Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 6, 202-207
Available online at http://pubs.sciepub.com/tjant/2/6/3
© Science and Education Publishing
DOI:10.12691/tjant-2-6-3
Hermite-Hadamard Type Inequalities for s-Convex
Stochastic Processes in the Second Sense
ERHAN SET, MUHARREM TOMAR*, SELAHATTIN MADEN
Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey
*Corresponding author: [email protected]
Received September 24, 2014; Revised November 10, 2014; Accepted November 23, 2014
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
Cite This Article: ERHAN SET, 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, vol. 2, no. 6 (2014): 202-207. doi: 10.12691/tjant-2-6-3.
where E [ X (t ,) ] denotes the expectation value of the
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 (Ω, A, P) be an arbitrary
probability space. A function X : Ω →  is called a
random value if it is A - measurable. A function
X : I × Ω →  , where I ⊂  is an interval, is called a
stochastic process if for every t ∈ I the function X (t ,) is
a random variable.
Recall that the stochastic process X : I × Ω →  is
called
(1) continuous in probability in interval I , if for all
t0 ∈ I
P − lim X (t ,) =
X (t0 ,)
t →t0
where P - lim denotes the limit in probability;
(2) mean - square continuous in the interval I, if for all
t0 ∈ I
P − lim E [ X (t ,) − X (t0 ,) ] =
0
t →t0
random variable X (t ,) ;
(3) increasing (decreasing) if for all t , s ∈ I such
that t < s ,
X (t ,) ≤ X ( s,), ( X (t ,) ≥ X ( s,) )
(4) monotonic if it is increasing or decreasing;
(5) differentiable at a point t ∈ I if there is a random
variable X ′(t ,) : Ω → 
X '(t ,)= P − lim
t →t0
X (t ,) − X (t0 ,)
t − t0
We say that a stochastic process X : I × Ω →  is
continuous (differentiable) if it is continuous
(differentiable) at every point of the interval I.
Definition 2. [2,3,8] Let (Ω, A, P) be a probability
space and I ⊂  be an interval. We say that a stochastic
process X : I × Ω →  is
(1) convex if
X ( λ s + (1 − λ ) t , ) ≤ λ X ( s,) + (1 − λ ) X (t ,)
if all s, t ∈ I and λ ∈ [ 0,1] . This class of stochastic
process are denoted by C.
(2) λ -convex (where λ is a fixed number from (0,1)) if
X ( λ s + (1 − λ ) t , ) ≤ λ X ( s,) + (1 − λ ) X (t ,)
if all s, t ∈ I and λ ∈ (0,1) . This class of stochastic
process are denoted by Cλ .
(3) Wright-convex if
X ( λ s + (1 − λ ) t , ) + X ( (1 − λ ) s + λ t , ) ≤ X ( s,) + X (t ,)
Turkish Journal of Analysis and Number Theory
203
if all s, t ∈ I and λ ∈ [ 0,1] . This class of stochastic
where
process are denoted by W.
(4) Jensen-convex if
1 b
s  a+b 
f ( x ) dx + (1 − t ) f 
, 
∫
a
b−a
 2

a
+
b
a+b 



X  tx + (1 − t )
,  + X  tb + (1 − t )
, 
2
2



H 2 (t ) = 
s +1
=
H1 ( t ) t s
 s + t  X ( s, ) + X ( t , )
X
,  ≤
2
 2 
This class of stochastic process are denoted by C 1 .
and t ∈ (0,1] ;
 ( t , ) : max { H ( t ) , H ( t )} , t ∈ [ 0,1] , then
Let 0 < s ≤ 1 . A function f :  + →  where =
iv: If H
1
2
=
[0,
∞
)
,
is
said
to
be
s-convex
function
in
the
second
+
 ( t ) ≤ t s f ( a ) + f ( b ) + (1 − t ) s 2 f  a + b  , t ∈ [ 0,1]
H


sense if:
s +1
s +1  2 
Theorem 4. [6] Let f : I ⊆  + →  + be a s-convex
(1.1)
f (α u + β v ) ≤ α s f ( u ) + β s f ( v )
mapping in the second sense, s ∈ (0,1] , a, b ∈ I
for all u, v ≥ 0 with α + β = 1. This class of functions
with a < b on [ a, b ] .
are denoted by K s2 . It can be easily seen that for s = 1 s1

1

convexity in the first sense reduce to ordinary convexity
F  s + = F  − s 
of functions defined on  + .
2

2

Throughout this paper, let I be an interval on  + .
 1
for all s ∈ 0,  and
Now, we present some theorems which was proved by
 2
Dargomir and Fitzpatrick [5] and H. Hudzik and L.
Maligranda [6] about some inequalities of HermiteF (=
t ) F (1 − t )
Hadamard type and basic properties for the s-convex
for all t ∈ [ 0,1] ;
functions in the second sense.
2
Proposition 1. [6] If f ∈ K s2 , then f is non-negatif on
[0, ∞) .
∈ K s2
Theorem 1. [6] Let f
. Then inequality (1.1)
holds for all u , v ∈  + and α , β ≥ 0 with α + β ≤ 1 if and
only if f ( 0 ) = 0 .
+
+
mapping in the second sense, s ∈ ( 0,1) and a, b ∈  with
a < b . If f ∈ L1 [ a, b ] , then one has the inequalities:
f ( a ) + f (b )
1 b
 a+b
2 s −1 f 
f ( x ) dx ≤
. (1.2)
≤
∫
a
s +1
 2  b−a
Theorem 3. [6] Now, suppose that f is Lebesque
integrable on [a,b] and consider the mapping
H : [ 0,1] →  given by
1 b 
a+b
f  tx + (1 − t )
 dx
∫
a
b−a
2 

Let f : I ⊆  + →  be a s-convex mapping in the
second sense on I, s ∈ (0,1] and Lebesque integrable on
[ a, b] ⊆ I , a < b . Then:
i: H is s-convex in the second sense on [0,1],
ii: We have the inequality:
 a+b
H ( t ) ≥ 2 s −1 f 

 2 
for all t ∈ [ 0,1] .
iii: We have inequality:
H ( t ) ≤ min { H1 ( t ) , H 2 ( t )} , t ∈ [ 0,1]
iii. We have the inequality:
1
1
21− s F ( t ) ≥ F   =
2
  ( b − a )2
b b
 x+ y
 dxdy,
2 
∫a ∫a F 
t ∈ [ 0,1]
+
Theorem 2. [5] Suppose that f :  →  is s-convex
=
H (t ) :
ii. F is s-convex in the second sense on [ 0,1] ;
iv. We have the inequality
 a+b
F ( t ) ≥ 2 s −1 H ( t ) ≥ 4 s −1 f 

 2 
for all t ∈ [ 0,1] .
v. We have the inequality:
b
 s

s 1
 t + (1 − t )  b − a ∫a f ( x ) dx, 




F ( t ) ≤ min  f ( a ) + f ( ta + (1 − t ) b )

 + f ( (1 − t ) a + tb ) + f ( b )



2


s
1
+
(
)


for all t ∈ [ 0,1] .
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. X : I × Ω →  stochastic process where
I ⊂  be an interval, is said to be s-convex in the second
sense if:
Turkish Journal of Analysis and Number Theory
204
X (α u + β v,)
X ( λ u + (1 − λ ) v, ) ≤ λ s X ( u , ) + (1 − λ ) X ( v, )
s
= X (aγ u + bγ v) ≤ a s X (γ u ,) + b s X (γ v,)
for all u , v ≥ 0 and s fixed in (0,1] . We denote this class
= a s X (γ u + (1 − γ )0,) + b s X (γ v + (1 − γ )0,)
of stochastic process by Cs2 .
≤ a s γ s X (u ,) + b s γ s X (v,) + (1 − γ ) s (a s + b s ) X (0,)
Remark 1. It can be easily seen that for s = 1 sconvexity in the second sense for stochastic processes
= a s γ s X (u ,) + b s γ sX (,)
reduce to ordinary convexity of stochastic processes
defined in Definition 2.
≤ a s X (u ,) + b s X (,)
2
Proposition 2. C ⊆ Cs for all λ ∈ [ 0,1] .
The following inequality is Hermite-Hadamard
Proof. To prove the proposition assume that X ∈ C and
inequality for s-convex stochastic processes in the second
take arbitrary points u , v ∈ I , s ∈ (0,1] , Since X is convex
sense.
Proposition 6. Let be a stochastic process
s
stochastic process and λ ≤ λ on [ 0,1] we have
X : I × Ω →  , it is integrate on at every point of the
interval ( 0,1) × Ω . Then one has equality
X λ u + (1 − λ ) v, ≤ λ X ( u , ) + (1 − λ ) X ( v, )
(
)
s
≤ λ s X ( u , ) + (1 − λ ) X ( v, )
1
Then X ∈ Cs2 .
Proposition 3.
C 1 ⊆ Cs2
2
Proof. Let us denote t * = tu + (1 − t ) v, t ∈ [ 0,1] . Then
for all λ ∈ (0,1) .
dt=
*
Proof. To prove the proposition assume that X ∈ C 1
2
and take arbitrary points u , v ∈ I , s ∈ (0,1] , Since X is
Jensen-convex stochastic process and
s
1 1
≤   we have
2 2
 u + v  X ( u , ) + X ( v, ) X ( u , ) + X ( v, )
≤
X
,  ≤
2
 2

2s
Then X
∈ Cs2
1
) dt ∫0 X ( (1 − t ) u + tv, ) dt. (2.1)
∫0 X ( tu + (1 − t ) v,=
( u − v ) dt
1
1
1 v
X ( t*, ) dt *.
v − u ∫u
=
Similarly, let us denote k * = ( (1 − t ) u + tv, ) , t ∈ [ 0,1] .
1
Proposition 5. If X ∈ Cs2 , then X is non-negative on I.
Proof. We have , for u ∈ I , s ∈ (0,1]
= 21− s X ( u , )
)
X (α u + β v, ) ≤ α s X ( u , ) + β s X ( v, )
Then
holds
inequality
for
all
u , v ∈ I and α , β ≥ 0 with α + β ≤ 1 if and only if
X(0,) = 0 .
Proof.
Necessity
is
obvious
by
taking
u= v= α= β= 0 , we obtain X(0,) ≤ 0 and as
X(0,) ≥ 0 proposition 5 , we get X(0,) = 0 .
Let u , v ∈ I and α , β ≥ 0 and with 0 < γ = α + β ≤ 1 .
Put a
=
α
β
α β
. Then a + b =
=
and b
+ and so.
γ
γ
γ γ
)
, dk *
1 v
u+v 
2 s −1 X 
,  ≤
∫ X ( t , ) dt
 2
 v−u u
X ( u , ) + X ( v, )
.
≤
s +1
(2.2)
As X stochastic process is s-convex in the second sense,
we have, for all t ∈ [ 0,1]
Hence, 21− s − 1 X ( u , ) ≥ 0 and so X ( u , ) ≥ 0 .
.
*
Theorem 6. If X : I × Ω →  + stochastic process is sconvex in the second sense, s ∈ (0,1) and u , v ∈ I , then
one has the inequalities:
1  X ( u , ) X ( u , )
1
+
X ( u , ) = X  u + u ,  ≤
2
2 

2s
2s
X ∈ Cs2
v
then equality is proved.
2
Let
and we have the change of variable
1
Proof. The proof is cleared owing to [3, Proposition 3]
and Proposition 3.
5.
( u − v ) dt
X (k
∫0 X ( (1 − t ) u + tv, ) dt =
v − u ∫u
Proposition 4. C ⊂ Cλ ⊂ C 1 ⊂ Cs2 for all λ ∈ (0,1) .
Theorem
u
X ( t*, ) dt *
∫0 X ( tu + (1 − t ) v, ) dt =
u − v ∫v
Then dk=
*
.
(
and we have the change of variable
X (α u + (1 − α ) v, ) ≤ α s X ( u , ) + (1 − α ) X ( v, )
s
Integrating this inequality over α on [0,1], we get
1
∫0 X (α u + (1 − α ) v, ) dα
1
1
0
0
s
≤ ∫ α s X ( u , ) dα + ∫ (1 − α ) X ( v, ) dα
1
1
0
0
s
= X ( u , ) ∫ α s dt + X ( v, ) ∫ (1 − α ) dα
=
X ( u , ) + X ( v , )
s +1
Turkish Journal of Analysis and Number Theory
H ( α x + β y , )
As the change of variable t = α u + (1 − α ) v gives us
that
1
1
v
=
∫0 X (α u + (1 − α ) v, ) dt =v − u ∫u X ( t , ) dt ,
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
 a + b  X (α , ) + X ( b, )
X
,  ≤
 2

2s
(2.3)
Now, let a = tu + (1 − t ) v and b =(1 − t ) u + tv with
t ∈ [ 0,1] . Then we get by (2.3) that:
205
u+v 
1 v 
X  (α x + β y ) t + 1 − (α x + β y ) 
,  dt
v − u ∫u 
2

u +v
 

 α xt + (1 − x )

2 
1 v  
 dt
=
X
v − u ∫u 
u + v 

, 
 + β  yt + (1 − y )
2  


 s 

u + v
 α X  xt + (1 − x )


2
1 v 


 dt
≤
X
v − u ∫u  s 
u + v 
, 
 + β X  yt + (1 − y )
2  


s
s
 u + v  X ( tu + (1 − t ) v, ) + X ( (1 − t ) u + tv, ) = α H ( x, ) + β H ( y, )
X
,  ≤
 2

2s
which shows that H stochastic process is s-convex in the
second sense.
for all t ∈ [ 0,1] . Integrating this inequality on [ 0,1] and
ii: Suppose that t ∈ (0,1] . Then a simple change of
take into accounting that the equality in Proposition 6, we
u+v
deduce the first part of (2.2).
gives us
variable θ = α t + (1 − α )
2
We are interested in pointing out some properties of
this process as in the classical convex stochastic processes.
u +v
α v +(1−α )
1
Theorem 7. Now, suppose that X stochastic process is,
2 X (θ , ) dθ
H ( t , ) =
α ( v − u ) ∫α u +(1−α ) u + v
s ∈ (0,1] and consider the stochastic process
2
H ( t , ) : [ 0,1] × Ω →  +
p
1
=
X (θ , ) dθ
p − q ∫q
1 v 
u+v 
, dt
H (α , ) :
X  α t + (1 − α )
=
2
v − u ∫u 

u+v
u+v
where p = α v + (1 − α )
.
and q = α v + (1 − α )
2
2
Let X : I × Ω →  + stochastic process be a s-convex in
Applying the left hand side of Hermite-Hadamard
the second sense on I × Ω, s ∈ ( 0,1] .
inequality for second sense s-convex stochastic process,
i: H is s-convex stochastic process in the secod sense.
we get
ii: We have the inequality:
p
1
u+v 
s −1  p + q 
2 s −1 X 
, 
∫q X (θ , ) dθ ≥ 2 X  2 ,  =
s −1  u + v 
p
−
q
 2

(2.4)
H ( t , ) ≥ 2 X 
, 
 2

Therefore inequality (2.4) is obtained.
iii: We have inequality:
iii: Applying the right hand side of Hermite-Hadamard
inequality for second sense s-convex stochastic process,
(2.5)
H ( t , ) ≤ min { H1 ( t ) , H 2 ( t )} , t ∈ [ 0,1]
we also have
where
1 v
u+v 
s
=
H1 ( t , ) t
X ( t , ) dt + (1 − t ) X 
, 
∫
u
v−u
 2

u+v 

X  α u + (1 − α )
, 
2


s
u+v 

+ X  α v + (1 − α )
, 
2


H 2 ( t , ) =
s +1
(2.6)
Proof. i: Let t1 , t2 ∈ [ 0,1] and α , β ≥ 0 with α + β =
1.
We have succesily
u+v 
u+v 


X  α u + (1 − α )
,  + X  α v + (1 − α )
, 
2
2



= 
s +1
= H 2 ( t , )
for all t ∈ [ 0,1] .
And t ∈ (0,1] ;
 ( t , ) : max { H ( t ) , H ( t )} , t ∈ [ 0,1] , then
iv. If H
=
1
2
 ( t , ) ≤ t s X ( u , ) + X ( v, )
H
s +1
u+v 
s 2
+ (1 − t )
X
,  , t ∈ [ 0,1]
s +1  2

X ( p, ) + X ( q, )
p
1
X (θ , ) dθ ≤
p − q ∫q
s +1
Note that if t = 0 , then
Proof.
2
u+v 
u+v 
X
X
,  =H ( 0, ) ≤ H 2 ( 0, ) =
, 
s +1  2
 2


is true as it is equivalent with
u+v 
,  ≤ 0
 2

( s − 1) X 
u+v 
and we know that for s ∈ (0,1] , X 
,  ≥ 0 .
 2

Turkish Journal of Analysis and Number Theory
206
iv: We have the inequality
On the other hand, it is obvious that
X (α t + (1 − α )
u+v
u+v
,) ≤ α s X ( t , ) + (1 − α ) s X (
, )
2
2
for all t ∈ [ 0,1] and t ∈ I .
Integrating this inequality on [a,b] we get (2.5).
iv: We have
s
u+v 
, 
F (α , ) ≥ 2 s −1 H (α , ) ≥ 4 s −1 X 
 2

for all α ∈ [ 0,1] .
v: We have the inequality:
v
 s

s 1
 α + (1 − α )  v − u ∫u X ( s, ) ds 


 X ( u , ) + X (α u + (1 − α ) v, )

F (α , ) ≤ min 
 (2.10)
 + X ( (1 − α ) u + α v, ) + X ( v, ) 


2


s
+
1
(
)


u+v 
, 
 2

u+v 
X
, 
 2

α s X ( u, ) + (1 − α ) X 
H 2 ( t , ) ≤
= αs
+α s X ( v, ) + (1 − α )
X ( u , ) + X ( v , )
s +1
s
s +1
+ (1 − α )
s
2
u+v 
X
, 
s +1  2

for all α ∈ [ 0,1] .
Proof. i. It is obvious owing to integrable properties.
ii.
for all t ∈ [0,1] .
On the other hand, we know that
F ( α x + β y , )
1
X ( u , ) + X ( v, )=
1 v
(2.7)
X ( t , ) dt ≤
∫
v −u u
s +1
and
u+v 
u+v 
s 2
,  ≤ (1 − α )
X
,  , t ∈ [ 0,1].
s +!  2
 2


v v
(v − u )
=
(1 − α )s X 
(2.9)
∫ ∫
2 u u
(
)
X (α x + β y ) s + (1 − (α x + β y ) ) t , dsdt
v v  α ( xs + (1 − x ) t , ) 
 dsdt
X
∫
2 u ∫u
 + β ( ys + (1 − y ) t , ) 
v−u
1
(
)




v v α X ( xs + (1 − x ) t , )
∫u ∫u + β s X ys + 1 − y t ,  dsdt
( ( ) )

s
1
≤
which gives us that
( v − u )2
X
u
,

+
X
v
,

( ) ( )
u+v 
s 2
H1 ( t , ) ≤ α s
+ (1 − α )
X
, 
α s F ( x , ) + β s F ( y , )
s +1
s +1  2 =

and the theorem is proved.
Now, suppose that X stochastic process is s - convex in
the second sense and integrable on [u , v ] × Ω .
Consider the stochastic process
F (α , ) :
=
1
v v
(v − u )
∫ ∫
2 u u
X (α s + (1 − α ) t , ) dsdt , α ∈ [ 0,1] .
The following theorem contains the main properties of
this stochastic process.
Theorem 8. Let X : I × Ω →  + s-convex in the second
sense
stochastic
process
be
integrable
on
I × Ω, s ∈ ( 0,1] , u , v ∈ I with u < v .
i.
1 

1

F  x + ,  = F  − x, 
2 

2

 1
for all s ∈ 0,  and
 2
F (α=
, ) F (1 − α , )
for all α ∈ [ 0,1] ;
ii: F is s-convex stochastic process in the secod sense
on [ 0,1] × Ω ;
iii: We have the inequality:
iii. By the fact that X is s-convex stochastic process in
the second sense on I × Ω . we have
 u + v  X ( tu + (1 − t ) v, ) + X ( (1 − t ) u + tv, )
X
,  ≤
 2

2s
for all t ∈ [ 0,1] and u , v ∈ [ a, b ] . Integrating this
2
inequality on [ a, b ] we have
b b
u+v 
, dudv
2

∫a ∫a X 
 b b

1  ∫a ∫a X ( tu + (1 − t ) v, ) dudv + 
≤

.
2 s  b b X ( (1 − t ) u + tv, ) dudv 
 ∫a ∫a

(2.11)
Since
b b
∫a ∫a X ( tu + (1 − t ) v, ) dudv =
b b
∫a ∫a X ( (1 − t ) u + tv, ) dudv
(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

v 1
v
1
F ( α , ) :
X (α s + (1 − α ) t , ) ds  dt.
=

∫
∫
1
( v − u ) u  ( v − u ) u

21− s F (α , ) ≥ F ( ,)
2
(2.8)
Now, for y fixed in [u , v] , we can consider the
v v
s+t
1

∈
F
dsdt
α
(
,
)
,
[0,1]
2 ∫u ∫u
stochastic process H t : [ 0,1] × Ω →  given by
2
(v − u )
Turkish Journal of Analysis and Number Theory
=
H t ( α , ) :
Integrating this inequality on [u; v ] over t, we deduce
1 v
X (α s + (1 − α ) t , ) ds.
v − u ∫u
 1 v

X (α v + (1 − α ) t , ) dt 
∫

u
1 v−u
F ( α , ) ≤

.
s +1  1 v
+
X α u + (1 − α ) s, ) ds 
 v − u ∫u (

As shown in the proof of Theorem 3, for α ∈ [ 0,1] we
have the inequality
H t (α , ) :=
207
p
1
X (θ , ) dθ
∫
q
p−q
A simple calculation shows that
1 v
X (α v + (1 − α ) t , ) dt
v − u ∫u
Hermite-Hadamard inequality for convex stochastic
X ( r , ) + X ( l , )
1 r
process in the second sense we get that
X (θ , ) dθ ≤
=
∫
l
r −1
s +1
p
1
s −1  p + q 


≥
θ
θ
X
d
X
,
2
,
(
)
X ( u , ) + X (α v + (1 − α ) u , )


p − q ∫q
2
 =

, α ∈ ( 0,1) ,
s +1

s −1  u + v
= 2 X α
+ (1 − α ) t , 
where r = v, l = α v + (1 − α ) u , α ∈ ( 0,1) ; and similarly,
2


p = α v + (1 − α ) t , q = α u + (1 − α ) s .
where
Applying
for all α ∈ ( 0,1) and t ∈ [u , v ] . Integrating on [u , v ] over
1 v
X (α u + (1 − α ) t , ) dt
v − u ∫u
X ( u , ) + X (α u + (1 − α ) v, )
≤
, α ∈ ( 0,1) ,
s +1
t, we easily deduce
F (α , ) ≥ 2 s −1 H ( (1 − α ) , )
for all α ∈ ( 0,1) . As F (α=
, ) F (1 − α , ) and by taking
into inequality (2.4) , the inequality (2.9) is proved for
α ∈ ( 0,1) . If t = 0 or t = 1 , 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
X (α u + (1 − α ) v, ) ≤ α s X ( u , ) + (1 − α ) X ( v, )
s
for all u , v ∈ [ a, b ] , α ∈ [ 0,1] . Integrating this inequality
2
on [u , v ] , we deduce the first part of the inequality (2.10).
Now, let us observe, by the second part of the HermiteHadamard inequality, that
H t (α , ) :=
≤
p
1
X (θ , ) dθ
p − q ∫q
which gives, by addition, the second inequality in (2.10).
If t = 0 or t = 1 , then this inequality also holds. We
shall omit the details.
References
[1]
[2]
[3]
[4]
[5]
[6]
X (α v + (1 − α ) t , ) + X (α u + (1 − α ) s, )
[7]
s +1
[8]
where p = α v + (1 − α ) t , q = α u + (1 − α ) s, α ∈ [ 0,1] .
K. Nikodem, on convex stochastic processes, Aequationes
Mathematicae 20 (1980) 184-197.
A. Skowronski, On some properties of J-convex stochastic
processes, Aequationes Mathematicae 44 (1992) 249-258.
A. Skowronski, On wright-convex stochastic processes, Annales
Mathematicae Silesianne 9(1995) 29-32.
D. Kotrys, Hermite-hadamart inequality for convex stochastic
processes, Aequationes Mathematicae 83 (2012) 143-151.
S.S. Dragomir and S. Fitzpatrick, The Hadamard's inequality for sconvex functions in the second sense, Demonstratio Math., 32 (4)
(1999), 687-696.
H. Hudzik and L. Maligranda, Some remarks on s-convex
functions, Aequationes Mathematicae, 48 (1994), 100-111.
S. Maden, M. Tomar and E. Set, s-convex stochastic processes in
the first sense, Pure and Applied Mathematics Letters, in press.
D. Kotrys, Hermite-hadamart inequality for convex stochastic
processes, Aequationes Mathematicae 83 (2012) 143-151.