# Estimates of Upper Bound for a Function Associated with Riemann-Liouville Fractional Integral viah-Convex Functions

Authors: Shan-He Wu; Muhammad Uzair Awan
Journal: Journal of Function Spaces
 (2019)

---

## Abstract

A new identity involving Riemann-Liouville fractional integral is proposed. The result is then used to obtain some estimates of upper bound for a function associated with Riemann-Liouville fractional integral viah-convex functions. An application for establishing the inequalities related to special means is also considered.


---

## Body

## 1. Introduction

A setK
⊂
R is said to be convex, if (1)
1
-
μ
x
+
μ
y
∈
K
,
∀
x
,
y
∈
K
,
μ
∈
0,1
.A functionf
:
K
→
R is said to be convex, if (2)
f
1
-
μ
x
+
μ
y
≤
1
-
μ
f
x
+
μ
f
y
,
∀
x
,
y
∈
K
,
μ
∈
0,1
.In 1978, Breckner [1] introduced the concept of s-convex functions as a generalization of convex functions, as follows.Definition 1 (see [1]).
Lets be a real number, s
∈
(
0,1
]. A function f
:
[
0
,
∞
)
→
[
0
,
∞
) is said to be s-convex, if(3)
f
1
-
μ
x
+
μ
y
≤
1
-
μ
s
f
x
+
μ
s
f
y
,
∀
x
,
y
∈
0
,
∞
,
μ
∈
0,1
.In recent years several new extensions of classical convexity have been proposed in the literature. Varošanec [2] investigated a more generalized class of convex functions named h-convex function, as follows.Definition 2 (see [2]).
Leth
:
J
→
R be a nonnegative function. A function f
:
I
→
R is said to be h-convex, if(4)
f
1
-
μ
x
+
μ
y
≤
h
1
-
μ
f
x
+
h
μ
f
y
,
∀
x
,
y
∈
I
,
μ
∈
0,1
.It has been observed that the class ofh-convex functions unifies several other classes of convexity; for example, if we take h
(
μ
)
=
μ
s, h
(
μ
)
=
μ
-
1, h
(
μ
)
=
μ
-
s, and h
(
μ
)
=
1, respectively, then we have the class of s-convex functions [1], the class of Q-functions [3], the class of s-Godunova-Levin type functions [4], and the class of P-functions [5]. For more details on convexity and its generalizations, see [6–9].Convexity of a function plays a vital role in theory of inequalities, because many inequalities can easily be obtained using the functions having convexity properties. Hermite and Hadamard’s result which is known as Hermite-Hadamard’s inequality is one of the most fascinating results in the field of integral inequalities. This inequality provides a lower and an upper estimate for the integral average of any convex function defined on an interval. This famous result reads as follows.Letf
:
[
a
,
b
]
→
R be a convex function, then (5)
f
a
+
b
2
≤
1
b
-
a
∫
a
b
f
x
d
x
≤
f
a
+
f
b
2
.Sarikaya et al. [10] gave a generalization of Hermite-Hadamard’s inequality using the h-convexity of the function as follows.Letf
:
[
a
,
b
]
→
R be h-convex function, then, for h
(
1
/
2
)
≠
0, we have (6)
1
2
h
1
/
2
f
a
+
b
2
≤
1
b
-
a
∫
a
b
f
x
d
x
≤
f
a
+
f
b
∫
0
1
h
μ
d
μ
.Although the fractional calculus has a long history, it plays significant role in different fields of pure and applied mathematics [11]. Up to now, the study of the fractional calculus is still very active. Sarikaya et al. [12] used the concepts of Riemann-Liouville integrals to obtain the fractional version of Hermite-Hadamard’s inequality. In fact, there are numerous new inequalities that have been obtained using the techniques of fractional calculus. For more details, see [12–15].In this paper, we present a new integral identity for differentiable functions involving fractional integrals. Then using this auxiliary result we establish our main results that are the estimates of upper bound for a function associated with Riemann-Liouville fractional integral viah-convex functions. At the end of the paper, we give an application of the obtained results to the special means.We begin with recalling the definition of Riemann-Liouville fractional integrals, as follows.Definition 3 (see [11]).
Letf
∈
L
1
[
a
,
b
]. The Riemann-Liouville integrals J
a
+
α
f and J
b
-
α
f of order α
>
0 with a
≥
0 are defined by (7)
J
a
+
α
f
x
=
1
Γ
α
∫
a
x
x
-
μ
α
-
1
f
μ
d
μ
,
x
>
a
,and(8)
J
b
-
α
f
x
=
1
Γ
α
∫
x
b
μ
-
x
α
-
1
f
μ
d
μ
,
x
<
b
,where (9)
Γ
α
=
∫
0
∞
μ
α
-
1
e
-
μ
d
μ
, is the Gamma function.The integral form of the hypergeometric function is(10)
F
1
2
x
,
y
;
c
;
z
=
1
B
y
,
c
-
y
∫
0
1
μ
y
-
1
1
-
μ
c
-
y
-
1
1
-
z
μ
-
x
d
μ
,
for
z
<
1
,
c
>
y
>
0
,where B
(
x
,
y
)
=
∫
0
1
‍
μ
x
-
1
(
1
-
μ
)
y
-
1
d
μ is the Beta function.


## 2. Main Results

In this section, we consider the estimates of upper bound for the function below, which is associated with Riemann-Liouville fractional integral. Consider the following:(11)
Ψ
a
,
b
;
n
;
α
f
=
n
+
1
α
-
1
Γ
α
+
1
b
-
a
α
J
n
a
+
b
/
n
+
1
-
α
f
a
+
J
a
+
n
b
/
n
+
1
+
α
f
b
-
1
n
+
1
f
n
a
+
b
n
+
1
+
f
a
+
n
b
n
+
1
.In order to establish the estimates of upper bound forΨ
(
a
,
b
;
n
;
α
)
(
f
), we first prove an auxiliary result which plays an important role in dealing with subsequent results.Lemma 4.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b. If f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N, then (12)
Ψ
a
,
b
;
n
;
α
f
=
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
-
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
+
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
=
n
+
1
α
-
1
Γ
α
+
1
b
-
a
α
J
n
a
+
b
/
n
+
1
-
α
f
a
+
J
a
+
n
b
/
n
+
1
+
α
f
b
-
1
n
+
1
f
n
a
+
b
n
+
1
+
f
a
+
n
b
n
+
1
.Proof.
Let(13)
I
=
∫
0
1
1
-
μ
α
-
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
+
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
=
-
∫
0
1
1
-
μ
α
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
d
μ
+
∫
0
1
1
-
μ
α
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
=
-
I
1
+
I
2
.
IntegratingI
1 gives(14)
I
1
=
∫
0
1
1
-
μ
α
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
d
μ
=
n
+
1
b
-
a
f
n
n
+
1
a
+
1
n
+
1
b
-
n
+
1
α
b
-
a
∫
a
n
/
n
+
1
a
+
1
/
n
+
1
b
n
+
1
α
-
1
u
-
a
b
-
a
α
-
1
f
u
n
+
1
b
-
a
d
u
=
n
+
1
b
-
a
f
n
n
+
1
a
+
1
n
+
1
b
-
n
+
1
α
+
1
Γ
α
+
1
b
-
a
α
+
1
J
n
/
n
+
1
a
+
1
/
n
+
1
b
-
α
f
a
.
Similarly integratingI
2, one has(15)
I
2
=
∫
0
1
1
-
μ
α
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
=
-
n
+
1
b
-
a
f
1
n
+
1
a
+
n
n
+
1
b
+
n
+
1
α
b
-
a
∫
1
/
n
+
1
a
+
n
/
n
+
1
b
b
n
+
1
α
-
1
b
-
v
b
-
a
α
-
1
f
v
n
+
1
b
-
a
d
v
=
-
n
+
1
b
-
a
f
1
n
+
1
a
+
n
n
+
1
b
+
n
+
1
α
+
1
Γ
α
+
1
b
-
a
α
+
1
J
1
/
n
+
1
a
+
n
/
n
+
1
b
+
α
f
b
.
Using (14) and (15) in (13) leads to the identity described in Lemma 4.Based on Lemma4, we are now in a position to establish our main results.Theorem 5.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b. If f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N and f
′ is h-convex function, then (16)
Ψ
a
,
b
;
n
;
α
f
≤
b
-
a
n
+
1
2
f
′
a
+
f
′
b
ψ
n
;
α
;
μ
,where(17)
ψ
n
;
α
;
μ
=
∫
0
1
1
-
μ
α
h
n
+
μ
n
+
1
+
h
1
-
μ
n
+
1
d
μ
.Proof.
Using Lemma4 and the fact that f
′ is h-convex function, we have (18)
Ψ
a
,
b
;
n
;
α
f
=
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
-
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
+
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
≤
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
d
μ
+
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
≤
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
d
μ
+
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
≤
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
h
n
+
μ
n
+
1
+
h
1
-
μ
n
+
1
d
μ
f
′
a
+
f
′
b
.
This completes the proof of Theorem5.We now discuss some special cases which can be deduced directly from Theorem5.(I) Putting h
μ
=
μin Theorem 5, we have the following.Corollary 6.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b. If f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N and f
′ is convex function, then (19)
Ψ
a
,
b
;
n
;
α
f
≤
b
-
a
n
+
1
2
α
+
1
f
′
a
+
f
′
b
.(II) Putting h
(
μ
)
=
μ
s
(
0
<
s
≤
1
) in Theorem 5, we have the following.Corollary 7.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b. If f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N and f
′ is s-convex function, then (20)
Ψ
a
,
b
;
n
;
α
f
≤
b
-
a
n
+
1
s
+
2
n
s
F
1
2
-
s
,
1,2
+
α
,
-
1
/
n
α
+
1
+
1
s
+
α
+
1
f
′
a
+
f
′
b
.(III) Taking h
(
μ
)
=
1 in Theorem 5, we have the following.Corollary 8.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b. If f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N and f
′ is P-function, then (21)
Ψ
a
,
b
;
n
;
α
f
≤
2
b
-
a
n
+
1
2
α
+
1
f
′
a
+
f
′
b
.(IV) Taking h
(
μ
)
=
μ
-
s
(
0
<
s
≤
1
) in Theorem 5, we have the following.Corollary 9.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b. If f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N and f
′ is s-Godunova-Levin type function, then (22)
Ψ
a
,
b
;
n
;
α
f
≤
b
-
a
n
+
1
2
-
s
n
-
s
F
1
2
s
,
1,2
+
α
,
-
1
/
n
α
+
1
+
1
α
-
s
+
1
f
′
a
+
f
′
b
.Theorem 10.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b, f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N, and let f
′
q be h-convex function, 1
/
p
+
1
/
q
=
1
,
p
,
q
>
0. Then (23)
Ψ
a
,
b
;
n
;
α
f
≤
b
-
a
n
+
1
2
1
α
p
+
1
1
/
p
f
′
a
q
∫
0
1
h
n
+
μ
n
+
1
d
μ
+
f
′
b
q
∫
0
1
h
1
-
μ
n
+
1
d
μ
1
/
q
+
f
′
a
q
∫
0
1
h
1
-
μ
n
+
1
d
μ
+
f
′
b
q
∫
0
1
h
n
+
μ
n
+
1
d
μ
1
/
q
.Proof.
Using Lemma4, the Hölder inequality and the fact that f
′
q is h-convex function, we have (24)
Ψ
a
,
b
;
n
;
α
f
=
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
-
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
+
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
≤
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
d
μ
+
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
d
μ
≤
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
p
d
μ
1
/
p
∫
0
1
f
′
n
+
μ
n
+
1
a
+
1
-
μ
n
+
1
b
q
d
t
1
/
q
+
b
-
a
n
+
1
2
∫
0
1
1
-
μ
α
p
d
μ
1
/
p
∫
0
1
f
′
1
-
μ
n
+
1
a
+
n
+
μ
n
+
1
b
q
d
t
1
/
q
≤
b
-
a
n
+
1
2
1
α
p
+
1
1
/
p
f
′
a
q
∫
0
1
h
n
+
μ
n
+
1
d
μ
+
f
′
b
q
∫
0
1
h
1
-
μ
n
+
1
d
μ
1
/
q
+
f
′
a
q
∫
0
1
h
1
-
μ
n
+
1
d
μ
+
f
′
b
q
∫
0
1
h
n
+
μ
n
+
1
d
μ
1
/
q
.
The proof of Theorem10 is complete.We give now four corollaries that follow from the special cases of Theorem10.(I) Choosing h
(
μ
)
=
μ in Theorem 10, we have the following.Corollary 11.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b, f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N, and let f
′
q be convex function, 1
/
p
+
1
/
q
=
1
,
p
,
q
>
0. Then(25)
Ψ
a
,
b
;
n
;
α
f
≤
b
-
a
n
+
1
2
+
1
/
q
α
p
+
1
1
/
p
2
n
+
1
2
f
′
a
q
+
1
2
f
′
b
q
1
/
q
+
1
2
f
′
a
q
+
2
n
+
1
2
f
′
b
1
/
q
.(II) Choosing h
(
μ
)
=
μ
s
(
0
<
s
≤
1
) in Theorem 10, we have the following.Corollary 12.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b, f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N, and let f
′
q be s-convex function, 1
/
p
+
1
/
q
=
1
,
p
,
q
>
0. Then(26)
Ψ
a
,
b
;
n
;
α
f
≤
b
-
a
n
+
1
2
+
s
/
q
α
p
+
1
1
/
p
n
+
1
s
+
1
-
n
s
+
1
s
+
1
f
′
a
q
+
1
s
+
1
f
′
b
q
1
/
q
+
1
s
+
1
f
′
a
q
+
n
+
1
s
+
1
-
n
s
+
1
s
+
1
f
′
b
q
1
/
q
.(III) Putting h
(
μ
)
=
1 in Theorem 10, we have the following.Corollary 13.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b, f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N, and let f
′
q be P-function, 1
/
p
+
1
/
q
=
1
,
p
,
q
>
0. Then(27)
Ψ
a
,
b
;
n
;
α
f
≤
2
b
-
a
n
+
1
2
α
p
+
1
1
/
p
f
′
a
q
+
f
′
b
q
1
/
q
.(IV) Putting h
(
μ
)
=
μ
-
s
(
0
<
s
≤
1
) in Theorem 10, we have the following.Corollary 14.
Letf
:
[
a
,
b
]
→
R be differentiable function on (
a
,
b
) with 0
<
a
<
b, f
′
∈
L
[
a
,
b
], α
>
0, and n
∈
N, and let f
′
q be s-Godunova-Levin type function, 1
/
p
+
1
/
q
=
1
,
p
,
q
>
0. Then(28)
Ψ
a
,
b
;
n
;
α
f
≤
b
-
a
n
+
1
2
-
s
/
q
α
p
+
1
1
/
p
n
+
1
1
-
s
-
n
1
-
s
1
-
s
f
′
a
q
+
1
1
-
s
f
′
b
q
1
/
q
+
1
1
-
s
f
′
a
q
+
n
+
1
1
-
s
-
n
1
-
s
1
-
s
f
′
b
q
1
/
q
.


## 3. Application to Special Means

In this section, we give an application of the obtained results to special means.Definition 15 (see [16]).
Recall the following definitions.(1)
For arbitrary positive numbersa
,
b (a
≠
b
), (29)
L
a
,
b
=
b
-
a
log
⁡
b
-
log
⁡
a
is called the logarithmic mean.
(2)
For arbitrary real numbersa
,
b, (30)
A
a
,
b
=
a
+
b
2
is called the arithmetic mean.
(3)
The extended logarithmic mean of two positive numbersa
,
b
(
a
≠
b
) is defined by(31)
L
p
a
,
b
=
b
p
+
1
-
a
p
+
1
p
+
1
b
-
a
1
/
p
,
p
≠
-
1,0
,
b
-
a
log
⁡
b
-
log
⁡
a
,
p
=
-
1
,
1
e
b
b
a
a
1
/
b
-
a
,
p
=
0
.We focus on the estimation of upper bound for the difference between logarithmic mean and arithmetic mean; we shall establish two inequalities related to these means.Proposition 16.
If0
<
a
<
b, 0
<
s
≤
1, then (32)
L
s
s
a
,
b
-
A
s
a
,
b
≤
ϑ
s
a
s
-
1
+
b
s
-
1
,where(33)
ϑ
s
=
s
b
-
a
2
s
+
2
F
1
2
-
s
,
1,3
,
-
1
2
+
1
s
+
2
.Proof.
We start by verifying thatφ
(
x
)
=
s
x
s
-
1
(
0
<
s
≤
1
) is s-convex on (
0
,
∞
).
In view of(34)
φ
′
′
x
=
s
s
-
1
s
-
2
x
s
-
3
≥
0
for
0
<
s
≤
1
,which implies that φ
(
x
) is convex on (
0
,
∞
), thus for all u
,
v
∈
(
0
,
∞
), t
∈
[
0,1
], and 0
<
s
≤
1 we have (35)
φ
1
-
t
u
+
t
v
=
s
1
-
t
u
+
t
v
s
-
1
≤
1
-
t
s
u
s
-
1
+
t
s
v
s
-
1
≤
1
-
t
s
s
u
s
-
1
+
t
s
s
v
s
-
1
=
1
-
t
s
φ
u
+
t
s
φ
v
.
Hence,φ
(
x
)
=
s
x
s
-
1
(
0
<
s
≤
1
) is s-convex on (
0
,
∞
)
.
Now, puttingf
(
x
)
=
x
s, s
∈
(
0,1
), α
=
1, n
=
1 in Corollary 7, we obtain(36)
Ψ
a
,
b
;
n
;
α
f
=
1
b
-
a
b
s
+
1
-
a
s
+
1
s
+
1
-
a
+
b
2
s
=
L
s
s
a
,
b
-
A
s
a
,
band(37)
b
-
a
n
+
1
s
+
2
n
s
F
1
2
-
s
,
1,2
+
α
,
-
1
/
n
α
+
1
+
1
s
+
α
+
1
f
′
a
+
f
′
b
=
s
b
-
a
2
s
+
2
F
1
2
-
s
,
1,3
,
-
1
2
+
1
s
+
2
a
s
-
1
+
b
s
-
1
.
Note that the functionf
′
x
=
s
x
s
-
1
(
0
<
s
≤
1
) is s-convex on (
0
,
∞
). Hence, the inequality (32) follows straightway from the inequality given in Corollary 7.Proposition 17.
If0
<
a
<
b, 1
/
p
+
1
/
q
=
1, p
>
0, q
>
0, and 0
<
s
≤
1, then(38)
L
s
s
a
,
b
-
A
s
a
,
b
≤
s
b
-
a
2
2
+
s
/
q
p
+
1
1
/
p
s
+
1
1
/
q
η
1
1
/
q
+
η
2
1
/
q
,where(39)
η
1
=
2
s
+
1
-
1
a
q
s
-
1
+
b
q
s
-
1
,
η
2
=
a
q
s
-
1
+
2
s
+
1
-
1
b
q
s
-
1
.Proof.
Takingf
(
x
)
=
x
s
(
0
<
s
≤
1
), α
=
1, n
=
1 in Corollary 12, we obtain(40)
Ψ
a
,
b
;
n
;
α
f
=
1
b
-
a
b
s
+
1
-
a
s
+
1
s
+
1
-
a
+
b
2
s
=
L
s
s
a
,
b
-
A
s
a
,
b
and(41)
b
-
a
n
+
1
2
+
s
/
q
α
p
+
1
1
/
p
n
+
1
s
+
1
-
n
s
+
1
s
+
1
f
′
a
q
+
1
s
+
1
f
′
b
q
1
/
q
+
1
s
+
1
f
′
a
q
+
n
+
1
s
+
1
-
n
s
+
1
s
+
1
f
′
b
q
1
/
q
=
s
b
-
a
2
2
+
s
/
q
p
+
1
1
/
p
s
+
1
1
/
q
2
s
+
1
-
1
a
q
s
-
1
+
b
q
s
-
1
1
/
q
+
a
q
s
-
1
+
2
s
+
1
-
1
b
q
s
-
1
1
/
q
.
It remains to prove thatϕ
(
x
)
=
s
q
x
q
(
s
-
1
)
(
0
<
s
≤
1
,
q
>
0
) is s-convex on (
0
,
∞
).
In fact, one has(42)
ϕ
′
′
x
=
s
q
q
s
-
1
q
s
-
1
-
1
x
q
s
-
1
-
2
≥
0
for
0
<
s
≤
1
,
q
>
0
.
It follows thatϕ
(
x
) is convex on (
0
,
∞
); thus for all u
,
v
∈
(
0
,
∞
), t
∈
[
0,1
], and 0
<
s
≤
1 we have (43)
ϕ
1
-
t
u
+
t
v
≤
1
-
t
ϕ
u
+
t
ϕ
v
≤
1
-
t
s
φ
u
+
t
s
φ
v
,which implies that ϕ
(
x
)
=
s
q
x
q
(
s
-
1
)
(
0
<
s
≤
1
,
q
>
0
) is s-convex on (
0
,
∞
).
Now, utilizing the fact thatf
′
x
q
=
s
q
x
q
(
s
-
1
)
(
0
<
s
≤
1
,
q
>
0
) is s-convex on (
0
,
∞
), we can deduce the desired inequality (38) from Corollary 12.



---

*Source: 9861790-2019-05-02.xml*