Some inequalities via <em>Ψ</em>-Riemann-Liouville fractional integrals

Naila Mehreen; Matloob Anwar; Naila Mehreen; Matloob Anwar

doi:10.3934/math.2019.5.1403

AIMS Mathematics

2019, Volume 4, Issue 5: 1403-1415. doi: 10.3934/math.2019.5.1403

Previous Article Next Article

Research article

Some inequalities via Ψ-Riemann-Liouville fractional integrals

Naila Mehreen ^,,
Matloob Anwar

School of Natural Sciences, National University of Sciences and Technology, H-12 Islamabad, Pakistan

Received: 08 June 2019 Accepted: 02 September 2019 Published: 17 September 2019
MSC : 26A33, 26A51, 26D07, 26D10, 26D15

In this paper, we establish some Hermite-Hadamard type inequalities via $\psi$-Riemann-Liouville fractional integrals for $s$-convex functions in second sense and the functions belongs to the class $P(I)$ $($that is, a class of non-negative functions $\curlyvee:I\rightarrow \mathbb{R}$ which satisfies the condition $\curlyvee(ra_1+(1-r)a_2)\leq \curlyvee(a_1)+\curlyvee(a_2)$, for all $a_1, a_2\in I$ and $r\in[0, 1])$. Some applications to special means are also investigated.
- Hermite-Hadamard inequalities,
- s-convex functions in second sense,
- non-negative functions P(I)
Citation: Naila Mehreen, Matloob Anwar. Some inequalities via Ψ-Riemann-Liouville fractional integrals[J]. AIMS Mathematics, 2019, 4(5): 1403-1415. doi: 10.3934/math.2019.5.1403

Related Papers:

Abstract

In this paper, we establish some Hermite-Hadamard type inequalities via $\psi$-Riemann-Liouville fractional integrals for $s$-convex functions in second sense and the functions belongs to the class $P(I)$ $($that is, a class of non-negative functions $\curlyvee:I\rightarrow \mathbb{R}$ which satisfies the condition $\curlyvee(ra_1+(1-r)a_2)\leq \curlyvee(a_1)+\curlyvee(a_2)$, for all $a_1, a_2\in I$ and $r\in[0, 1])$. Some applications to special means are also investigated.

References

[1]	M. Avci, H. Kavurmaci, M. E. Ozdemir, New inequalities of Hermite-Hadamard type via sconvex functions in the second sense with applications, Appl. Math. Comput., 217 (2011), 5171-5176.
[2]	F. Chen, Extension of the Hermite-Hadamard inequality for harmonically convex functions via fractional integrals, Appl. Math. Comput., 268 (2015), 121-128.
[3]	F. Chen, S. Wu, Several complementary inequalities to inequalities of Hermite-Hadamard type for s-convex functions, J. Nonlinear Sci. Appl., 9 (2016), 705-716. doi: 10.22436/jnsa.009.02.32
[4]	S. S. Dragomir, S. Fitzpatrick, The Hadamard's inequality for s-convex functions in the second sense, Demonstratio Math., 32 (1999), 687-696.
[5]	S. S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21 (1995), 335-341.
[6]	J. Hadamard, Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann, J. Math. Pures Appl., (1893), 171-215.
[7]	Ch. Hermite, Sur deux limites d'une integrale denie, Mathesis., 3 (1883), 82.
[8]	H. Hudzik, L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48 (1994), 100-111. doi: 10.1007/BF01837981
[9]	I. Iscan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 43 (2014), 935-942.
[10]	I. Iscan, Hermite-Hadamard type inequalities for p-convex functions, Int. J. Ana. Appl., 11 (2016), 137-145.
[11]	M. Jleli, D. O'Regan, B. Samet, On Hermite-Hadamard type inequalities via generalized fractional integrals, Turk. J. Math., 40 (2016), 1221-1230. doi: 10.3906/mat-1507-79
[12]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier Science, 2006.
[13]	K. Liu, J. Wang, D. O'Regan, On the Hermite-Hadamard type inequality for Ψ-Riemann-Liouville fractional integrals via convex functions, J. Inequal. Appl., 2019 (2019), 27.
[14]	C. P. Niculescu, L. E. Persson, Convex Functions and Their Applications. A Contemporary Approach, 2 Eds., New York: Springer, 2018.
[15]	N. Mehreen, M. Anwar, Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for p-convex functions via new fractional conformable integral operators, J. Math. Compt. Sci., 19 (2019), 230-240. doi: 10.22436/jmcs.019.04.02
[16]	N. Mehreen, M. Anwar, Hermite-Hadamard type inequalities via exponentially p-convex functions and exponentially s-convex functions in second sense with applications, J. Inequal. Appl., 2019 (2019), 92.
[17]	N. Mehreen, M. Anwar, Integral inequalities for some convex functions via generalized fractional integrals, J. Inequal. Appl., 2018 (2018), 208.
[18]	M. Z. Sarikaya, E. Set, H. Yaldiz, et al., Hermite-Hadamard's inequlities for fractional integrals and related fractional inequalitis, Math. Comput. Modelling, 57 (2013), 2403-2407. doi: 10.1016/j.mcm.2011.12.048
[19]	E. Set, M. Z. Sarikaya, A. Gozpinar, Some Hermite-Hadamard type inequalities for convex functions via conformable fractional integrals and related inequalities, Creat. Math. Inform., Accepted paper, 2017.
[20]	E. Set, M. Z. Sarikaya, M. E. Ozdemir, et al., The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results, J. Appl. Math. Stat. Inform., 10 (2014), 69-83. doi: 10.2478/jamsi-2014-0014
[21]	J. V. C. Sousa, E. C. Oliveira, On the Ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simulat., 60 (2018), 72-91. doi: 10.1016/j.cnsns.2018.01.005

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)