale définie. Mathesis 3, 82 (1883)
Hadamard, J: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl. 58, 171-215 (1893)
Niculescu, CP, Persson, L-E: Convex Functions and Their Applications. Springer, New York (2006)
Wang, M-K, Li, Y-M, Chu, Y-M: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. (2017). doi:10.1007/s11139-017-9888-3
Wang, M-K, Chu, Y-M: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607-622 (2017)
Adil Khan, M, Khurshid, Y, Ali, T: Hermite-Hadamard inequality for fractional integrals via η-convex functions. Acta Math. Univ. Comen. 86(1), 153-164 (2017)
Adil Khan, M, Khurshid, Y, Ali, T, Rehman, N: Inequalities for three times differentiable functions. Punjab Univ. J. Math. 48(2), 35-48 (2016)
Chu, Y-M, Adil Khan, M, Khan, TU, Ali, T: Generalizations of Hermite-Hadamard type inequalities for MT-convex functions. J. Nonlinear Sci. Appl. 9(6), 4305-4316 (2016)
Wu, Y, Qi, F, Niu, D-W: Integral inequalities of Hermite-Hadamard type for the product of strongly logarithmically convex and other convex functions. Maejo Int. J. Sci. Technol. 9(3), 394-402 (2015)
Noor, MA, Noor, KI, Awan, MU: Hermite-Hadamard inequalities for relative semi-convex functions and applications. Filomat 28(2), 221-230 (2014)
Bai, R-F, Qi, F, Xi, B-Y: Hermite-Hadamard type inequalities for the m- and \((\alpha, m)\)-logarithmically convex functions. Filomat 27(1), 1-7 (2013)
Matłoka, M: On some Hadamard-type inequalities for \((h_{1}, h_{2})\)-preinvex functions on the co-ordinates. J. Inequal. Appl. 2013, Article ID 227 (2013)
Zhang, X-M, Chu, Y-M, Zhang, X-H: The Hermite-Hadamard type inequality of GA-convexity functions. J. Inequal. Appl. 2010, Article ID 507560 (2010)
Chu, Y-M, Wang, G-D, Zhang, X-H: Schur convexity and Hadamard’s inequality. Math. Inequal. Appl. 13(4), 725-731 (2010)
Bombardelli, M, Varošanec, S: Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities. Comput. Math. Appl. 58(9), 1869-1877 (2009)
Sarikaya, MZ, Saglam, A, Yildirim, H: On some Hadamard-type inequalities for h-convex functons. J. Math. Inequal. 2(3), 335-341 (2008)
Kirmaci, US, Klaričić Bakula, M, Özdemir, ME, Pečarić, J: Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193(1), 26-35 (2007)
Noor, MA: On Hadamard integral inequalities involving two log-preinvex functions. JIPAM. J. Inequal. Pure Appl. Math. 8(3), Article ID 75 (2007)
Dragomir, SS, McAndrew, A: Refinments of the Hermite-Hadamard inequality for convex functions. JIPAM. J. Inequal. Pure Appl. Math. 6(5), Article ID 140 (2005)
Dragomir, SS, Fitzpatrick, S: The Hadamard inequalities for s-convex functions in the second sense. Demonstr. Math. 32(4), 687-696 (1999)
Dragomir, SS, Pečarić, JE, Persson, L-E: Some inequalities of Hadamard type. Soochow J. Math. 21(5), 335-341 (1995)
Dragomir, SS: Two mappings in connection to Hadamard’s inequalities. J. Math. Anal. Appl. 167(1), 49-56 (1992)
Dragomir, SS, Agarwal, RP: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 11(5), 91-95 (1998)
Pearce, CEM, Pečarić, JE: Inequalities for differentiable mapping with application to special means and quadrature formulae. Appl. Math. Lett. 13(2), 51-55 (2000)
Iyiola, OS, Nwaeze, ER: Some new results on the new conformable fractional calculus with application using D’Alambert approach. Prog. Fract. Differ. Appl. 2(2), 115-122 (2016)
Abdeljawad, T: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57-66 (2015)
Hammad, MA, Khalil, R: Conformable fractional heat differential equations. Int. J. Pure Appl. Math. 94(2), 215-221 (2014)
Hammad, MA, Khalil, R: Abel’s formula and Wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl. 13(3), 177-183 (2014)
Khalil, R, Al Horani, M, Yousef, A, Sababheh, M: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65-70 (2014)
Cheng, J-F, Chu, Y-M: On the fractional difference equations of order \((2, q)\). Abstr. Appl. Anal. 2011, Article ID 497259 (2011)
Cheng, J-F, Chu, Y-M: Solution to the linear fractional differential equation using Adomian decomposition method. Math. Probl. Eng. 2011, Article ID 587068 (2011)
Cheng, J-F, Chu, Y-M: Fractional difference equations with real variable. Abstr. Appl. Anal. 2012, Article ID 918529 (2012)
Anderson, DR: Taylor’s formula and integral inequalities for conformable fractional derivatives. In: Contributions in Mathematics and Engineering, in Honor of Constantin Carathéodory. Springer, Berlin (2016)
Acknowledgements
The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 61374086, 11371125, 11401191) and the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Chu, YM., Adil Khan, M., Ali, T. et al. Inequalities for α-fractional differentiable functions. J Inequal Appl 2017, 93 (2017). https://doi.org/10.1186/s13660-017-1371-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-017-1371-6
MSC
- 26D15
- 26A51
- 26A33
Keywords
- convex function
- Hermite-Hadamard inequality
- fractional derivative
- fractional integral
- special mean
- trapezoidal formula