New results on Bullen-type inequalities for coordinated convex functions obtained by using conformable fractional integrals

  • Fatih Hezenci Department of Mathematics, Faculty of Science and Arts, Duzce University, Turkey
  • Hasan Kara Department of Mathematics, Faculty of Science and Arts, Duzce University, Turkey
  • Hüseyin Budak Department of Mathematics, Faculty of Science and Arts, Duzce University, Turkey
Keywords: Bullen type inequality, Fractional integral operators, conformable fractional integrals, Co-ordinated convex function

Abstract

UDC 517.9

Our aim is to investigate novel Bullen-type inequalities for coordinated convex mappings by employing conformable fractional integrals. Initially, an identity incorporating the conformable fractional integrals was established to serve for this purpose.  By using this identity, new inequalities аre derived expanding the scope of previously established results obtained with the help of Riemann–Liouville integrals by making specific  choices of variable and applying the Hölder inequality and the power-mean inequality.

References

A. A. Abdelhakim, The flaw in the conformable calculus: it is conformable because it is not fractional, Fract. Calc. and Appl. Anal., 22, 242–254 (2019). DOI: https://doi.org/10.1515/fca-2019-0016

T. Abdeljawad, On conformable fractional calculus, J. Comput. and Appl. Math., 279, 57–66 (2015). DOI: https://doi.org/10.1016/j.cam.2014.10.016

S. Aslan, A. O. Akdemir, M. A. Dokuyucu, Exponentially $ m $- and $ (α, m)$-convex functions on the coordinates and related inequalities, Turk. J. Sci., 7, № 3, 231–244 (2022).

H. Budak, F. Hezenci, H. Kara, On generalized Ostrowski, Simpson and Trapezoidal type inequalities for coordinated convex functions via generalized fractional integrals, Adv. Difference Equat., 2021, № 1, 1–32 (2021). DOI: https://doi.org/10.1186/s13662-021-03463-0

P. S. Bullen, Error estimates for some elementary quadrature rules, Publ. Elektrotehn. fakulteta. Ser. Mat. i Fiz., 602/633, 97–103 (1978).

H. Budak, S. K. Yildirim, H. Kara, H. Yildirim, On new generalized inequalities with some parameters for coordinated convex functions via generalized fractional integrals, Math. Methods Appl. Sci., 44, № 17, 13069–13098 (2021). DOI: https://doi.org/10.1002/mma.7610

M. Bozkurt, A. Akkurt, H. Yildirim, Conformable derivatives and integrals for the functions of two variables, Konuralp J. Math., 9, № 1, 49–59 (2021).

M. Çakmak, The differentiable $h$-convex functions involving the Bullen inequality, Acta Univ. Apulensis Math. Inform., 65, 29 (2021).

M. Çakmak, Refinements of Bullen-type inequalities for $s$-convex functions via Riemann–Liouville fractional integrals involving Gauss hypergeometric function, J. Interdiscip. Math., 22, № 6, 975–989 (2019). DOI: https://doi.org/10.1080/09720502.2019.1698803

M. Çakmak, On some Bullen-type inequalities

via conformable fractional integrals, J. Sci. Perspectives, 3, № 4, 285–298 (2019). DOI: https://doi.org/10.26900/jsp.3.030

M. Çakmak, Some Bullen-type inequalities for

conformable fractional integrals, General Math., 28, № 2, 3–17 (2020). DOI: https://doi.org/10.2478/gm-2020-0011

S. S. Dragomir, On Hadamard's inequality for convex

functions on the coordinates in a rectangle from the plane, Taiwanese J. Math., 4, 775–788 (2001).

T. S. Du, Y. Peng, Hermite–Hadamard type inequalities for multiplicative Riemann–Liouville fractional integrals, J. Comput. and Appl. Math., 440, Article 115582 (2024). DOI: https://doi.org/10.1016/j.cam.2023.115582

T. S. Du, X. M. Yuan, On the parameterized fractal integral inequalities and related applications, Chaos, Solitons, Fractals, 170, Article 113375 (2023). DOI: https://doi.org/10.1016/j.chaos.2023.113375

S. S. Dragomir, S. O. N. G. Wang, A generalization of Bullen's inequality for convex mappings and its applications, Soochow J. Math., 24, № 2, 97–103 (1998).

S. S. Dragomir, On Hadamard's inequality for convex functions on the coordinates in a rectangle from the plane, Taiwanese J. Math., 4, 775–788 (2001). DOI: https://doi.org/10.11650/twjm/1500574995

S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite–Hadamard inequalities and applications, RGMIA Monographs, Victoria Univ. (2000).

T. Du, C. Luo, Z. Cao, On the Bullen-type inequalities via generalized fractional integrals and their applications, Fractals, 29, № 7, Article 2150188 (2021). DOI: https://doi.org/10.1142/S0218348X21501887

S. Erden, M. Z. Sarikaya, Generalized Bullen-type inequalities for local fractional integrals and its applications, Palest. J. Math., 9, № 2, 945–956 (2020).

F. Hezenci, A note on fractional midpoint type inequalities for coordinated $(s_{1},s_{2})$-convex functions, Cumhuriyet Sci. J., 43, № 3, 477–491 (2022). DOI: https://doi.org/10.17776/csj.1088703

F. Hezenci, H. Budak, Novel results on trapezoid-type inequalities for conformable fractional integrals, Turk. J. Math., 47, № 2, 425–438 (2023). DOI: https://doi.org/10.55730/1300-0098.3371

D. Y. Hwang, K. L. Tseng, G. S. Yang, Some Hadamard's inequality for coordinated convex functions in a rectangle from the plane, Taiwanese J. Math., 11, 63–731 (2007). DOI: https://doi.org/10.11650/twjm/1500404635

H. R. Hwang, K. L. Tseng, K. C. Hsu, New inequalities for fractional integrals and their applications, Turk. J. Math., 40, № 3, 471–486 (2016). DOI: https://doi.org/10.3906/mat-1411-61

A. Hyder, A. H. Soliman, A new generalized $θ $-conformable calculus and its applications in mathematical physics, Physica Scripta, 96, Article 015208 (2020). DOI: https://doi.org/10.1088/1402-4896/abc6d9

I. İşcan, T. Toplu, F. Yetgin, Some new inequalities on generalization of Hermite–Hadamard and Bullen type inequalities, applications to trapezoidal and midpoint formula, Kragujevac J. Math., 45, № 4, 647–657 (2021). DOI: https://doi.org/10.46793/KgJMat2104.647I

F. Jarad, E. Uǧurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Difference Equat., 2017, 247 (2017). DOI: https://doi.org/10.1186/s13662-017-1306-z

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. and Comput., 218, № 3, 860–865 (2011). DOI: https://doi.org/10.1016/j.amc.2011.03.062

H. Kara, H. Budak, F. Hezenci, New extensions of the parameterized inequalities based on Riemann–Liouville fractional integrals, Mathematics, 10, № 18, 3374 (2022). DOI: https://doi.org/10.3390/math10183374

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud., 204, Elsevier Sci. B.V., Amsterdam (2006).

S. Khan, H. Budak, On fractional Simpson type integral inequalities for coordinated convex functions, J. Inequal. and Appl., 2022, 94 (2022). DOI: https://doi.org/10.1186/s13660-022-02830-z

R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. and Appl. Math., 264, 65–70 (2014). DOI: https://doi.org/10.1016/j.cam.2014.01.002

M. A. Latif, S. S. Dragomir, On some new inequalities for differentiable coordinated convex functions, J. Inequal. and Appl., 2012, № 1, 1–13 (2012). DOI: https://doi.org/10.1186/1029-242X-2012-28

M. Matić, J. Pečarić, A. Vukelić, On generalization of Bullen–Simpson's inequality, Rocky Mountain J. Math., 1727–1754 (2005). DOI: https://doi.org/10.1216/rmjm/1181069660

J. Park, Generalizations of the Simpson-like type inequalities for coordinated $s$-convex mappings, Far East J. Math. Sci., 54, № 2, 225–236 (2011).

J. Park, On the Hermite–Hadamard-type inequalities for coordinated $(s,r)$-convex mappings, Int. J. Pure and Appl. Math., 74, № 2, 251–263 (2012).

C. Peng, C. Zhou, T. S. Du, Riemann–Liouville fractional Simpson's inequalities through generalized ($m,h_{1},h_{2}$)-preinvexity, Ital. J. Pure and Appl. Math., 38, 345–367 (2017). DOI: https://doi.org/10.4067/S0716-09172018000200345

M. Z. Sarıkaya, On the Hermite–Hadamard-type

inequalities for coordinated convex function via fractional integrals, Integral Transforms and Spec. Funct., 25, № 2, 134–147 (2014). DOI: https://doi.org/10.1080/10652469.2013.824436

M. Z. Sarikaya, H. Budak, Some integral inequalities for local fractional integrals, Int. J. Anal. and Appl., 14, № 1, 9–19 (2017).

M. Z. Sarikaya, E. Set, M. E. Ozdemir, S. S. Dragomir, New some Hadamard's type inequalities for coordinated convex functions, Tamsui Oxf. J. Inf. and Math. Sci., 28, № 2, 137–152 (2012).

K. L. Tseng, S. R. Hwang, K. C. Hsu, Hadamard-type and Bullen-type inequalities for Lipschitzian functions and their applications, Comput. Math. and Appl., 64, № 4, 651–660 (2012). DOI: https://doi.org/10.1016/j.camwa.2011.12.076

T. Tunç, M. Z. Sarikaya, H. Yaldiz, Fractional Hermite–Hadamard's type inequality for coordinated convex functions, TWMS J. Pure and Appl. Math., 11, № 1, 3–29 (2020).

C. Unal, F. Hezenci, H. Budak, Some remarks on parameterized inequalities involving conformable fractional operators, Turk. J. Math., 47, № 2, 590–607 (2023). DOI: https://doi.org/10.55730/1300-0098.3381

X. You, F. Hezenci, H. Budak, H. Kara, New Simpson type inequalities for twice differentiable functions via generalized fractional integrals, AIMS Mathematics, 7, № 3, 3959–3971 (2021). DOI: https://doi.org/10.3934/math.2022218

L. L. Zhang, Y. Peng, T. S. Du, On multiplicative Hermite–Hadamard and Newton-type

inequalities for multi-plicatively $ (P,m) $-convex functions, J. Math. Anal. and Appl., 534, № 2, Article 128117 (2024).

D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54, 903–917 (2017). DOI: https://doi.org/10.1007/s10092-017-0213-8

T. C. Zhou, Z. R. Yuan, T. S. Du, On the fractional integral inclusions having exponential kernels for interval-valued convex functions, Math. Sci., 17, № 2, 107–120 (2023). DOI: https://doi.org/10.1007/s40096-021-00445-x

Published
29.11.2024
How to Cite
HezenciF., KaraH., and BudakH. “New Results on Bullen-Type Inequalities for Coordinated Convex Functions Obtained by Using Conformable Fractional Integrals”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, no. 11, Nov. 2024, pp. 1691 -2, doi:10.3842/umzh.v76i11.7989.
Section
Research articles