On Hermite-Hadamard type inequalities for $n$-polynomial convex stochastic processes

1.   Introduction and Preliminaries

In 1980, Nikodem ^[1] proposed convex stochastic processes and examined the regularity properties of these processes. Skowroński in 1992, obtained some further results on Jensen-convex and Wright-convex stochastic processes ^[2,3]. Some interesting properties of convex and Jensen-convex processes are also presented in ^[4]. Moreover, Kotrys ^[5] in 2011 developed Hermite-Hadamard type inequalities for convex stochastic processes. In recent years, many studies have been done in the literature on some types of convexity for stochastic processes see ^[9,10,11] and Hermite-Hadamard inequalities for related convex stochastic processes ^{[12,13,14,15,16,17,18]}.

A stochastic process $X:I\times \varOmega \rightarrow \mathbb{R}$ is said to be convex if we have

for all $c, d \in I$, $t \in [0, 1]$. $X$ is said to be concave on $I\neq \emptyset$, if above inequality reverses.

The notion of stochastic processes for convexity is of great importance in optimization and also useful for numerical approximations when there exist probabilistic quantities in the literature ^[19]. In ^[20], the authors studied stochastic optimization problem under constraints in a general framework including fnancial models with constrained portfolios, labor income and large investor models and reinsurance models (see ^{[21,22,23,24,25,26,27]} for more details). A constrained stochastic successive convex approximation (CSSCA) algorithm is proposed in ^[28] to find a stationary point for a general non-convex stochastic optimization problem, whose objective and constraint functions are non-convex and involve expectations over random states. The algorithm solved a sequence of convex objective/feasibility optimization problems obtained by replacing the objective/constraint functions in the original problems with some convex surrogate functions. For more on applications of convex stochastic processes see ^{[29,30,31,32,33,34,35,36]}.

Let $X:I\times \varOmega \rightarrow \mathbb{R}$ be the Jensen convex and mean-square continuous in the interval $I\times \varOmega$ then

for any $c, d \in I$, $c < d$. If the stochastic process $X$ is concave then both inequalities hold in the reversed direction. This double inequality is well known in the literature as Hermite-Hadamard inequality ^[5]. In this article, we establish new refinements of Hölder and power mean integral inequality. Also, we present the counterpart of the research made by T. Toplu et al. in ^[6] for stochastic processes.

Let us present some important and useful definitions for this research.

Let $(\Omega, \mathcal{A}, P)$ be a probability space. A function $X:\Omega \rightarrow \mathbb{R}$ is a random variable if it is $\mathcal{A}$-measurable. A function $X: I \times \Omega \rightarrow \mathbb{R}$, where $I \subseteq \mathbb{R}$ is an interval, is a stochastic process if for every $x \in I$, the function $X(x, .)$ is a random variable. Readers can look at ^[7,8] for the defnitions and basic properties.

Definition 1.1. ^[8] Consider a stochastic process $X(t, .)$ such that the expectation squared is bounded, i.e. $\mathbb{E}[X(t, .)]^2 < \infty$ for all $t \in I$. The stochastic process $X$ is defined:

(i) continuous in probability on $I$, if

for all $c_{\circ} \in I$, where $\mu - \lim$ represents limit in the probability;

(ii) mean-square continuous in $I$, if

for all $c_{\circ} \in I$;

(iii) increasing (decreasing) if for all $c, d \in I$, $c < v$

(iv) mean-square differentiable at a point $c \in I$, if there is a stochastic process $X^{'}$ (derivative of $X$) such that

The stochastic process $X:I\times \varOmega \rightarrow \mathbb{R}$ is continuous (differentiable) if it is continuous (differentiable) at every point of interval $I$.

Definition 1.2. ^[8] Assume a stochastic process $X : I\times \varOmega \rightarrow \mathbb{R}$ with $\mathbb{E}\left[X(a)^2\right] < \infty$, where $a\in I$. A random variable $Y: \varOmega \rightarrow \mathbb{R}$ is said to be the mean-square integral of the process $X$ on $[c, d]$ if for every normal sequence of partitions of $[c, d]\subseteq I$, $c = t_{\circ} < t_1 < ... < t_n = d$ and for all $\varTheta_k \in [t_{k-1}, t_k]$, k = 1, ..., n, we have

Then we write

For the existence of the mean-square integral it is enough to assume the mean-square continuity of the stochastic process $X$.

The paper is organized as follow: In section $2$, we derive Hölder-İşcan and improved power mean integral inequality, whereas in section $3$, we introduce the notion of $n$-polynomial convex stochastic process and present some interesting properties for these processes. Furthermore, section $4$ and section $5$ are devoted to state and prove Hermite-Hadamard inequality and some new type of Hermite-Hadamard inequalities respectively for $n$-polynomial convex stochastic processes.

2.   Hölder-İşcan and improved power mean integral inequality

In this section, we establish new refinements of Hölder and power mean integral inequality.

Theorem 2.1. (Hölder-İşcan integral inequality) Let $X, Y: [c, d] \times \Omega\rightarrow \mathbb{R}$ be real stochastic processes and $|X^{'}|^p$, $|Y^{'}|^p$ be mean square integrable on $[c, d]$. If $p > 1$ and $\frac{1}{p}+\frac{1}{q} = 1$, then we have almost everywhere:

Proof. Using the Hölder integral inequality, we obtain

Theorem 2.2. (Improved power mean integral inequality) Let $X, Y: [c, d] \times \Omega\rightarrow \mathbb{R}$ be real stochastic processes and $|X|$, $|X||Y|^q$ be mean square integrable on $[c, d]$. If $q\geq 1$, then the following inequality holds almost everywhere:

Proof. Firstly, let $q > 1$. Using Hölder inequality, we easily see that

For $q = 1$, the inequality (2.2) holds trivially.

3.   Main definition and basic properties

In this section, we introduce our main definition and present some results concerning to the basic properties of $n$-polynomial convex stochastic processes.

Definition 3.1. Let $X:I \times \Omega\rightarrow \mathbb{R}$ be a non-negative stochastic process and $n \in \mathbb{N}$. Then $X$ is said to be $n$-polynomial convex if

for every $c, d \in I$ and $t\in[0, 1]$.

Remark 3.2. It is worth pointing out that for

1. $n = 1$ in (3.1), the $1$-polynomial convexity implies classical convexity.

2. $h(t) = \frac{1}{n}\sum\limits_{k = 1}^{n}[1-(1-t)^k]$ in (3.1), we obtain an $h$-convex stochastic process (see ^[10]).

Remark 3.3. Let $X:I \times \Omega\rightarrow \mathbb{R}$ be a $2$-polynomial convex stochastic process, then

for every $c, d \in I$ and $t \in [0, 1]$. It is clear that for all $t \in [0, 1]$

This shows that every non-negative convex stochastic process is also a $2$-polynomial convex stochastic process. More generally, every non-negative convex stochastic process is also an $n$-polynomial convex stochastic process.

Proposition 3.4. Let $X, Y: I \times \Omega\rightarrow \mathbb{R}$ be $n$-polynomial convex stochastic processes and $\gamma \in \mathbb{R}$ $(\gamma \geq 0)$, then $X+Y$ and $\gamma X$ are also $n$-polynomial convex stochastic processes.

Proof. Consider $c, d \in I$ and $t \in (0, 1)$ arbitrary.

Now consider $\gamma \geq 0$. Then,

Proposition 3.5. If $X:I \times \Omega\rightarrow \mathbb{R}$ is convex and $Y: I \times \Omega\rightarrow \mathbb{R}$ is an $n$-polynomial convex and non-decreasing stochastic process, then $Y\circ X: I \times \Omega\rightarrow \mathbb{R}$ is $n$-polynomial convex stochastic process.

Proof. By definition of composition, we have

which completes the prove.

Theorem 3.6. Let $X_\alpha: [a, b]\times \Omega\rightarrow \mathbb{R}$, $b > 0$, be an arbitrary collection of n-polynomial convex stochastic processes and define a stochastic process $Y(c, .) = \sup \limits_\alpha X_\alpha(c, .)$. If $J = \left\{v \in [a, b]:Y(v, .) < \infty\right\}$is non-empty, then $J$ is an interval and $Y$ is an n-polynomial convex stochastic process on $J \times \Omega$.

Proof. Let $t \in [0, 1]$ and $c, d \in J$ be arbitrary. Then

This implies that $J$ is an interval and $Y$ is an $n$-polynomial convex stochastic process on $J \times \Omega$.

4.   Hermite-Hadamard inequality

In this section, we will establish Hermite-Hadamard inequality for $n$-polynomial convex, mean square integrable stochastic process.

Theorem 4.1. Let $X: I\times \Omega\rightarrow \mathbb{R}$ be an $n$-polynomial convex, mean square integrable stochastic process. Then for every $c, d \in I$, $c < d$, we have almost everywhere

Proof. By using inequality (3.1), we have

By integrating last inequality with respect to $t \in [0, 1]$, we get

implies

Now, changing the variable as $x = tc+(1-t)d$, and using the inequality (3.1), we have

where

Combining (4.2) and (4.3) yields (4.1).

Remark 4.2. For $n = 1$, the inequality (4.1) coincides with the inequality (1.1).

5.   New inequalities of Hermite-Hadamard type

In this section, we purpose to develop new Hermite-Hadamard type inequalities for stochastic processes whose first derivative in absolute value, raised to a certain power is $n$-polynomial convex stochastic process. In order to prove these inequalities it is necessary to use the following lemma:

Lemma 5.1. Let $X: I\times \Omega\rightarrow \mathbb{R}$ be a mean square differentiable stochastic process on $I^\circ$ and $X^{'}$ is mean square integrable on $[c, d]$, where $c, d \in I$, $c < d$. Then we have almost everywhere

Proof. Using integration by parts, we get

This implies that

Theorem 5.2. Let $X: I\times \Omega\rightarrow \mathbb{R}$ be a mean square differentiable stochastic process on $I^\circ$ and $X^{'}$ be mean square integrable on $[c, d]$. If $|X^{'}|$ is an n-polynomial convex stochastic process on $[c, d]$, then we have almost everywhere:

where $A$ is the arithmetic mean.

Proof. Using Lemma 5.1 and the property of $n$-polynomial convexity of stochastic process $|X^{'}|$, we have

where $A$ is the arithmetic mean and

Corollary 5.3. For $n = 1$ in (5.2), we obtain the following inequality for convex stochastic processes:

Theorem 5.4. Let $X: I\times \Omega\rightarrow \mathbb{R}$ be a mean square differentiable stochastic process on $I^\circ$ with $q > 1$, $\frac{1}{p}+\frac{1}{q} = 1$ and assume that $X^{'}$ be mean square integrable on $[c, d]$. If $|X^{'}|^q$ is an n-polynomial convex stochastic process on $[c, d]$, then the following inequality holds almost everywhere for $t\in [0, 1]$:

Proof. By using Lemma 5.1, Hölder's integral inequality and the definition of $n$-polynomial convexity of stochastic process $|X^{'}|^q$, we get

where $A$ is the arithmetic mean and

Corollary 5.5. For $n = 1$ in (5.3), we obtain the following inequality for convex stochastic processes:

Theorem 5.6. Let $X: I\times \Omega\rightarrow \mathbb{R}$ be a mean square differentiable stochastic process on $I^\circ$ with $q\geq1$ and $X^{'}$ be mean square integrable on $[c, d]$. If $|X^{'}|^q$ is an n-polynomial convex stochastic process on $[c, d]$, then the following inequality holds almost everywhere for $t\in [0, 1]$:

Proof. Assume $q > 1$. By using Lemma 5.1, Hölder's integral inequality and the property of $n$-polynomial convexity of stochastic process $|X^{'}|^q$, we get

where

For $q = 1$, we use the estimates from the proof of Theorem 5.2. Thus, the proof of Theorem 5.6 is completed.

Remark 5.7. If we take $q = 1$ in Theorem 5.6, then we get the conclusion of Theorem 5.2.

Corollary 5.8. For $n = 1$ in (5.4), we obtain the following inequality for convex stochastic processes:

Now, we will prove the Theorem 5.4 by using Hölder-İşcan integral inequality. Then we will compare the results obtained in this Theorem and Theorem 5.4.

Theorem 5.9. Let $X: I\times \Omega\rightarrow \mathbb{R}$ be mean square differentiable on $I^\circ$ with $c < d$, $q > 1$, $\frac{1}{p}+\frac{1}{q} = 1$. Assume that $X^{'}$ be mean square integrable on $[c, d]$ and $|X^{'}|^q$ be an n-polynomial convex stochastic process on $[c, d]$, then the following inequality holds almost everywhere for $t\in [0, 1]$:

Proof. By using Lemma 5.1, Hölder-İşcan integral inequality and the property of $n$-polynomial convexity of stochastic process $|X^{'}|^q$, we obtain

implies

which gives

where

and

Hence, the proof of Theorem 5.9 is completed..

Corollary 5.10. For $n = 1$ in (5.5), we obtain the following inequality for convex stochastic processes:

Remark 5.11. The inequality (5.5) gives better result than (5.3). We will prove it by showing that

If we use the concavity of the stochastic process $Y:[0, \infty)\times \Omega\rightarrow \mathbb{R}$, defined by $Y(u, .) = u^\lambda$, $0 < \lambda\leq 1$, we get

which is required.

Theorem 5.12. Let $X: I\times \Omega\rightarrow \mathbb{R}$ be mean square differentiable on $I^\circ$ with $c < d$, $q\geq1$. Assume that $X^{'}$ be mean square integrable on $[c, d]$ and $|X^{'}|^q$ be an n-polynomial convex stochastic process on $[c, d]$, then the following inequality holds almost everywhere for $t\in [0, 1]$:

where

Proof. Assume first that $q > 1$. From Lemma 5.1, improved power-mean integral inequality and the definition of n-polynomial convexity of stochastic process $|X^{'}|^q$, we have

where

For $q = 1$, we use the estimates from the proof of Theorem 5.2. Thus, the proof of Theorem 5.12 is completed.

Corollary 5.13. For $n = 1$ in (5.6), we obtain the following inequality for convex stochastic processes:

Remark 5.14. The inequality (5.6) gives better estimate than (5.4). Let us prove it by showing

By using concavity of the stochastic process $Y:[0, \infty)\times \Omega\rightarrow \mathbb{R}$, defined by $Y(u, .) = u^\lambda$, $0 < \lambda\leq 1$, we obtain

where

which completes the proof of remark.

6.   Applications for special means

For two positive numbers $c, d$ with $c < d$, define

These means are respectively called the arithmetic, geometric, harmonic, identric, logarithmic and $p$-logarithmic means of two positive numbers $c$ and $d$.

Applying Theorem 4.1 to $X(u, .) = -lnu$, $u \in (0, 1]$ results in the following inequality for means:

Corollary 6.1. Assume $c, d \in (0, 1]$ with $c < d$, then

Applying Theorem 4.1 to $X(u, .) = u^m$, $u \in [0, \infty)$ results in the following inequality for means.

Corollary 6.2. Assume $m \in (-\infty, 0)\cup[1, \infty)\setminus\left\{-1\right\}$ and $c, d \in [0, \infty)$ with $c < d$, then

Taking $X(u, .) = u^{-1}$, $u \in (0, \infty)$ in Theorem 4.1 results in the inequality for means as follows.

Corollary 6.3. Assume $c, d \in (0, \infty)$ with $c < d$, then

7.   Conclusions

The more generalized class of convex stochastic processes named as $n$-polynomial convex stochastic process is introduced in the present note. Hermite-Hadamard inequality and some of its refined forms hold for this new generalization. Some existing results in literature became the particular cases of these results as mentioned in remarks. All the results and inequalities derived here are new, interesting and important in the field of integral inequalities.

Acknowledgment

This work was sponsored in part by NSFC China, Grant (11571067).

Conflict of interest

The authors declare that they do not have any competing interests.