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New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions

Year 2024, Volume: 7 Issue: 1, 30 - 37, 18.03.2024

Henok Desalegn Desta Hüseyin Budak Hasan Kara

https://doi.org/10.32323/ujma.1397051
Cited By: 2

Abstract

This paper delves into an inquiry that centers on the exploration of fractional adaptations of Milne-type inequalities by employing the framework of twice-differentiable convex mappings. Leveraging the fundamental tenets of convexity, H\"{o}lder's inequality, and the power-mean inequality, a series of novel inequalities are deduced. These newly acquired inequalities are fortified through insightful illustrative examples, bolstered by rigorous proofs. Furthermore, to lend visual validation, graphical representations are meticulously crafted for the showcased examples.

Keywords

Convex function Fractional integrals Milne type inequalities Twice differentiable

References

  • [1] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
  • [2] S. S. Dragomir, R. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95.
  • [3] S. S. Dragomir, On trapezoid quadrature formula and applications, Kragujevac J. Math., 23 (2001), 25–36.
  • [4] M. Z. Sarıkaya, N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Model., 54(9-10) (2011), 2175–2182.
  • [5] M. Z. Sarıkaya, H. Budak, Some Hermite-Hadamard type integral inequalities for twice differentiable mappings via fractional integrals, Facta Univ. Ser. Math. Inform., 29(4) (2014), 371–384.
  • [6] M. Z. Sarıkaya, E. Set, H. Yaldiz, N. Basak, Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57(9-10) (2013), 2403–2407
  • [7] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147(5) (2004), 137–146.
  • [8] M. Z. Sarikaya, A. Saglam, H. Yıldırım, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex, Int. J. Open Probl. Comput. Sci. Math., 5(3) (2012), 1–14.
  • [9] M. Iqbal, M. I. Bhatti, K. Nazeer, Generalization of inequalities analogous to Hermite–Hadamard inequality via fractional integrals, Bull. Korean Math. Soc., 52(3) (2015), 707–716.
  • [10] M.Z. Sarikaya, H. Yildirim, On Hermite–Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17(2)(2016), 1049–1059.
  • [11] S. Hussain, S. Qaisar, More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings, SpringerPlus, 5(1) (2016), 1–9.
  • [12] J. Nasir, S. Qaisar, S. I. Butt, A. Qayyum, Some Ostrowski type inequalities for mappings whose second derivatives are preinvex function via fractional integral operator, AIMS Mathematics, 7(3) (2022), 3303–3320.
  • [13] M. Z. Sarikaya, E. Set, M. E. Özdemir, On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex, J. Appl. Math. Stat. Inform., 9(1) (2013), 37–45.
  • [14] H. Budak, P. K¨osem, H. Kara, On new Milne-type inequalities for fractional integrals, J. Inequal. Appl., (2023), Art. 10.
  • [15] A. D. Booth, Numerical methods, 3rd Ed., Butterworths, California, 1966.
  • [16] H. Budak, A. A. Hyder, Enhanced bounds for Riemann-Liouville fractional integrals: Novel variations of Milne inequalities, AIMS Mathematics, 8(12) (2023), 30760–30776.
  • [17] P. Bosch, J. M. Rodr´ıguez, J. M Sigarreta, On new Milne-type inequalities and applications, J. Inequal. Appl., (2023), Art. 3.
  • [18] B. Meftah, A. Lakhdari, W. Saleh, A. Kilic¸man, Some new fractal Milne-type integral inequalities via generalized convexity with applications, Fractal Fract., 7(2) (2023), Art. 166.
  • [19] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B. V., Amsterdam, 2006.
  • [20] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Wien: Springer-Verlag, 1997, 223–276.
  • [21] S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.

Year 2024, Volume: 7 Issue: 1, 30 - 37, 18.03.2024

Henok Desalegn Desta Hüseyin Budak Hasan Kara

https://doi.org/10.32323/ujma.1397051
Cited By: 2

Abstract

References

  • [1] S. S. Dragomir, C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000.
  • [2] S. S. Dragomir, R. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95.
  • [3] S. S. Dragomir, On trapezoid quadrature formula and applications, Kragujevac J. Math., 23 (2001), 25–36.
  • [4] M. Z. Sarıkaya, N. Aktan, On the generalization of some integral inequalities and their applications, Math. Comput. Model., 54(9-10) (2011), 2175–2182.
  • [5] M. Z. Sarıkaya, H. Budak, Some Hermite-Hadamard type integral inequalities for twice differentiable mappings via fractional integrals, Facta Univ. Ser. Math. Inform., 29(4) (2014), 371–384.
  • [6] M. Z. Sarıkaya, E. Set, H. Yaldiz, N. Basak, Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57(9-10) (2013), 2403–2407
  • [7] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula, Appl. Math. Comput., 147(5) (2004), 137–146.
  • [8] M. Z. Sarikaya, A. Saglam, H. Yıldırım, New inequalities of Hermite-Hadamard type for functions whose second derivatives absolute values are convex and quasi-convex, Int. J. Open Probl. Comput. Sci. Math., 5(3) (2012), 1–14.
  • [9] M. Iqbal, M. I. Bhatti, K. Nazeer, Generalization of inequalities analogous to Hermite–Hadamard inequality via fractional integrals, Bull. Korean Math. Soc., 52(3) (2015), 707–716.
  • [10] M.Z. Sarikaya, H. Yildirim, On Hermite–Hadamard type inequalities for Riemann-Liouville fractional integrals, Miskolc Math. Notes, 17(2)(2016), 1049–1059.
  • [11] S. Hussain, S. Qaisar, More results on Simpson’s type inequality through convexity for twice differentiable continuous mappings, SpringerPlus, 5(1) (2016), 1–9.
  • [12] J. Nasir, S. Qaisar, S. I. Butt, A. Qayyum, Some Ostrowski type inequalities for mappings whose second derivatives are preinvex function via fractional integral operator, AIMS Mathematics, 7(3) (2022), 3303–3320.
  • [13] M. Z. Sarikaya, E. Set, M. E. Özdemir, On new inequalities of Simpson’s type for functions whose second derivatives absolute values are convex, J. Appl. Math. Stat. Inform., 9(1) (2013), 37–45.
  • [14] H. Budak, P. K¨osem, H. Kara, On new Milne-type inequalities for fractional integrals, J. Inequal. Appl., (2023), Art. 10.
  • [15] A. D. Booth, Numerical methods, 3rd Ed., Butterworths, California, 1966.
  • [16] H. Budak, A. A. Hyder, Enhanced bounds for Riemann-Liouville fractional integrals: Novel variations of Milne inequalities, AIMS Mathematics, 8(12) (2023), 30760–30776.
  • [17] P. Bosch, J. M. Rodr´ıguez, J. M Sigarreta, On new Milne-type inequalities and applications, J. Inequal. Appl., (2023), Art. 3.
  • [18] B. Meftah, A. Lakhdari, W. Saleh, A. Kilic¸man, Some new fractal Milne-type integral inequalities via generalized convexity with applications, Fractal Fract., 7(2) (2023), Art. 166.
  • [19] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B. V., Amsterdam, 2006.
  • [20] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Wien: Springer-Verlag, 1997, 223–276.
  • [21] S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Optimisation
Journal Section Articles
Authors

Henok Desalegn Desta 0000-0003-0395-4857

Hüseyin Budak 0000-0001-8843-955X

Hasan Kara 0000-0002-2075-944X

Early Pub Date January 16, 2024
Publication Date March 18, 2024
Submission Date November 28, 2023
Acceptance Date January 16, 2024
Published in Issue Year 2024 Volume: 7 Issue: 1

Cite

APA Desta, H. D., Budak, H., & Kara, H. (2024). New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions. Universal Journal of Mathematics and Applications, 7(1), 30-37. https://doi.org/10.32323/ujma.1397051
AMA Desta HD, Budak H, Kara H. New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions. Univ. J. Math. Appl. March 2024;7(1):30-37. doi:10.32323/ujma.1397051
Chicago Desta, Henok Desalegn, Hüseyin Budak, and Hasan Kara. “New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions”. Universal Journal of Mathematics and Applications 7, no. 1 (March 2024): 30-37. https://doi.org/10.32323/ujma.1397051.
EndNote Desta HD, Budak H, Kara H (March 1, 2024) New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions. Universal Journal of Mathematics and Applications 7 1 30–37.
IEEE H. D. Desta, H. Budak, and H. Kara, “New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions”, Univ. J. Math. Appl., vol. 7, no. 1, pp. 30–37, 2024, doi: 10.32323/ujma.1397051.
ISNAD Desta, Henok Desalegn et al. “New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions”. Universal Journal of Mathematics and Applications 7/1 (March 2024), 30-37. https://doi.org/10.32323/ujma.1397051.
JAMA Desta HD, Budak H, Kara H. New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions. Univ. J. Math. Appl. 2024;7:30–37.
MLA Desta, Henok Desalegn et al. “New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions”. Universal Journal of Mathematics and Applications, vol. 7, no. 1, 2024, pp. 30-37, doi:10.32323/ujma.1397051.
Vancouver Desta HD, Budak H, Kara H. New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions. Univ. J. Math. Appl. 2024;7(1):30-7.

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