1. Home
  2. Issues
  3. Volume 5, Issue 4 (2018)
  4. On generalized stochastic fractional int ...

Modern Stochastics: Theory and Applications


On generalized stochastic fractional integrals and related inequalities
Volume 5, Issue 4 (2018), pp. 471–481
Pub. online: 24 September 2018      Type: Research Article      Open accessOpen Access
Received
29 May 2018
Revised
13 September 2018
Accepted
13 September 2018
Published
24 September 2018

Abstract

The generalized mean-square fractional integrals ${\mathcal{J}_{\rho ,\lambda ,u+;\omega }^{\sigma }}$ and ${\mathcal{J}_{\rho ,\lambda ,v-;\omega }^{\sigma }}$ of the stochastic process X are introduced. Then, for Jensen-convex and strongly convex stochastic proceses, the generalized fractional Hermite–Hadamard inequality is establish via generalized stochastic fractional integrals.

References

[1] 
Agahi, H., Babakhani, A.: On fractional stochastic inequalities related to Hermite–Hadamard and Jensen types for convex stochastic processes. Aequationes mathematicae 90(5), 1035–1043 (2016) MR3547706. https://doi.org/10.1007/s00010-016-0425-z
[2] 
Barráez, D., González, L., Merentes, N., Moros, A.: On h-convex stochastic processes. Mathematica Aeterna 5(4), 571–581 (2015)
[3] 
Budak, H., Sarikaya, M.Z.: A new Hermite-Hadamard inequality for h-convex stochastic processes. RGMIA Research Report Collection 19, 30 (2016)
[4] 
González, L., Merentes, N., Valera-López, M.: Some estimates on the Hermite-Hadamard inequality through convex and quasi-convex stochastic processes. Mathematica Aeterna 5(5), 745–767 (2015)
[5] 
Hafiz, F.M.: The fractional calculus for some stochastic processes. Stochastic analysis and applications 22(2), 507–523 (2004) MR2038026. https://doi.org/10.1081/SAP-120028609
[6] 
Kotrys, D.: Hermite–Hadamard inequality for convex stochastic processes. Aequationes mathematicae 83(1), 143–151 (2012) MR2885506. https://doi.org/10.1007/s00010-011-0090-1
[7] 
Kotrys, D.: Remarks on strongly convex stochastic processes. Aequationes mathematicae 86(1–2), 91–98 (2013) MR3094634. https://doi.org/10.1007/s00010-012-0163-9
[8] 
Luo, M.-J., Raina, R.K.: A note on a class of convolution integral equations. Honam Math. J 37(4), 397–409 (2015) MR3445221. https://doi.org/10.5831/HMJ.2015.37.4.397
[9] 
Maden, S., Tomar, M., Set, E.: Hermite-Hadamard type inequalities for s-convex stochastic processes in first sense. Pure and Applied Mathematics Letters 2015, 1 (2015)
[10] 
Materano, J., N., M., Valera-Lopez, M.: Some estimates on the Simpson’s type inequalities through
[11] 
Nikodem, K.: On convex stochastic processes. Aequationes mathematicae 20(1), 184–197 (1980) MR0577487. https://doi.org/10.1007/BF02190513
[12] 
Parmar, R.K., Luo, M., Raina, R.K.: On a multivariable class of Mittag-Leffler type functions. Journal of Applied Analysis and Computation 6(4), 981–999 (2016) MR3502352
[13] 
Raina, R.K.: On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian mathematical journal 21(2), 191–203 (2005)
[14] 
Sarikaya, M.Z., Yaldiz, H., Budak, H.: Some integral inequalities for convex stochastic processes. Acta Mathematica Universitatis Comenianae 85(1), 155–164 (2016) MR3456531
[15] 
Set, E., Sarıkaya, M.Z., Tomar, M.: Hermite-Hadamard type inequalities for coordinates convex stochastic processes. Mathematica Aeterna 5(2), 363–382 (2015)
[16] 
Set, E., Tomar, M., Maden, S.: Hermite Hadamard type inequalities for s-convex stochastic processes in the second sense. Turkish Journal of Analysis and Number Theory 2(6), 202–207 (2014)
[17] 
Skowroński, A.: On some properties ofj-convex stochastic processes. Aequationes Mathematicae 44(2), 249–258 (1992) MR1181272. https://doi.org/10.1007/BF01830983
[18] 
Sobczyk, K.: Stochastic Differential Equations: with Applications to Physics and Engineering vol. 40. Springer (2013)
[19] 
Tomar, M., Set, E., Bekar, N.O.: Hermite-Hadamard type inequalities for strongly-log-convex stochastic processes. Journal of Global Engineering Studies 1, 53–61 (2014)
[20] 
Tomar, M., Set, E., Maden, S.: Hermite-Hadamard type inequalities for log-convex stochastic processes. Journal of New Theory 2, 23–32 (2015)

Full article Cited by PDF XML
Full article Cited by PDF XML

Copyright
© 2018 The Author(s). Published by VTeX
Open access article under the CC BY license.

Keywords
Hermite–Hadamard inequality stochastic fractional integrals convex stochastic process

MSC2010
26D15 26A51 60G99

Metrics
since March 2018
605

Article info
views

449

Full article
views

438

PDF
downloads

165

XML
downloads


Share


RSS