Journal of Probability and Statistics

Journal of Probability and Statistics / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 958980 | https://doi.org/10.1155/2015/958980

Abdullah Akkurt, Zeynep Kaçar, Hüseyin Yildirim, "Generalized Fractional Integral Inequalities for Continuous Random Variables", Journal of Probability and Statistics, vol. 2015, Article ID 958980, 7 pages, 2015. https://doi.org/10.1155/2015/958980

Generalized Fractional Integral Inequalities for Continuous Random Variables

Academic Editor: Z. D. Bai
Received05 Oct 2014
Accepted09 Dec 2014
Published01 Jan 2015

Abstract

Some generalized integral inequalities are established for the fractional expectation and the fractional variance for continuous random variables. Special cases of integral inequalities in this paper are studied by Barnett et al. and Dahmani.

1. Introduction

Integral inequalities play a fundamental role in the theory of differential equations, functional analysis, and applied sciences. Important development in this theory has been achieved in the last two decades. For these, see [18] and the references therein. Moreover, the study of fractional type inequalities is also of vital importance. Also see [913] for further information and applications. The first one is given in [14]; in their paper, using Korkine identity and Holder inequality for double integrals, Barnett et al. established several integral inequalities for the expectation and the variance of a random variable having a probability density function (p.d.f.) . In [1517] the authors presented new inequalities for the moments and for the higher order central moments of a continuous random variable. In [17, 18] Dahmani and Miao and Yang gave new upper bounds for the standard deviation , for the quantity , , and for the absolute deviation of a random variable . Recently, Anastassiou et al. [9] proposed a generalization of the weighted Montgomery identity for fractional integrals with weighted fractional Peano kernel. More recently, Dahmani and Niezgoda [17, 19] gave inequalities involving moments of a continuous random variable defined over a finite interval. Other papers dealing with these probability inequalities can be found in [2022].

In this paper, we introduce new concepts on “generalized fractional random variables.” We obtain new generalized integral inequalities for the generalized fractional dispersion and the generalized fractional variance functions of a continuous random variable having the probability density function (p.d.f.) by using these concepts. Our results are extension of [12, 14, 17].

2. Preliminaries

Definition 1 (see [23]). Let . The Riemann-Liouville fractional integrals and of order are defined by respectively, where is Gamma function and .

We give the following properties for the :

Definition 2 (see [24]). Consider the space () of those real-valued Lebesgue measurable functions on for which

Definition 3 (see [24]). Consider the space of those real-valued Lebesgue measurable functions on for which and for the case In particular, when () the space coincides with the -space and also if we take () the space coincides with the classical -space.

Definition 4 (see [24]). Let . The generalized Riemann-Liouville fractional integrals and of orders and are defined by Here is Gamma function and . Integral formulas (7) and (8) are called right generalized Riemann-Liouville integral and left generalized Riemann-Liouville fractional integral, respectively.

Definition 5. The fractional expectation function of orders and , for a random variable with a positive p.d.f. defined on , is defined as In the same way, we define the fractional expectation function of by what follows.

Definition 6. The fractional expectation function of orders , , and , for a random variable , is defined as where is the p.d.f. of .

For , we introduce the following concept.

Definition 7. The fractional expectation of orders , , and , for a random variable with a positive p.d.f. defined on , is defined as

For the fractional variance of , we introduce the following two definitions.

Definition 8. The fractional variance function of orders , , and , for a random variable having a p.d.f. , is defined as where is the classical expectation of .

Definition 9. The fractional variance of order , for a random variable with a p.d.f. , is defined as We give the following important properties.(1)If we take and in Definition 5, we obtain the classical expectation .(2)If we take and in Definition 7, we obtain the classical variance .(3)If we take in Definitions 59, we obtain Definitions  2.2–2.6 in [17].(4)For , the p.d.f. satisfies .(5)For , we have the well known property .

3. Main Results

Theorem 10. Let be a continuous random variable having a p.d.f. . Then(a) for all , , and , provided that ;(b) the inequality is also valid for all , , and .

Proof. Let us define the quantity for p.d.f. and : Taking a function , multiplying (16) by , and then integrating the resulting identity with respect to from to , we have Similarly, multiplying (17) by , , and integrating the resulting identity with respect to over , we can write If, in (18), we take and , , then we have On the other hand, we have Thanks to (19) and (20), we obtain part (a) of Theorem 10.
For part (b), we have Then, by (19) and (21), we get the desired inequality (14).

We give also the following corollary.

Corollary 11. Let be a continuous random variable with a p.d.f. defined on . Then(i)if , then for any and , one has (ii)the inequality is also valid for any and .

Remark 12. (r1) Taking and in (i) of Corollary 11, we obtain the first part of Theorem 1 in [14].
(r2) Taking and in (ii) of Corollary 11, we obtain the last part of Theorem 1 in [14].

We will further generalize Theorem 10 by considering two fractional positive parameters.

Theorem 13. Let X be a continuous random variable having a p.d.f. . Then one has the following.(a) For all , , , and , where .(b) The inequality is also valid for any , , , and .

Proof. Using (15), we can write Taking and , , in the above identity, yields We have also Thanks to (27) and (28), we obtain (a).
To prove (b), we use the fact that . We obtain And, by (27) and (29), we get (25).

Remark 14. (r1) Applying Theorem 13 for , we obtain Theorem 10.

We give also the following fractional integral result.

Theorem 15. Let be the p.d.f. of on . Then for all , , and , one has

Proof. Using Theorem 1 of [25], we can write Taking and , , then and . Hence, (30) allows us to obtain This implies that Theorem 15 is thus proved.

For , we propose the following interesting inequality.

Corollary 16. Let be the p.d.f. of on . Then for any and , one has

Remark 17. Taking in Corollary 16, we obtain Theorem 2 of [14].

We also present the following result for the fractional variance function with two parameters.

Theorem 18. Let be the p.d.f. of the random variable on . Then for all , , , and , one has

Proof. Thanks to Theorem 4 of [25], we can state that In (35), we take and , . We obtain Combining (27) and (37) and taking into account the fact that the left-hand side of (27) is positive, we get Therefore, Substituting the values of and in (33), then a simple calculation allows us to obtain (35). Theorem 18 is thus proved.

To finish, we present to the reader the following corollary.

Corollary 19. Let be the p.d.f. of on . Then for all , , and , the inequality is valid.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. N. S. Barnett, P. Cerone, S. S. Dragomir, and J. Roumeliotis, “Some inequalities for the expectation and variance of a random variable whose PDF is n-time differentiable,” Journal of Inequalities in Pure and Applied Mathematics, vol. 1, no. 2, article 21, 2000. View at: Google Scholar | MathSciNet
  2. P. L. Chebyshev, “Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites,” Proceedings of the Mathematical Society of Kharkov, vol. 2, pp. 93–98, 1882. View at: Google Scholar
  3. S. S. Dragomir, “A generalization of Grussís inequality in inner product spaces and applications,” Journal of Mathematical Analysis and Applications, vol. 237, no. 1, pp. 74–82, 1999. View at: Publisher Site | Google Scholar | MathSciNet
  4. D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, Dordrecht, The Netherlands, 1993. View at: Publisher Site | MathSciNet
  5. B. G. Pachpatte, “On multidimensional Gruss type integral inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 32, pp. 1–15, 2002. View at: Google Scholar
  6. F. Qi, A. J. Li, W. Z. Zhao, D. W. Niu, and J. Cao, “Extensions of several integral inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 3, pp. 1–6, 2006. View at: Google Scholar
  7. F. Qi, “Several integral inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 1, no. 2, pp. 1–9, 2000. View at: Google Scholar
  8. M. Z. Sarikaya, N. Aktan, and H. Yildirim, “On weighted Čebyšev-Grüss type inequalities on time scales,” Journal of Mathematical Inequalities, vol. 2, no. 2, pp. 185–195, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  9. G. A. Anastassiou, M. R. Hooshmandasl, A. Ghasemi, and F. Moftakharzadeh, “Montgomery identities for fractional integrals and related fractional inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 4, pp. 1–6, 2009. View at: Google Scholar | MathSciNet
  10. G. A. Anastassiou, Fractional Differentiation Inequalities, Springer Science, 2009. View at: Publisher Site | MathSciNet
  11. S. Belarbi and Z. Dahmani, “On some new fractional integral inequalities,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 3, pp. 1–12, 2009. View at: Google Scholar | MathSciNet
  12. Z. Dahmani, “New inequalities in fractional integrals,” International Journal of Nonlinear Science, vol. 9, no. 4, pp. 493–497, 2010. View at: Google Scholar | MathSciNet
  13. Z. Dahmani, “On Minkowski and Hermite-Hadamad integral inequalities via fractional integration,” Annals of Functional Analysis, vol. 1, no. 1, pp. 51–58, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  14. N. S. Barnett, P. Cerone, S. S. Dragomir, and J. Roumeliotis, “Some inequalities for the dispersion of a random variable whose PDF is defined on a finite interval,” Journal of Inequalities in Pure and Applied Mathematics, vol. 2, no. 1, pp. 1–18, 2001. View at: Google Scholar
  15. P. Kumar, “Moment inequalities of a random variable defined over a finite interval,” Journal of Inequalities in Pure and Applied Mathematics, vol. 3, no. 3, pp. 1–24, 2002. View at: Google Scholar
  16. P. Kumar, “Inequalities involving moments of a continuous random variable defined over a finite interval,” Computers and Mathematics with Applications, vol. 48, no. 1-2, pp. 257–273, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  17. Z. Dahmani, “Fractional integral inequalities for continuous random variables,” Malaya Journal of Matematik, vol. 2, no. 2, pp. 172–179, 2014. View at: Google Scholar
  18. Y. Miao and G. Yang, “A note on the upper bounds for the dispersion,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 3, pp. 1–13, 2007. View at: Google Scholar
  19. M. Niezgoda, “New bounds for moments of continuous random variables,” Computers & Mathematics with Applications, vol. 60, no. 12, pp. 3130–3138, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  20. A. M. Acu, F. Sofonea, and C. V. Muraru, “Gruss and Ostrowski type inequalities and their applications,” Scientific Studies and Research: Series Mathematics and Informatics, vol. 23, no. 1, pp. 5–14, 2013. View at: Google Scholar
  21. T. F. Mori, “Sharp inequalities between centered moments,” Journal of Mathematical Analysis and Applications, vol. 10, no. 4, pp. 1–19, 2009. View at: Google Scholar
  22. R. Sharma, S. Devi, G. Kapoor, S. Ram, and N. S. Barnett, “A brief note on some bounds connecting lower order moments for random variables defined on a finite interval,” International Journal of Theoretical & Applied Sciences, vol. 1, no. 2, pp. 83–85, 2009. View at: Google Scholar
  23. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
  24. H. Yıldırım and Z. Kırtay, “Ostrowski inequality for generalized fractional integral and related inequalities,” Malaya Journal of Matematik, vol. 2, no. 3, pp. 322–329, 2014. View at: Google Scholar
  25. Z. Dahmani and L. Tabharit, “On weighted Gruss type inequalities via fractional integrals,” Journal of Advanced Research in Pure Mathematics, vol. 2, no. 4, pp. 31–38, 2010. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2015 Abdullah Akkurt et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views1424
Downloads998
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.