Research Article
BibTex RIS Cite

HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS

Year 2016, Volume: 4 Issue: 1, 108 - 113, 01.04.2016
https://izlik.org/JA93LG67YE

Abstract

Fractional calculus is as important as calculus. This paper is due to presentation of Hadamard and Fejer-Hadamard inequalities for fractional calculus. We prove Hadamard and Fejer-Hadamard inequalities for general- ized fractional integral involving Mittag-Lefter function. Also, inequalities for special cases are obtained.

References

  • [1] R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Di erential and Di erence Equations, Kluwer Academic Publishers, Dordrecht, Boston, London 1995.
  • [2] G. A. Anastassiou, Advanced inequalities, 11, World Scienti c, 2011.
  • [3] A. G. Azpeitia, Convex functions and the Hadamard inequality, Revista Colombina Mat. 28 (1994)7-12.
  • [4] M. K. Bakula, J. Pecaric, Note on some Hadamard type inequalities, J. ineq. Pure Appl. Math. 5 (3)(2004) Art. 74.
  • [5] L. Curiel, L. Galue, A generalization of the integral operators involving the Gauss hypergeo- metric function, Revista Tecnica de la Facultad de Ingenieria Universidad del Zulia, 19 (1) (1996), 17{22.
  • [6] S. S. Dragomir, R. P. Agarwal, Two inequalities for di erentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett. 11 (5)(1998), 91-95.
  • [7] S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. Math. Sic. Marh. Roum., 47 (2004), 3-14.
  • [8] G. Farid, J. Pecaric and Z. Tomovski, Opial-type inequalities for fractional integral operator involving Mittag-Leer function, Fractional Di er. Calc., Vol. 5 , No. 1 (2015), 93-106.
  • [9] L. Fejer, Uberdie Fourierreihen, II, Math. Naturwise. Anz Ungar. Akad., Wiss, 24 (1906), 369-390, (in Hungarian).
  • [10] A. A. Kilbas, M. Saigo, R.K. Saxena, Generalized MittagLeer function and generalized fractional calculus operators, Integral Transform. Spec. Funct. 15 (2004) 3149.
  • [11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of fractional derivatial Equations, North-Holland Mathematics Studies, 204, Elsevier, New York-London, 2006.
  • [12] K. Miller and B. Ross, An introduction to the fractional calculus and fractional di erential Equations, John Wiley and Sons Inc., New York, 1993.
  • [13] K. Oldham and J. Spanier The fractional calculus, Academic Press, New York - London, 1974.
  • [14] M. E. Ozdemir, M Avci, E. Set, On some inequalities of Harmite-Hadamard type via m- convexity, Appl. Math. Lett. 23 (9)(2010), 1065-1070.
  • [15] E. Set, M. E. Ozdemir, S. S. Dragomir, On the Harmite-Hadamard inequality and other integral inequalities involving two functions, J. Inequal. Appl. (2010) 9. Article ID 148102.
  • [16] E. Set, M. E. Ozdemir, S. S. Dragomir, On Hadamard-type inequalities involving several kinds of convexity, J. Inequal. Appl. (2010) 12. Article ID 286845.
  • [17] T. R. Prabhakar, A singular integral equation with a generalized Mittag{Leer function in the kernel, Yokohama Math. J. 19 (1971) 715.
  • [18] I. Scan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional inte- grals, arXiv preprint arXiv: 1404. 7722 (2014).
  • [19] T. O. Salim, and A. W. Faraj, A Generalization of Mittag{Leer function and integral operator associated with fractional calculus, J. Fract. Calc. Appl. Vol. 3 July 2012, No. 5, pp. 1 - 13.
  • [20] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math., Comp., Modelling, 57 (2013), 2403-2407.
  • [21] H. M. Srivastava, and Z. Tomovski, Fractional calculus with an integral operator con- taining generalized Mittag{Leer function in the kernal, Appl. Math. Comput. (2009), doi:10.1016/j.amc.2009.01.055
  • [22] D. V. Widder, The Laplace transform, Princeton Uni. Press, New Jersey, 1941.

Year 2016, Volume: 4 Issue: 1, 108 - 113, 01.04.2016
https://izlik.org/JA93LG67YE

Abstract

References

  • [1] R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Di erential and Di erence Equations, Kluwer Academic Publishers, Dordrecht, Boston, London 1995.
  • [2] G. A. Anastassiou, Advanced inequalities, 11, World Scienti c, 2011.
  • [3] A. G. Azpeitia, Convex functions and the Hadamard inequality, Revista Colombina Mat. 28 (1994)7-12.
  • [4] M. K. Bakula, J. Pecaric, Note on some Hadamard type inequalities, J. ineq. Pure Appl. Math. 5 (3)(2004) Art. 74.
  • [5] L. Curiel, L. Galue, A generalization of the integral operators involving the Gauss hypergeo- metric function, Revista Tecnica de la Facultad de Ingenieria Universidad del Zulia, 19 (1) (1996), 17{22.
  • [6] S. S. Dragomir, R. P. Agarwal, Two inequalities for di erentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett. 11 (5)(1998), 91-95.
  • [7] S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. Math. Sic. Marh. Roum., 47 (2004), 3-14.
  • [8] G. Farid, J. Pecaric and Z. Tomovski, Opial-type inequalities for fractional integral operator involving Mittag-Leer function, Fractional Di er. Calc., Vol. 5 , No. 1 (2015), 93-106.
  • [9] L. Fejer, Uberdie Fourierreihen, II, Math. Naturwise. Anz Ungar. Akad., Wiss, 24 (1906), 369-390, (in Hungarian).
  • [10] A. A. Kilbas, M. Saigo, R.K. Saxena, Generalized MittagLeer function and generalized fractional calculus operators, Integral Transform. Spec. Funct. 15 (2004) 3149.
  • [11] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of fractional derivatial Equations, North-Holland Mathematics Studies, 204, Elsevier, New York-London, 2006.
  • [12] K. Miller and B. Ross, An introduction to the fractional calculus and fractional di erential Equations, John Wiley and Sons Inc., New York, 1993.
  • [13] K. Oldham and J. Spanier The fractional calculus, Academic Press, New York - London, 1974.
  • [14] M. E. Ozdemir, M Avci, E. Set, On some inequalities of Harmite-Hadamard type via m- convexity, Appl. Math. Lett. 23 (9)(2010), 1065-1070.
  • [15] E. Set, M. E. Ozdemir, S. S. Dragomir, On the Harmite-Hadamard inequality and other integral inequalities involving two functions, J. Inequal. Appl. (2010) 9. Article ID 148102.
  • [16] E. Set, M. E. Ozdemir, S. S. Dragomir, On Hadamard-type inequalities involving several kinds of convexity, J. Inequal. Appl. (2010) 12. Article ID 286845.
  • [17] T. R. Prabhakar, A singular integral equation with a generalized Mittag{Leer function in the kernel, Yokohama Math. J. 19 (1971) 715.
  • [18] I. Scan, Hermite-Hadamard-Fejer type inequalities for convex functions via fractional inte- grals, arXiv preprint arXiv: 1404. 7722 (2014).
  • [19] T. O. Salim, and A. W. Faraj, A Generalization of Mittag{Leer function and integral operator associated with fractional calculus, J. Fract. Calc. Appl. Vol. 3 July 2012, No. 5, pp. 1 - 13.
  • [20] M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math., Comp., Modelling, 57 (2013), 2403-2407.
  • [21] H. M. Srivastava, and Z. Tomovski, Fractional calculus with an integral operator con- taining generalized Mittag{Leer function in the kernal, Appl. Math. Comput. (2009), doi:10.1016/j.amc.2009.01.055
  • [22] D. V. Widder, The Laplace transform, Princeton Uni. Press, New Jersey, 1941.
There are 22 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Article
Authors

G. Farıd

Submission Date July 10, 2014
Publication Date April 1, 2016
IZ https://izlik.org/JA93LG67YE
Published in Issue Year 2016 Volume: 4 Issue: 1

Cite

APA Farıd, G. (2016). HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS. Konuralp Journal of Mathematics, 4(1), 108-113. https://izlik.org/JA93LG67YE
AMA 1.Farıd G. HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS. Konuralp J. Math. 2016;4(1):108-113. https://izlik.org/JA93LG67YE
Chicago Farıd, G. 2016. “HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS”. Konuralp Journal of Mathematics 4 (1): 108-13. https://izlik.org/JA93LG67YE.
EndNote Farıd G (April 1, 2016) HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS. Konuralp Journal of Mathematics 4 1 108–113.
IEEE [1]G. Farıd, “HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS”, Konuralp J. Math., vol. 4, no. 1, pp. 108–113, Apr. 2016, [Online]. Available: https://izlik.org/JA93LG67YE
ISNAD Farıd, G. “HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS”. Konuralp Journal of Mathematics 4/1 (April 1, 2016): 108-113. https://izlik.org/JA93LG67YE.
JAMA 1.Farıd G. HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS. Konuralp J. Math. 2016;4:108–113.
MLA Farıd, G. “HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS”. Konuralp Journal of Mathematics, vol. 4, no. 1, Apr. 2016, pp. 108-13, https://izlik.org/JA93LG67YE.
Vancouver 1.G. Farıd. HADAMARD AND FEJER-HADAMARD INEQUALITIES FOR GENERALIZED FRACTIONAL INTEGRALS INVOLVING SPECIAL FUNCTIONS. Konuralp J. Math. [Internet]. 2016 Apr. 1;4(1):108-13. Available from: https://izlik.org/JA93LG67YE
Creative Commons License
The published articles in KJM are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.