International Journal of Analysis and Applications

Volume 17, Number 5 (2019), 793-802

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-793

HERMITE-HADAMARD TYPE INEQUALITIES FOR m-CONVEX AND

(α,m)-CONVEX STOCHASTIC PROCESSES

SERAP ÖZCAN∗

Department of Mathematics, Faculty of Arts and Sciences, Kırklareli University, 39100 Kırklareli, Turkey

∗Corresponding author: serapozcann@yahoo.com

Abstract. In this paper, the concepts of m-convex and (α,m)-convex stochastic processes are introduced.

Several new inequalities of Hermite-Hadamard type for differentiable m-convex and (α,m)-convex stochastic

processes are established. The results obtained in this work are the generalizations of the known results.

1. Introduction

Stochastic convexity and its applications is of great importance in statistics and probability, because it

provides numerical approximations for existing probabilistic quantities.

In 1980, Nikodem [10] defined convex stochastic processes and investigated their properties. In 1988,

Shaked et al. [16] defined stochastic convexity and gave its applications. In 1992, Skowronski [17] introduced

some new types of convex stochastic processes and obtained some further results on these processes. In

2012, Kotrys [6] extended classical Hermite-Hadamard inequality to convex stochastic processes. In recent

years, there have been many studies on the above mentioned processes. For recent generalizations and

improvements on convex stochastic processes, please refer to [4]- [8], [11]- [15], [19].

Received 2019-05-24; accepted 2019-06-14; published 2019-09-02.

2010 Mathematics Subject Classification. 26D15, 26A51, 60G99.

Key words and phrases. convex stochastic process; m-convex stochastic process; (α,m)-convex stochastic process; mean-

square integral; Hermite-Hadamard type inequality.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

793

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-793


Int. J. Anal. Appl. 17 (5) (2019) 794

2. Preliminaries

Let (Ω,κ,P) be a probability space. A function X : Ω → R is called a random variable if it is κ-

measurable. Let I ⊂ R be an interval. Then, a function X : I × Ω → R is called a stochastic process if for

every t ∈ I the function X (t, ·) is a random variable.

Let P − lim and E [X (t, ·)] denote the limit in probability and the expectation value of random variable

X (t, ·), respectively. Then, a stochastic process X : I × Ω → R is called

(1) continuous in probability in the interval I, if

P − lim
t→t0

X (t, ·) = X (t0, ·)

for all t0 ∈ I.

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

lim
t→t0

E
[
(X (t, ·) −X (t0, ·))

2
]

= 0

for all t0 ∈ I.

(3) mean-square differentiable at a point t ∈ I if there is a random variable X′ (t, ·) : I × Ω → R such

that

X′ (t, ·) = P − lim
t→t0

X (t, ·) −X (t0, ·)
t− t0

.

Let X : I × Ω → R be a stochastic process with E
[
(X (t, ·))2

]
< ∞ for all t ∈ I. Let u = t0 < t1 < t2 <

... < tn = b be a partition of [u,v] if the identity

lim
n→∞

E
[
(
∑
X (Θk) (tk − tk−1) −Y )

2
]

= 0

holds for all normal sequences of partitions of the interval [u,v] and for all

Θk ∈ [tk−1, tk], k = 1, 2, ...,n. Then, we can write

Y (·) =
∫ v
u

X (t, ·) dt (a.e.).

The assumption of the mean-square continuity of the stochastic process X is enough for the mean-square

integral to exist.

Definition 2.1. [10] The stochastic process X : I × Ω → R is convex if for all λ ∈ [0, 1] and u,v ∈ I the

inequality

X (λu + (1 −λ) v, ·) ≤ λX (u, ·) + (1 −λ) X (v, ·) (a.e.) (2.1)

is satisfied. If the inequality (2.1) is assumed only for λ = 1
2

, then the stochastic process X is called Jensen-

convex or 1
2

-convex.

In [6], Kotrys defined convex stochastic processes as following:



Int. J. Anal. Appl. 17 (5) (2019) 795

Theorem 2.1. Let X : I × Ω → R be a Jensen-convex stochastic process and mean-square continuous in

the interval I. Then the following inequality holds for all u,v ∈ I, u < v.

X

(
u + v

2
, ·
)
≤

1

v −u

∫ v
u

X (t, ·) dt ≤
X (u, ·) + X (v, ·)

2
(a.e.). (2.2)

Definition 2.2. [18] Let m ∈ [0, 1]. The function f : [0,c] → R, c > 0, is said to be m-convex, if

f (tx + m (1 − t) y) ≤ tf (x) + m (1 − t) f (y)

is satisfied for every x,y ∈ [0,c] and t ∈ [0, 1].

Definition 2.3. [9] Let α,m ∈ [0, 1]. The function f : [0,c] → R, c > 0, is said to be (α,m)-convex, if

f (tx + m (1 − t) y) ≤ tαf (x) + m (1 − tα) f (y)

is satisfied for every x,y ∈ [0,c] and t ∈ [0, 1].

For further information about m-convex and (α,m)-convex functions, please refer to [1], [2], [5], [13].

Theorem 2.2. [3] Let a,b ∈ R with a < b and let f : [a,b] → R be a differentiable function on (a,b). If

|f′| is convex on [a,b], then∣∣∣∣∣f (a) + f (b)2 − 1b−a
∫ b
a

f (x) dx

∣∣∣∣∣ ≤ (b−a) (|f
′ (a)| + |f′ (b)|)

8
.

Theorem 2.3. [3] Let a,b ∈ R with a < b and let f : [a,b] → R be a differentiable function on (a,b).

Suppose p ∈ R with p > 1. If |f′|q is convex on [a,b] for q ∈ R with q > 1, then∣∣∣∣∣f (a) + f (b)2 − 1b−a
∫ b
a

f (x) dx

∣∣∣∣∣ ≤ b−a2 (p + 1) 1p
[
|f′ (a)|q + |f′ (b)|q

2

]1
q

,

where 1
p

+ 1
q

= 1.

3. Main Results

In order to establish our main results we give the following definitions and lemma:

Definition 3.1. The stochastic process X : [a,b] × Ω → R is said to be m-convex where m ∈ [0, 1], if

X (λu + m (1 −λ) v, ·) ≤ λX (u, ·) + m (1 −λ) X (v, ·)

holds for all u,v ∈ [a,b] and λ ∈ [0, 1].

Definition 3.2. The stochastic process X : [a,b]×Ω → R is said to be (α,m)-convex where (α,m) ∈ [0, 1]2,

if

X (λu + m (1 −λ) v, ·) ≤ λαX (u, ·) + m (1 −λα) X (v, ·)

holds for all u,v ∈ [a,b] and λ ∈ [0, 1].



Int. J. Anal. Appl. 17 (5) (2019) 796

Lemma 3.1. [11] Let X : I◦ ⊆ R × Ω → R be a mean-square differentiable stochastic process on I◦ and

u,v ∈ I◦ with u < v. If X′ is mean-square integrable on [u,v], then the following inequality holds almost

everywhere:

X (u, ·) + X (v, ·)
2

−
1

v −u

∫ v
u

X (t, ·) dt

=
v −u

2

∫ 1
0

(1 − 2λ) X′ (λu + (1 −λ) v, ·) dλ.

Now we obtain results for stochastic processes whose derivatives absolute values raise to some certain

power are m-convex and (α,m)-convex.

Theorem 3.1. Suppose b∗ > 0. Let X : I ⊂ [a,b∗] × Ω → R be a differentiable stochastic process on I◦ and

let X′ be mean-square integrable on [u,v] where u,v ∈ I with u < v. If |X′| is m-convex stochastic process

on [u,v] for m ∈ (0, 1], then the following inequality holds almost everywhere:∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣ ≤ v −u8

[
|X′ (u, ·)| + m

∣∣∣X′( v
m
, ·
)∣∣∣] .

Proof. From Lemma 3.1, we obtain∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

∫ 1
0

|1 − 2λ| |X′ (λu + (1 −λ) v, ·)|dλ.

Since |X′| is m-convex stochastic process on [u,v] for all u,v ∈ I, λ ∈ [0, 1] and m ∈ (0, 1], we have

|X′ ((λu + (1 −λ) v) , ·)| =
∣∣∣X′(λu + m (1 −λ) v

m
, ·
)∣∣∣

≤ λ |X′ (u, ·)| + m (1 −λ)
∣∣∣X′( v

m
, ·
)∣∣∣ .

Hence we have ∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

[
|X′ (u, ·)|

∫ 1
0

|1 − 2λ|(1 −λ) dλ + m
∣∣∣X′( v

m
, ·
)∣∣∣∫ 1

0

|1 − 2λ|λdλ
]
.

Since ∫ 1
0

|1 − 2λ|(1 −λ) dλ =
∫ 1
0

|1 − 2λ|λdλ =
1

4
,

we obtain the desired result. �

Remark 3.1. For m = 1, Theorem 3.1 becomes to Theorem 5 in [11].



Int. J. Anal. Appl. 17 (5) (2019) 797

Theorem 3.2. Suppose b∗ > 0. Let X : I ⊂ [a,b∗] × Ω → R be a differentiable stochastic process on I◦ and

let X′ be mean-square integrable on [u,v] where u,v ∈ I with u < v. If |X′|q is m-convex stochastic process

on [u,v] for q > 1 and 1
p

+ 1
q

= 1, then the following inequality holds almost everywhere:

∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2 (p + 1)
1
p

[
|X′ (u, ·)|q + m

∣∣X′( v
m
, ·
)∣∣q

2

]1
q

. (3.1)

Proof. By Lemma 3.1 and using well known Hölder’s inequality, we have∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

(∫ 1
0

|1 − 2λ|p dλ
)1

p
(∫ 1

0

|X′ (λu + (1 −λ) v, ·)|dλ
)1

q

. (3.2)

Since |X′|q is m-convex stochastic process on [u,v] for all u,v ∈ I with u < v, λ ∈ [0, 1] and m ∈ (0, 1], we

have

|X′ (λu + (1 −λ) v, ·)|q ≤ λ |X′ (u·)|q + m (1 −λ)
∣∣∣X′( v

m
, ·
)∣∣∣q .

Thus we obtain

∫ 1
0

|X′ (λu + (1 −λ) v, ·)|q dλ ≤
∫ 1
0

[
λ |X′ (u, ·)|q + m (1 −λ)

∣∣∣X′( v
m
, ·
)∣∣∣q]dλ

=
1

2
|X′ (u, ·)|q +

m

2

∣∣∣X′( v
m
, ·
)∣∣∣q . (3.3)

Moreover, since

∫ 1
0

|1 − 2λ|p dλ =
∫ 1/2
0

(1 − 2λ)p dλ +
∫ 1
1/2

(2λ− 1)p dλ =
1

p + 1
, (3.4)

utilizing inequalities (3.3) and (3.4) in (3.2), we get the inequality (3.1). �

Remark 3.2. For m = 1, Theorem 3.2 becomes to Corollary 6 in [11].

Theorem 3.3. Suppose b∗ > 0. Let X : I ⊂ [a,b∗] × Ω → R be a differentiable stochastic process on I◦ and

let X′ be mean-square integrable on [u,v] where u,v ∈ I with u < v. If |X′|q is m-convex stochastic process

on [u,v] for m ∈ (0, 1], q ≥ 1, then the following inequality holds almost everywhere :∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

4

[
|X′ (u, ·)|q + m

∣∣X′( v
m
, ·
)∣∣q

2

]1
q

. (3.5)



Int. J. Anal. Appl. 17 (5) (2019) 798

Proof. For q = 1, the proof is the same as that of Theorem 3.1. Suppose that q > 1. From Lemma 3.1 and

using well known power-mean inequality, we have∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

(∫ 1
0

|1 − 2λ|dλ
)1−1

q
(∫ 1

0

|1 − 2λ| |X′ (λu + (1 −λ) v, ·)|q dλ
)1

q

.

(3.6)

Using m-convexity of the stochastic process |X′|q on [u,v] in the second integral on the right side of the

inequality (3.6), we have∫ 1
0

|1 − 2λ| |X′ (λu + (1 −λ) v, ·)|q dλ

≤
∫ 1
0

|1 − 2λ|
[
λ |X′ (u, ·)|q + m (1 −λ)

∣∣∣X′( v
m
, ·
)∣∣∣q]dλ

= |X′ (u, ·)|q
∫ 1
0

λ |1 − 2λ|dλ + m
∣∣∣X′( v

m
, ·
)∣∣∣q ∫ 1

0

(1 −λ) |1 − 2λ|dλ

=
1

4
|X′ (u, ·)|q +

m

4

∣∣∣X′( v
m
, ·
)∣∣∣q .

A usage of the last inequality in (3.6) gives the desired result. �

Remark 3.3. For q = 1, the inequality (3.5) reduces to the inequality proved in Theorem 3.1. If q =

p
p−1 (p > 1), then one has 4

p > p + 1 and so 1
4
< 1

2(p+1)
1
p

. This shows that the inequality (3.5) is better than

the one given by (3.1) in Theorem 3.2.

Now we establish our results for (α,m)-convex stochastic processes.

Theorem 3.4. Suppose b∗ > 0. Let X : I ⊂ [a,b∗] × Ω → R be a differentiable stochastic process on I◦

and let X′ be mean-square integrable on [u,v] where u,v ∈ I with u < v. If |X′| is (α,m)-convex stochastic

process on [u,v] for m ∈ (0, 1], q ≥ 1, then the following inequality holds almost everywhere:∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

[
M1 |X′ (u, ·)| + mM2

∣∣∣X′( v
m
, ·
)∣∣∣] (3.7)

where

M1 =
1 + α2α

2α (1 + α) (2 + α)
, (3.8)

M2 =
1

2
−M1. (3.9)



Int. J. Anal. Appl. 17 (5) (2019) 799

Proof. From Lemma 3.1, we have ∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

∫ 1
0

|1 − 2λ| |X′ (λu + (1 −λ) v, ·)|dλ. (3.10)

Since |X′| is (α,m)-convex stochastic process on [u,v] for all u,v ∈ I with u < v, (α,m) ∈ (0, 1]2 and

λ ∈ [0, 1], we have ∫ 1
0

|1 − 2λ| |X′ (λu + (1 −λ) v, ·)|dλ

≤ |X′ (u, ·)|
∫ 1
0

|1 − 2λ|λαdλ + m
∣∣∣X′( v

m
, ·
)∣∣∣∫ 1

0

|1 − 2λ|(1 −λα) dλ

= M1 |X′ (u, ·)| + m
(

1

2
−M1

)∣∣∣X′( v
m
, ·
)∣∣∣ (3.11)

where ∫ 1
0

|1 − 2λ|λαdλ =
1 + α2α

2α (1 + α) (2 + α)
= M1,

and ∫ 1
0

|1 − 2λ|(1 −λα) dλ =
1

2
−

1 + α2α

2α (1 + α) (2 + α)
=

1

2
−M1 = M2.

Using the inequality (3.11) in the inequality (3.10), we get the required result. �

Remark 3.4. For (α,m) = (1, 1), Theorem 3.4 becomes to Theorem 5 in [11].

Theorem 3.5. Suppose b∗ > 0. Let X : I ⊂ [a,b∗] × Ω → R be a differentiable stochastic process on I◦

and let X′ be mean-square integrable on [u,v] where u,v ∈ I with u < v. If |X′|q is (α,m)-convex stochastic

process on [u,v] for (α,m) ∈ (0, 1]2, q ≥ 1, then the following inequality holds almost everywhere:∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2 (p + 1)
1
p

[
α |X′ (u, ·)|q + m

∣∣X′( v
m
, ·
)∣∣q

1 + α

]1
q

(3.12)

where 1
p

+ 1
q

= 1.

Proof. Using Lemma 3.1 and Hölder’s inequality, we have∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

(∫ 1
0

|1 − 2λ|p dλ
)1

p
(∫ 1

0

|X′ (λu + (1 −λ) v, ·)|q dλ
)1

q

. (3.13)

By (α,m)-convexity of the stochastic processes |X′|q on [u,v], we have for every λ ∈ [0, 1]

|X′ ((λu + (1 −λ) v) , ·)|q ≤ λα |X′ (u, ·)|q + m(1 −λα)
∣∣∣X′( v

m
, ·
)∣∣∣q



Int. J. Anal. Appl. 17 (5) (2019) 800

for (α,m) ∈ (0, 1]2. Hence

∫ 1
0

|X′ (λu + (1 −λ) v, ·)|

≤ |X′ (u, ·)|q
∫ 1
0

λαdλ + m
∣∣∣X′( v

m
, ·
)∣∣∣q ∫ 1

0

(1 −λα) dλ

=
1

1 + α
|X′ (u, ·)|q +

mα

1 + α

∣∣∣X′( v
m
, ·
)∣∣∣q .

Utilizing of the above inequality in (3.13) and the fact

∫ 1
0

|1 − 2λ|p dλ =
1

p + 1

completes the proof. �

Remark 3.5. For (α,m) = (1, 1), Theorem 3.5 becomes to Corollary 6 in [11].

Theorem 3.6. Suppose b∗ > 0. Let X : I ⊂ [a,b∗] × Ω → R be a differentiable stochastic process on I◦

and let X′ be mean-square integrable on [u,v] where u,v ∈ I with u < v. If |X′|q is (α,m)-convex stochastic

process on [u,v] for (α,m) ∈ (0, 1]2, q ≥ 1, then the following inequality holds almost everywhere:

∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

(
1

2

)1−1
q [
M1 |X′ (u, ·)|

q
+ mM2

∣∣∣X′( v
m
, ·
)∣∣∣q]1q ,

where

M1 =
1 + α2α

2α (1 + α) (2 + α)
,

M2 =
1

2
−M1.

Proof. For q = 1, the proof is similar to that of Theorem 3.4. Now suppose that q > 1. Using Lemma 3.1

and power-mean inequality, we have

∣∣∣∣X (u, ·) + X (v, ·)2 − 1v −u
∫ v
u

X (t, ·) dt
∣∣∣∣

≤
v −u

2

(∫ 1
0

|1 − 2λ|dλ
)1−1

q
(∫ 1

0

|1 − 2λ| |X′ (λu + (1 −λ) v, ·)|q dλ
)1

q

.

(3.14)



Int. J. Anal. Appl. 17 (5) (2019) 801

Since |X′|q is (α,m)-convex stochastic process on [u,v] for every λ ∈ [0, 1] and (α,m) ∈ (0, 1]2, we have∫ 1
0

|1 − 2λ| |X′ (λu + (1 −λ) v, ·)|q dλ

≤
∫ 1
0

|1 − 2λ|
[
λα |X′ (u, ·)|q + m (1 −λα)

∣∣∣X′( v
m
, ·
)∣∣∣q]dλ

= |X′ (u, ·)|q
∫ 1
0

|1 − 2λ|λαdλ +
∣∣∣X′( v

m
, ·
)∣∣∣q ∫ 1

0

|1 − 2λ|(q −λα) dλ

= M1 |X′ (u, ·)|
q

+ M2

∣∣∣X′( v
m
, ·
)∣∣∣q . (3.15)

Using the inequality (3.15) in the inequality (3.14) we get the desired result. �

References

[1] M. K. Bakula, M. E. Özdemir and J. Pec̆arić, Hadamard type inequalities for m-convex and (α,m)-convex functions, J.

Inequal. Pure Appl. Math., 9(4) (2008), Article 96.

[2] M. K. Bakula, J. Pec̆arić and M. Ribićić, Companion inequalities to Jensen’s inequality for m-convex and (α,m)-convex

functions, J. Inequal. Pure Appl. Math., 7(5) (2006), Article 194.

[3] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real

numbers and trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91–95.

[4] L. Gonzalez, N. Merentes and M. Valera-Lopez, Some estimates on the Hermite-Hadamard inequality through convex and

quasi-convex stochastic processes, Math. Eterna, 5(5) (2015), 745–767.

[5] İ. İşcan, H. Kadakal and M. Kadakal, Some new integral inequalities for functions whose nth derivatives in absolute value

are (α,m)-convex functions, New Trends Math. Sci., 5(2) (2017), 180–185.

[6] D. Kotrys, Hermite-Hadamard inequality for convex stochastic processes, Aequationes Math., 83 (2012), 143–151.

[7] D. Kotrys, Remarks on strongly convex stochastic processes, Aequationes Math., 86 (2013), 91–98.

[8] L. Li and Z. Hao, On Hermite-Hadamard inequality for h-convex stochastic processes, Aequationes Math., 91 (2017),

909–920.

[9] V. G. Miheşan, A generalization of the convexity, Seminer on Functional Equations, Approximation and Convexity, Cluj-

Napoca, Romania, 1993.

[10] K. Nikodem, On convex stochastic processes, Aequationes Math., 20 (1980), 184–197.

[11] N. Okur, İ. İşcan and E. Yuksek Dizdar, Hermite-Hadamard type inequalities for p-convex stochastic processes, Int. J.

Optim. Control, Theor. Appl., 9(2) (2019), 148–153.

[12] M. Z. Sarıkaya, H. Yaldız and H. Budak, Some integral inequalities for convex stochastic processes, Acta Math. Univ.

Comenianae, 85 (2016), 155–164.

[13] E. Set, M Sardari, M. E. Özdemir and J. Rooin, On generalizations of the Hadamard inequality for (α,m)-convex functions,

Kyungpook Math. J., 52 (2012), 307–317.

[14] E. Set, M. Tomar and S. Maden, Hermite-Hadamard type inequalities for s-convex stochastic processes in the second sense,

Turk. J. Anal. Numb. Theory, 2(6) (2016), 202–207.

[15] E. Set, M. Z. Sarıkaya and M. Tomar, Hermite-Hadamard type inequalities for coordinates convex stochastic processes,

Math. Aeterna, 5(2) (2015), 363–382.

[16] M. Shaked and J. G. Shanthikumar, Stochastic convexity and its applications, Adv. Appl. Probab., 20 (1988), 427–446.

[17] A. Skowronski, On some properties of J-convex stochastic processes, Aequationes Math., 44 (1992), 249–258.



Int. J. Anal. Appl. 17 (5) (2019) 802

[18] G. Toader, Some generalizations of the convexity, Proc. Colloq. Approx. Optim., Univ. Cluj-Napoca, Cluj-Napoca, Roma-

nia, (1985), 329–338.

[19] M. Tomar, E. Set and S. Maden, Hermite-Hadamard type inequalities for log-convex stochastic processes, J. New Theory,

2 (2015), 23–32.


	1. Introduction
	2. Preliminaries
	3. Main Results
	References