Some New Estimates of Hermite–Hadamard, Ostrowski and Jensen-Type Inclusions for h-Convex Stochastic Process via Interval-Valued Functions
Abstract
:1. Introduction
2. Preliminaries and Background
- (Associativie w.r.t addition) ∀
- (Commutative w.r.t addition) ∀
- (Additive element) ∀
- (Law of Cancellation) ∀
- (Associative w.r.t multiplication) ∀
- (Commutative w.r.t multiplication) ∀
- (Unity element) ∀
3. Stochastic Process
Properties of the Stochastic Process
- continuous over interval ℑ if , one has
- For the continuity in mean square sense over interval ℑ, if , one has
- For the differentiability in mean square sense at any arbitrary point , if there is a random variable , then this is true.
- For the mean-square integral over ℑ, if , and . Let , is a partition of . Let . A random variable is mean-square integrable over , and if this holds true,
4. Main Results
- (i)
- If , Definition 5 incorporates the output in the sense of a stochastic process for the P-function.
- (ii)
- If , Definition 5 incorporates the output in the sense of a stochastic process for the function.
- (iii)
- If , Definition 5 incorporates the output in the sense of a stochastic process for the usual convex function.
- (iv)
- If , Definition 5 incorporates the output in the sense of a stochastic process for the s-convex function.
4.1. Stochastically Hermite–Hadamard Inclusions
4.2. Stochastically Ostrowski-Type Inclusions
4.3. Stochastically Jensen-Type Inclusion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Afzal, W.; Prosviryakov, E.Y.; El-Deeb, S.M.; Almalki, Y. Some New Estimates of Hermite–Hadamard, Ostrowski and Jensen-Type Inclusions for h-Convex Stochastic Process via Interval-Valued Functions. Symmetry 2023, 15, 831. https://doi.org/10.3390/sym15040831
Afzal W, Prosviryakov EY, El-Deeb SM, Almalki Y. Some New Estimates of Hermite–Hadamard, Ostrowski and Jensen-Type Inclusions for h-Convex Stochastic Process via Interval-Valued Functions. Symmetry. 2023; 15(4):831. https://doi.org/10.3390/sym15040831
Chicago/Turabian StyleAfzal, Waqar, Evgeniy Yu. Prosviryakov, Sheza M. El-Deeb, and Yahya Almalki. 2023. "Some New Estimates of Hermite–Hadamard, Ostrowski and Jensen-Type Inclusions for h-Convex Stochastic Process via Interval-Valued Functions" Symmetry 15, no. 4: 831. https://doi.org/10.3390/sym15040831