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On the Generalized Hermite-Hadamard Inequalities Involving Beta Function

Year 2021, Volume: 9 Issue: 1, 112 - 118, 28.04.2021

Abstract

In this paper, we establish new generalized fractional integral inequalities of Hermite-Hadamard type which cover the previously published result such as Riemann integral, Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral.

References

  • [1] A. Akkurt, Z. Kacar, H. Yildirim, Generalized Fractional Integral Inequalities for Continuous Random Variables, Journal of Probability and Statistics 2015(2015), Article ID 958980.
  • [2] A. Akkurt, M. E. Yildirim, H. Yildirim, On some integral inequalities for (k,h)-Riemann-Liouville fractional inte- gral, New Trends in Mathematical Sciences (NTMSCI) 4 (1), 138-146, 2016.
  • [3] S. S. Dragomir and 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.
  • [4] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional di¤ erential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
  • [5] U. S. Kirmaci, Inequalities for di¤ erentiable mappings and applications to special means of real numbers and to midpoint formula, Applied Mathematics and Computation 147 (2004) 137–-46
  • [6] R. Goren‡o and F. Mainardi, Fractional calculus: integral and di¤ erential equations of fractional order, Springer Verlag, Wien (1997), 223-276.
  • [7] J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction considree par, Rie- mann, J. Math. Pures. et Appl. 58 (1893), 171-215.
  • [8] S. Miller and B. Ross, An introduction to the fractional calculus and fractional di¤ erential equations, John Wiley & Sons, USA, 1993, p.2.
  • [9] R. K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2) (2005), 191-203.
  • [10] M. Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite-Hadamard 0s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57 (2013) 2403-2407.
  • [11] M. Z. Sarikaya and H. Yaldiz, On generalization integral inequalities for fractional integrals, Nihonkai Math. J., Vol.25(2014), 93-104.
  • [12] M. Z. Sarikaya, H. Yaldiz and N. Basak, New fractional inequalities of Ostrowski-Grüss type, Le Matematiche, Vol. LXIX (2014)-Fasc. I, pp. 227-235.
  • [13] M. Z. Sarikaya and F. Ertu¼gral, On the generalized Hermite-Hadamard inequalities, Accepted in Annals of the University of Craiova - Mathematics and Computer Science Series), 2019.
  • [14] M. E. Ozdemir, S.S. Dragomir and C. Yildiz, The Hadamard’s inequality for convex function via fractional integrals, Acta Mathematica Scientia 2013,33B(5):1293-1299.
  • [15] T. Ali, M. A. Khan and Y. Khurshidi, Hermite-Hadamard inequality for fractional integrals via eta-convex functions, Acta Mathematica Universitatis Comenianae, 86(1), (2017), 153-164.
  • [16] M. Kunt and ·I. ·I¸scan, Hermite-Hadamard-Fejér type inequalities for p-convex functions. Arab J. Math. Sci. 23 (2017), no. 2, 215-230.
  • [17] M. Kunt and ·I. ·I¸scan, Hermite-Hadamard type inequalities for harmonically ( , m)-convex functions by using fractional integrals, Konuralp J. Math. 5 (2017), no. 1, 201-213.
  • [18] M. Kirane, B. T. Torebek, Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte Type Inequalities for Convex Functions via Fractional Integrals, arXiv:1701.00092.
  • [19] M. Iqbal, M. I. Bhatti, and K. Nazeer, Generalization of Inequalities Analogous to Hermite–Hadamard Inequality via Fractional Integrals, Bull. Korean Math. Soc, 52 (2015), No. 3, 707-716.
  • [20] S. Mubeen and G. M Habibullah, k-Fractional integrals and application, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 2, 89 - 94.
  • [21] G. Farid, A. Rehman and M. Zahra, On Hadamard inequalities for k-fractional integrals, Nonlinear Functional Analysis and Applications Vol. 21, No. 3 (2016), pp. 463-478.
  • [22] R. Hussain, A. Ali, G. Gulshan, A. Latif and M. Muddassar, Generalized co-ordinated integral inequalities for convex functions by way of k -fractional derivatives, Miskolc Mathematical Notesa Publications of the university of Miskolc. (Submitted)
  • [23] R. Hussain, A. Ali, A. Latif, G. Gulshan, Some k–fractional associates of Hermite–Hadamard’s inequality for quasi–convex functions and applications to special means, Fractional Di¤erential Calculus, Volume 7, Number 2 (2017), 301-309.
Year 2021, Volume: 9 Issue: 1, 112 - 118, 28.04.2021

Abstract

References

  • [1] A. Akkurt, Z. Kacar, H. Yildirim, Generalized Fractional Integral Inequalities for Continuous Random Variables, Journal of Probability and Statistics 2015(2015), Article ID 958980.
  • [2] A. Akkurt, M. E. Yildirim, H. Yildirim, On some integral inequalities for (k,h)-Riemann-Liouville fractional inte- gral, New Trends in Mathematical Sciences (NTMSCI) 4 (1), 138-146, 2016.
  • [3] S. S. Dragomir and 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.
  • [4] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional di¤ erential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
  • [5] U. S. Kirmaci, Inequalities for di¤ erentiable mappings and applications to special means of real numbers and to midpoint formula, Applied Mathematics and Computation 147 (2004) 137–-46
  • [6] R. Goren‡o and F. Mainardi, Fractional calculus: integral and di¤ erential equations of fractional order, Springer Verlag, Wien (1997), 223-276.
  • [7] J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction considree par, Rie- mann, J. Math. Pures. et Appl. 58 (1893), 171-215.
  • [8] S. Miller and B. Ross, An introduction to the fractional calculus and fractional di¤ erential equations, John Wiley & Sons, USA, 1993, p.2.
  • [9] R. K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2) (2005), 191-203.
  • [10] M. Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite-Hadamard 0s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57 (2013) 2403-2407.
  • [11] M. Z. Sarikaya and H. Yaldiz, On generalization integral inequalities for fractional integrals, Nihonkai Math. J., Vol.25(2014), 93-104.
  • [12] M. Z. Sarikaya, H. Yaldiz and N. Basak, New fractional inequalities of Ostrowski-Grüss type, Le Matematiche, Vol. LXIX (2014)-Fasc. I, pp. 227-235.
  • [13] M. Z. Sarikaya and F. Ertu¼gral, On the generalized Hermite-Hadamard inequalities, Accepted in Annals of the University of Craiova - Mathematics and Computer Science Series), 2019.
  • [14] M. E. Ozdemir, S.S. Dragomir and C. Yildiz, The Hadamard’s inequality for convex function via fractional integrals, Acta Mathematica Scientia 2013,33B(5):1293-1299.
  • [15] T. Ali, M. A. Khan and Y. Khurshidi, Hermite-Hadamard inequality for fractional integrals via eta-convex functions, Acta Mathematica Universitatis Comenianae, 86(1), (2017), 153-164.
  • [16] M. Kunt and ·I. ·I¸scan, Hermite-Hadamard-Fejér type inequalities for p-convex functions. Arab J. Math. Sci. 23 (2017), no. 2, 215-230.
  • [17] M. Kunt and ·I. ·I¸scan, Hermite-Hadamard type inequalities for harmonically ( , m)-convex functions by using fractional integrals, Konuralp J. Math. 5 (2017), no. 1, 201-213.
  • [18] M. Kirane, B. T. Torebek, Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte Type Inequalities for Convex Functions via Fractional Integrals, arXiv:1701.00092.
  • [19] M. Iqbal, M. I. Bhatti, and K. Nazeer, Generalization of Inequalities Analogous to Hermite–Hadamard Inequality via Fractional Integrals, Bull. Korean Math. Soc, 52 (2015), No. 3, 707-716.
  • [20] S. Mubeen and G. M Habibullah, k-Fractional integrals and application, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 2, 89 - 94.
  • [21] G. Farid, A. Rehman and M. Zahra, On Hadamard inequalities for k-fractional integrals, Nonlinear Functional Analysis and Applications Vol. 21, No. 3 (2016), pp. 463-478.
  • [22] R. Hussain, A. Ali, G. Gulshan, A. Latif and M. Muddassar, Generalized co-ordinated integral inequalities for convex functions by way of k -fractional derivatives, Miskolc Mathematical Notesa Publications of the university of Miskolc. (Submitted)
  • [23] R. Hussain, A. Ali, A. Latif, G. Gulshan, Some k–fractional associates of Hermite–Hadamard’s inequality for quasi–convex functions and applications to special means, Fractional Di¤erential Calculus, Volume 7, Number 2 (2017), 301-309.
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mehmet Zeki Sarıkaya

Fatih Ata This is me

Publication Date April 28, 2021
Submission Date March 18, 2021
Acceptance Date April 9, 2021
Published in Issue Year 2021 Volume: 9 Issue: 1

Cite

APA Sarıkaya, M. Z., & Ata, F. (2021). On the Generalized Hermite-Hadamard Inequalities Involving Beta Function. Konuralp Journal of Mathematics, 9(1), 112-118.
AMA Sarıkaya MZ, Ata F. On the Generalized Hermite-Hadamard Inequalities Involving Beta Function. Konuralp J. Math. April 2021;9(1):112-118.
Chicago Sarıkaya, Mehmet Zeki, and Fatih Ata. “On the Generalized Hermite-Hadamard Inequalities Involving Beta Function”. Konuralp Journal of Mathematics 9, no. 1 (April 2021): 112-18.
EndNote Sarıkaya MZ, Ata F (April 1, 2021) On the Generalized Hermite-Hadamard Inequalities Involving Beta Function. Konuralp Journal of Mathematics 9 1 112–118.
IEEE M. Z. Sarıkaya and F. Ata, “On the Generalized Hermite-Hadamard Inequalities Involving Beta Function”, Konuralp J. Math., vol. 9, no. 1, pp. 112–118, 2021.
ISNAD Sarıkaya, Mehmet Zeki - Ata, Fatih. “On the Generalized Hermite-Hadamard Inequalities Involving Beta Function”. Konuralp Journal of Mathematics 9/1 (April 2021), 112-118.
JAMA Sarıkaya MZ, Ata F. On the Generalized Hermite-Hadamard Inequalities Involving Beta Function. Konuralp J. Math. 2021;9:112–118.
MLA Sarıkaya, Mehmet Zeki and Fatih Ata. “On the Generalized Hermite-Hadamard Inequalities Involving Beta Function”. Konuralp Journal of Mathematics, vol. 9, no. 1, 2021, pp. 112-8.
Vancouver Sarıkaya MZ, Ata F. On the Generalized Hermite-Hadamard Inequalities Involving Beta Function. Konuralp J. Math. 2021;9(1):112-8.
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