1. Introduction

First, the idea of convex functions was put forward, and many studies were made in this area. Later, in 1980, Nikodem ^[1] introduced the convex stochastic processes in his article. In 1995, Skowronski ^[3] presented some further results on convex stochastic processes. Moreover, in 2011, Kotrys ^[4] derived some Hermite-Hadamard type inequalities for convex stochastic processes. In 2014 Maden et. al. ^[7] introduced the convex stochastic processes in the first sense and extended some well-known results concerning convex functions in the first sense, such as Hermite-Hadamard type inequalities and some inequalities to convex stochastic processes.

We shall introduce the following definitions which are used throughout this paper.

Definition 1. ^{[1, 2, 8]} Let be an arbitrary probability space. A function is called a random value if it is A - measurable. A function , where is an interval, is called a stochastic process if for every the function is a random variable.

Recall that the stochastic process is called

(1) continuous in probability in interval I , if for all

where P - lim denotes the limit in probability;

(2) mean - square continuous in the interval I, if for all

where denotes the expectation value of the random variable ;

(3) increasing (decreasing) if for all such that,

(4) monotonic if it is increasing or decreasing;

(5) differentiable at a point if there is a random variable

We say that a stochastic process is continuous (differentiable) if it is continuous (differentiable) at every point of the interval I.

Definition 2. ^{[2, 3, 8]} Let be a probability space and be an interval. We say that a stochastic process is

(1) convex if

if all and . This class of stochastic process are denoted by C.

(2) -convex (where is a fixed number from (0,1)) if

if all and . This class of stochastic process are denoted by .

(3) Wright-convex if

if all and . This class of stochastic process are denoted by W.

(4) Jensen-convex if

This class of stochastic process are denoted by .

Let . A function where , is said to be s-convex function in the second sense if:

(1.1)

for all with + = 1. This class of functions are denoted by . It can be easily seen that for s = 1 s-convexity in the first sense reduce to ordinary convexity of functions defined on .

Throughout this paper, let I be an interval on .

Now, we present some theorems which was proved by Dargomir and Fitzpatrick ^[5] and H. Hudzik and L. Maligranda ^[6] about some inequalities of Hermite-Hadamard type and basic properties for the s-convex functions in the second sense.

Proposition 1. ^[6] If , then f is non-negatif on .

Theorem 1. ^[6] Let . Then inequality (1.1) holds for all and with if and only if .

Theorem 2. ^[5] Suppose that is s-convex mapping in the second sense, and with . If , then one has the inequalities:

(1.2)

Theorem 3. ^[6] Now, suppose that f is Lebesque integrable on [a,b] and consider the mapping given by

Let be a s-convex mapping in the second sense on I, and Lebesque integrable on . Then:

i: H is s-convex in the second sense on [0,1],

ii: We have the inequality:

for all .

iii: We have inequality:

where

and ;

iv: If , then

Theorem 4. ^[6] Let be a s-convex mapping in the second sense, , with on .

for all and

for all ;

ii. F is s-convex in the second sense on ;

iii. We have the inequality:

iv. We have the inequality

for all .

v. We have the inequality:

for all .

The aim of this paper is to introduce convex stochastic processes in the second sense and to obtain basic properties and Hermite-Hadamard type inequalities for this procesess.

2. s-convex Stochastic Process in the Second Sense

Definition 3. stochastic process where be an interval, is said to be s-convex in the second sense if:

for all and s fixed in . We denote this class of stochastic process by .

Remark 1. It can be easily seen that for s = 1 s-convexity in the second sense for stochastic processes reduce to ordinary convexity of stochastic processes defined in Definition 2.

Proposition 2. for all .

Proof. To prove the proposition assume that and take arbitrary points , Since X is convex stochastic process and on we have

Then .

Proposition 3. for all .

Proof. To prove the proposition assume that and take arbitrary points , Since X is Jensen-convex stochastic process and we have

Then .

Proposition 4. for all .

Proof. The proof is cleared owing to [3, Proposition 3] and Proposition 3.

Proposition 5. If , then X is non-negative on I.

Proof. We have , for

Hence, and so .

Theorem 5. Let . Then inequality holds for all and with if and only if .

Proof. Necessity is obvious by taking , we obtain and as proposition 5 , we get .

Let and and with . Put . Then and so.

The following inequality is Hermite-Hadamard inequality for s-convex stochastic processes in the second sense.

Proposition 6. Let be a stochastic process , it is integrate on at every point of the interval . Then one has equality

(2.1)

Proof. Let us denote . Then and we have the change of variable

Similarly, let us denote . Then and we have the change of variable

then equality is proved.

Theorem 6. If stochastic process is s-convex in the second sense, and , then one has the inequalities:

(2.2)

As X stochastic process is s-convex in the second sense, we have, for all

Integrating this inequality over on [0,1], we get

As the change of variable gives us that

the second inequality in (2.2) is proved. To prove the first inequality in (2.2), we observe that for all a,b∈I we have

(2.3)

Now, let and with . Then we get by (2.3) that:

for all . Integrating this inequality on and take into accounting that the equality in Proposition 6, we deduce the first part of (2.2).

We are interested in pointing out some properties of this process as in the classical convex stochastic processes.

Theorem 7. Now, suppose that X stochastic process is, and consider the stochastic process

Let stochastic process be a s-convex in the second sense on .

i: H is s-convex stochastic process in the secod sense.

ii: We have the inequality:

(2.4)

iii: We have inequality:

(2.5)

where

And ;

iv. If , then

(2.6)

Proof. i: Let and with . We have succesily

which shows that H stochastic process is s-convex in the second sense.

ii: Suppose that . Then a simple change of variable gives us

where .

Applying the left hand side of Hermite-Hadamard inequality for second sense s-convex stochastic process, we get

Therefore inequality (2.4) is obtained.

iii: Applying the right hand side of Hermite-Hadamard inequality for second sense s-convex stochastic process, we also have

for all .

Note that if , then

Proof.

is true as it is equivalent with

and we know that for , .

On the other hand, it is obvious that

for all and .

Integrating this inequality on [a,b] we get (2.5).

iv: We have

for all .

On the other hand, we know that

(2.7)

and

which gives us that

and the theorem is proved.

Now, suppose that X stochastic process is s - convex in the second sense and integrable on .

Consider the stochastic process

The following theorem contains the main properties of this stochastic process.

Theorem 8. Let s-convex in the second sense stochastic process be integrable on with .

i.

for all and

for all ;

ii: F is s-convex stochastic process in the secod sense on ;

iii: We have the inequality:

(2.8)

iv: We have the inequality

(2.9)

for all .

v: We have the inequality:

(2.10)

for all .

Proof. i. It is obvious owing to integrable properties.

ii.

iii. By the fact that X is s-convex stochastic process in the second sense on . we have

for all and . Integrating this inequality on we have

(2.11)

Since

(2.12)

the (2.11) inequality and (2.12) equality gives us the desired result (2.8).

iv. First of all, let us observe that

Now, for y fixed in , we can consider the stochastic process given by

As shown in the proof of Theorem 3, for we have the inequality

where . Applying Hermite-Hadamard inequality for convex stochastic process in the second sense we get that

for all and . Integrating on over t, we easily deduce

for all . As and by taking into inequality (2.4) , the inequality (2.9) is proved for . If or , then this inequality also holds. We shall omit the details.

v. By the Definition of s-convex stochastic process in the second sense, we have

for all , . Integrating this inequality on , we deduce the first part of the inequality (2.10).

Now, let us observe, by the second part of the Hermite-Hadamard inequality, that

where .

Integrating this inequality on over t, we deduce

A simple calculation shows that

where and similarly,

which gives, by addition, the second inequality in (2.10).

If or , then this inequality also holds. We shall omit the details.

References

[1]	K. Nikodem, on convex stochastic processes, Aequationes Mathematicae 20 (1980) 184-197.
	In article		CrossRef
[2]	A. Skowronski, On some properties of J-convex stochastic processes, Aequationes Mathematicae 44 (1992) 249-258.
	In article		CrossRef
[3]	A. Skowronski, On wright-convex stochastic processes, Annales Mathematicae Silesianne 9(1995) 29-32.
	In article
[4]	D. Kotrys, Hermite-hadamart inequality for convex stochastic processes, Aequationes Mathematicae 83 (2012) 143-151.
	In article		CrossRef
[5]	S.S. Dragomir and S. Fitzpatrick, The Hadamard's inequality for s-convex functions in the second sense, Demonstratio Math., 32 (4) (1999), 687-696.
	In article
[6]	H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Mathematicae, 48 (1994), 100-111.
	In article		CrossRef
[7]	S. Maden, M. Tomar and E. Set, s-convex stochastic processes in the first sense, Pure and Applied Mathematics Letters, in press.
	In article
[8]	D. Kotrys, Hermite-hadamart inequality for convex stochastic processes, Aequationes Mathematicae 83 (2012) 143-151.
	In article		CrossRef