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Hermite-Hadamard Type Inequalities for *s*-Convex Stochastic Processes in the Second Sense

**ERHAN SET**^{1}, **MUHARREM TOMAR**^{1,}, **SELAHATTIN MADEN**^{1}

^{1}Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey

### Abstract

In this study, s-convex stochastic processes in the second sense are presented and some well-known results concerning s-convex functions are extended to s-convex stochastic processes in the second sense. Also, we investigate relation between s-convex stochastic processes in the second sense and convex stochastic processes.

**Keywords:** Hermite Hadamard Inequality, s-convex functions, convex stochastic process, s-convex stochastic process

*Turkish Journal of Analysis and Number Theory*, 2014 2 (6),
pp 202-207.

DOI: 10.12691/tjant-2-6-3

Received September 24, 2014; Revised November 10, 2014; Accepted November 23, 2014

**Copyright**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- SET, ERHAN, MUHARREM TOMAR, and SELAHATTIN MADEN. "Hermite-Hadamard Type Inequalities for
*s*-Convex Stochastic Processes in the Second Sense."*Turkish Journal of Analysis and Number Theory*2.6 (2014): 202-207.

- SET, E. , TOMAR, M. , & MADEN, S. (2014). Hermite-Hadamard Type Inequalities for
*s*-Convex Stochastic Processes in the Second Sense.*Turkish Journal of Analysis and Number Theory*,*2*(6), 202-207.

- SET, ERHAN, MUHARREM TOMAR, and SELAHATTIN MADEN. "Hermite-Hadamard Type Inequalities for
*s*-Convex Stochastic Processes in the Second Sense."*Turkish Journal of Analysis and Number Theory*2, no. 6 (2014): 202-207.

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### 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 **fi**xed 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 | ||