On some Hermite–Hadamard inequalities for fractional integrals and their applications

  • S.-R. Hwang China Univ. Sci. and Technology, Nankang, Taipei, Taiwan
  • S.-Y. Yeh Aletheia Univ., Tamsui, New Taipei city, Taiwan
  • K.-L. Tseng Aletheia Univ., Tamsui, New Taipei city, Taiwan

Abstract

UDC 517.5

We establish some new extensions of Hermite\,--\,Hadamard inequality for fractional integrals and present several applications for the Beta function.

References

Alomari, Mohammad; Darus, Maslina. On the Hadamard's inequality for log-convex functions on the coordinates. J. Inequal. Appl. 2009, Art. ID 283147 13 pp. doi: 10.1155/2009/283147

Dragomir, Sever Silvestru. Two mappings in connection to Hadamard's inequalities. J. Math. Anal. Appl. 167 (1992), no. 1, 49--56. doi: 10.1016/0022-247X(92)90233-4

Dragomir, S. S. On the Hadamard's inequality for convex functions on the co-ordinates in a rectangle from the plane. Taiwanese J. Math. 5 (2001), no. 4, 775--788. doi: 10.11650/twjm/1500574995

Dragomir, S. S.; Agarwal, R. P. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 11 (1998), no. 5, 91--95. doi: 10.1016/S0893-9659(98)00086-X

Dragomir, S. S.; Cho, Y. J.; Kim, S. S. Inequalities of Hadamard's type for Lipschitzian mappings and their applications. J. Math. Anal. Appl. 245 (2000), no. 2, 489--501. doi: 10.1006/jmaa.2000.6769

L. Fejer,́ Uber die Fourierreihen, II, Math. Naturwiss. Anz Ungar. Akad. Wiss.̈, 24, 369 – 390 (1906) (in Hungarian).

J. Hadamard, Etude sur les propríet́es des fonctions entíeres en particulier d’une fonction consid`eŕee par Riemanń, J. Math. Pures et Appl.,58, 171 – 215 (1893).

Hwang, Shiow-Ru; Tseng, Kuei-Lin; Hsu, Kai-Chen. New inequalities for fractional integrals and their applications. Turkish J. Math. 40 (2016), no. 3, 471--486. doi: 10.3906/mat-1411-61

Hwang, Shiow-Ru; Tseng, Kuei-Lin. New Hermite-Hadamard-type inequalities for fractional integrals and their applications. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 112 (2018), no. 4, 1211--1223. doi: 10.1007/s13398-017-0419-2

Hwang, Shiow-Ru; Hsu, Kai-Chen; Tseng, Kuei-Lin. Hadamard-type inequalities for Lipschitzian functions in one and two variables with applications. J. Math. Anal. Appl. 405 (2013), no. 2, 546--554. doi: 10.1016/j.jmaa.2013.04.032

Hwang, Shiow-Ru; Yeh, Shu-Ying; Tseng, Kuei-Lin. Refinements and similar extensions of Hermite-Hadamard inequality for fractional integrals and their applications. Appl. Math. Comput. 249 (2014), 103--113. doi: 10.1016/j.amc.2014.10.032

Kirmaci, Uǧur S. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 147 (2004), no. 1, 137--146. doi: 10.1016/S0096-3003(02)00657-4

Kirmaci, U. S.; Özdemir, M. E. On some inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula. Appl. Math. Comput. 153 (2004), no. 2, 361--368. doi: 10.1016/S0096-3003(03)00637-4

M. Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, Hermite – Hadamard’s inequalities for fractional integrals and relatedfractional inequalities, Math. and Comput. Model.,57, 2403 – 2407 (2013).

Tseng, Kuei-Lin; Hwang, Shiow-Ru; Dragomir, S. S. Fejér-type inequalities (I). J. Inequal. Appl. 2010, Art. ID 531976, 7 pp. doi: 10.1155/2010/531976

Yang, Gou-Sheng; Tseng, Kuei-Lin. On certain integral inequalities related to Hermite-Hadamard inequalities. J. Math. Anal. Appl. 239 (1999), no. 1, 180--187. doi: 10.1006/jmaa.1999.6506

Yang, Gou-Sheng; Tseng, Kuei-Lin. Inequalities of Hadamard's type for Lipschitzian mappings. J. Math. Anal. Appl. 260 (2001), no. 1, 230--238. doi: 10.1006/jmaa.2000.7460

C. Zhu, M. Feckan, J.-R. Wang,ˇFractional integral inequalities for differentiable convex mappings and applicationsto special means and a midpoint formula, J. Amer. Math. Soc., 8, No 2, 21 – 28 (2012)

Published
28.03.2020
How to Cite
HwangS.-R., YehS.-Y., and TsengK.-L. “On Some Hermite–Hadamard Inequalities for Fractional Integrals and Their Applications”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 3, Mar. 2020, pp. 407-24, doi:10.37863/umzh.v72i3.6043.
Section
Research articles