Mean Square Integral Inequalities for Generalized Convex Stochastic Processes via Beta Function

The integral inequalities have become a very popular area of research in recent years. The present paper deals with some important generalizations of convex stochastic processes. Several mean square integral inequalities are derived for this generalization. The involvement of the beta function in the results makes the inequalities more convenient for applied sciences.


Introduction
Just as the probability theory is regarded as the study of mathematical models of random phenomena, the theory of stochastic processes plays an important role in the investigation of random phenomena depending on time. A random phenomenon that arises through a process which is developing in time and controlled by some probability law is called a stochastic process. Thus, stochastic processes can be referred to as the dynamic part of the probability theory. We will now give a formal definition of a stochastic process.
Various collections of random variables Xðl, ·Þ, l ∈ J, have the property in some sense that XðlÞ is stochastically convex (or −Xðl, ·Þ is stochastically concave). The stochastic process with convexity properties has a large number of applications. In [1], the authors demonstrated the use of a stochastically convex function in different areas of probability and statistics.
In queueing theory, the convexity of steady-state waiting time is used in [2]. More in [1], the authors used the convexity of payoff in the success rate to obtain an imperfect repair.
The Hermite-Hadamard inequality for the convex stochastic process is defined as follows: Let X : J × Ω ⟶ ℝ be a convex and mean square continuous in the interval T × Ω; then, the inequality holds almost everywhere: Definition 2 (see [13]). A family F l of α-fields on Ω parametrized by l ∈ J, where J ⊂ ℝ, is said to be a filtration if for any s, l ∈ J such that s ≤ l.
Definition 3 (see [13]). A stochastic process XðlÞ parametrized by l ∈ T is said to be a martingale (supermartingale, submartingale) with respect to a filtration F l if (1) XðlÞ is integrable for each l ∈ J (2) XðlÞ is F l -measurable for each l ∈ J (3) XðsÞ = EðXðlÞ | F s Þ (respectively, ≤ or ≥) for every s, l ∈ J such that s ≤ l.
Definition 4 (see [7]). Let ðΩ, A, PÞ be an arbitrary probability space and J ⊂ ℝ be an interval. A stochastic process X : Ω ⟶ ℝ is called as follows: where P − lim denotes the limit in probability.
(4) Monotonic if it is increasing or decreasing (5) If there exists a random variable X ′ ðl, ·Þ: J × Ω ⟶ ℝ, then we say that it is differentiable at a point l ∈ J, such that A stochastic process X : J × Ω ⟶ ℝ is continuous (differentiable) if it is continuous (differentiable) at every point of interval J.
Next, we write some basic definitions which will be used in this work: Definition 6 (see [4]). Let ðΩ, A, PÞ be a probability space and J ⊆ R be an interval. A stochastic process X : J × Ω ⟶ R is called a convex stochastic process; then, the inequality holds almost everywhere: ∀r, s ∈ J and ϑ ∈ ½0, 1.
Definition 7 (see [17]). A process X : J × Ω ⟶ ℝ is said to be a p-convex stochastic process, if the following inequality holds: for all r, s ∈ J and ϑ ∈ ½0, 1.
Obviously, by taking hðϑÞ = ϑ in (13), then the definition of the h-convex stochastic process reduces to the definition of the convex stochastic process [4]. Journal of Function Spaces Definition 9 (see [9]). Let c : Ω ⟶ ℝ be a positive random variable. A stochastic process X : J × Ω ⟶ ℝ is known as strongly convex with modulus cð·Þ > 0, if the following inequality holds: for all r, s ∈ J and ϑ ∈ ½0, 1.
For more details on the strongly convex stochastic process, we refer to [9], and for some interesting properties of some special function, see [19,20]. Obviously, if we omit the term cð·Þϑð1 − ϑÞðr − sÞ 2 in (14), then we get the definition of a convex stochastic process (see [4]). On the other hand, if we set c = 0, then we get it from (14) in limit case. Also, we use the beta function in this present work which is expressed as

Main Results
Lemma 10 (see [21]). Suppose that X : J × Ω ⟶ ℝ be a mean square continuous and mean square integrable stochastic process. Then, the following equality holds almost everywhere: for some fixed μ, ν > 0.
Lemma 11. Suppose that X : J × Ω ⟶ ℝ be a mean square continuous and mean square integrable stochastic process. Then, the following equality holds almost everywhere: for some fixed μ, ν > 0.
The following results are derived for p-convex stochastic processes.
Theorem 13. Suppose that X : J × Ω ⟶ ℝ be a mean square continuous and mean square integrable stochastic process. If jXj is p-convex on ½r, s, where r, s ∈ J with r < s, and μ, ν > 0 is taken, then the inequality holds almost everywhere: Proof. By using Lemma 11, the definition of the p-convexity of jXj and the beta function yield that which completes the proof. ☐ Remark 14. If we take p = 1 in Theorem 13, then we obtain Theorem 3.1 of [21].
Theorem 15. Suppose that X : J × Ω ⟶ ℝ be a mean square continuous and mean square integrable stochastic process. If jXj q is p-convex on ½r, s for q > 1 with 1/κ + 1/q = 1, where r , s ∈ J, r < s, and μ, ν > 0 is taken, then the inequality holds almost everywhere: Since jXj q is a p-convex stochastic process, one can yield that and by the definition of the beta function, we can write Inserting (23) and (24) in (22) yields the required inequality (21). ☐ Remark 16. If we take p = 1 in Theorem 15, then we get Theorem 3.2 of [21].
Theorem 17. Let X : J × Ω ⟶ ℝ be a mean square continuous and mean square integrable stochastic process. If jXj q is p -convex on ½r, s for q > 1, where r, s ∈ J with r < s, and μ, ν > 0 is taken, then the inequality holds almost everywhere: Proof. Making use of Lemma 11 and the power-mean integral inequality for κ ≥ 1 yields that By using the p-convexity of the stochastic process jXj q and by the definition of the beta function, we have ða:e:Þ which completes the proof. ☐ Remark 18. If we take p = 1 in Theorem 17, then we obtain Theorem 3.3 of [21].
The following results are derived for h-convex stochastic processes.
Theorem 19. Suppose that X : J × Ω ⟶ ℝ be a mean square continuous and mean square integrable stochastic process. If jXj is h-convex on ½r, s, where r, s ∈ J with r < s, and μ, ν > 0 is taken, then the inequality holds almost everywhere: where Proof. By Lemma 10, the definition of the h-convexity of |X | and the beta function yield that 4 Journal of Function Spaces which completes the proof. ☐ Remark 20. If we take hðϑÞ = ϑ in Theorem 19, then we obtain Theorem 3.1 of [21].
Theorem 21. Suppose that X : J × Ω ⟶ ℝ be a mean square continuous and mean square integrable stochastic process. If jXj q is h-convex on ½r, s for q > 1 with 1/κ + 1/q = 1, where r , s ∈ J, r < s, and μ, ν > 0 is taken, then the inequality holds almost everywhere: where β h ðϑÞ = Ð 1 0 hðϑÞdϑ and β h ð1 − ϑÞ = Proof. Employing Lemma 10 and Hölder's integral inequality, we have ða:e:Þ Since jXj q is an h-convex stochastic process, one can yield that and by the definition of the beta function, we can write Inserting (34) and (35) in (33) yields the desired inequality (32). ☐ Remark 22. If we take hðϑÞ = ϑ in Theorem 21, then we obtain Theorem 3.2 of [21].
Theorem 23. Let X : J × Ω ⟶ ℝ be a mean square continuous and mean square integrable stochastic process. If jXj q is h -convex on ½r, s for q > 1, where r, s ∈ J with r < s, and μ, ν > 0 is taken, then the inequality holds almost everywhere: where Proof. By Lemma 10 and the power-mean integral inequality for κ ≥ 1, one can yield that which completes the proof. ☐