Abstract
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.
Highlights
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
Since jXjq is an h-convex stochastic process, one can yield that ð1 jXðθr + ð1 − θÞs, ·Þjqdθ
By Lemma 10 and the power-mean integral inequality for κ ≥ 1, one can yield that ðs ðω − rÞμðs − ωÞνXðω, ·Þdω r ð1
Summary
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 process X : J × Ω ⟶ R is said to be a p-convex stochastic process, if the following inequality holds: X 1⁄2θrp + ð1 − θÞsp1/p, · ≤ θXðr, ·Þ + ð1 − θÞXðs, ·Þða:e:Þ, ð12Þ A stochastic process X : J × Ω ⟶ R is known as strongly convex with modulus cð·Þ > 0, if the following inequality holds: Xðθr + ð1 − θÞs, ·Þ ≤ θXðr, ·Þ + ð1 − θÞXðs, ·Þ − cð·Þθð1 − θÞðr − sÞ2ða:e:Þ, ð14Þ
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