Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications
Abstract
:1. Introduction
2. Preliminaries
- ()
- The RL-fractional integral operator defined as in [25] with ;
- ()
- The Hadamard fractional integral operator defined as in [25] with ;
- ()
- The Katugampola fractional integral operator defined as in [26] with for ;
- ()
- The conformable fractional integral operator defined as in [27] with for ;
- ()
- The generalized conformable fractional integral operator defined as in [28] with .
- ()
- If we input , and in Definition 4, it yields that the classical expectation ;
- ()
- If we input , and in Definition 7, it yields that the classical variance ;
- ()
- If we input , and in Definition 9, it yields that the classical moment of order r: ;
- ()
- ()
- If we input in Definitions 4–8, it yields Definitions – in [40].
- ()
- For , the h satisfies
- ()
- For and , we have the well-known property .
3. Main Results
- For any and ,
- The integral inequality
- For every , , and , then
- The inequality
- For every α, and ,
- The integral inequality
- ()
- By setting in Theorem 2, Theorem 2 deduces to Theorem 1;
- ()
- By setting and in Theorem 2, then Theorem 2 deduces to Theorem in [34];
- ()
- By setting and in of Theorem 2 then Theorem 2 deduces the first part of Theorem 1 as in [33];
- ()
- By setting and in of Theorem 2, then Theorem 2 deduces the last part of Theorem 1 as in [33];
- ()
- If we input , we have Theorem in [40].
4. Some Examples
- (i)
- The proportional of order : From Definition 4, we obtain thatClearly, if and , then (43) reduces to the classical expectation of
- (ii)
- The proportional of order : By Definition 9, one hasClearly, if and , then (44) reduces to the classical moment of order 2
- (iii)
- The proportional of order : By applying Theorem 1, we obtainBy direct computation with Definitions 3 and 4, we haveClearly, if and then (48) reduces to the classical varience of
- (iv)
- The proportional of order :By Definition 9 and binomial expression,This implies that
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
FC | Fractional calculus |
RL | Riemann–Liouville |
Continuous random variable | |
Probability density function | |
Fractional expectation | |
Fractional variance | |
Fractional moment |
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Sudsutad, W.; Jarasthitikulchai, N.; Thaiprayoon, C.; Kongson, J.; Alzabut, J. Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications. Mathematics 2022, 10, 573. https://doi.org/10.3390/math10040573
Sudsutad W, Jarasthitikulchai N, Thaiprayoon C, Kongson J, Alzabut J. Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications. Mathematics. 2022; 10(4):573. https://doi.org/10.3390/math10040573
Chicago/Turabian StyleSudsutad, Weerawat, Nantapat Jarasthitikulchai, Chatthai Thaiprayoon, Jutarat Kongson, and Jehad Alzabut. 2022. "Novel Generalized Proportional Fractional Integral Inequalities on Probabilistic Random Variables and Their Applications" Mathematics 10, no. 4: 573. https://doi.org/10.3390/math10040573