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

Yıl 2024, Cilt: 7 Sayı: 1, 30 - 37, 18.03.2024

Henok Desalegn Desta Hüseyin Budak Hasan Kara

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

Öz

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.

Anahtar Kelimeler

Convex function Fractional integrals Milne type inequalities Twice differentiable

Kaynakça

  • [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.

Yıl 2024, Cilt: 7 Sayı: 1, 30 - 37, 18.03.2024

Henok Desalegn Desta Hüseyin Budak Hasan Kara

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

Öz

Kaynakça

  • [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.
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematikte Optimizasyon
Bölüm Makaleler
Yazarlar

Henok Desalegn Desta 0000-0003-0395-4857

Hüseyin Budak 0000-0001-8843-955X

Hasan Kara 0000-0002-2075-944X

Erken Görünüm Tarihi 16 Ocak 2024
Yayımlanma Tarihi 18 Mart 2024
Gönderilme Tarihi 28 Kasım 2023
Kabul Tarihi 16 Ocak 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 7 Sayı: 1

Kaynak Göster

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. Mart 2024;7(1):30-37. doi:10.32323/ujma.1397051
Chicago Desta, Henok Desalegn, Hüseyin Budak, ve Hasan Kara. “New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions”. Universal Journal of Mathematics and Applications 7, sy. 1 (Mart 2024): 30-37. https://doi.org/10.32323/ujma.1397051.
EndNote Desta HD, Budak H, Kara H (01 Mart 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, ve H. Kara, “New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions”, Univ. J. Math. Appl., c. 7, sy. 1, ss. 30–37, 2024, doi: 10.32323/ujma.1397051.
ISNAD Desta, Henok Desalegn vd. “New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions”. Universal Journal of Mathematics and Applications 7/1 (Mart 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 vd. “New Perspectives on Fractional Milne-Type Inequalities: Insights from Twice-Differentiable Functions”. Universal Journal of Mathematics and Applications, c. 7, sy. 1, 2024, ss. 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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