# On Harmonically(

Authors: Shan-He Wu; Imran Abbas Baloch; İmdat İşcan
Journal: Journal of Function Spaces
 (2017)

---

## Abstract

We define a new generalized class of harmonically preinvex functions named harmonically(
p
,
h
,
m
)-preinvex functions, which includes harmonic (
p
,
h
)-preinvex functions, harmonic p-preinvex functions, harmonic h-preinvex functions, and m-convex functions as special cases. We also investigate the properties and characterizations of harmonically (
p
,
h
,
m
)-preinvex functions. Finally, we establish some integral inequalities to show the applications of harmonically (
p
,
h
,
m
)-preinvex functions.


---

## Body

## 1. Introduction

Theory of convex functions had not only stimulated new and deep results in many branches of mathematical and engineering sciences, but also provided us with a unified and general framework to study a wide class of unrelated problems. For applications, generalizations, and other aspects of convex functions, see [1–10] and the references therein.It is well known that a function is a convex function on an intervalI, if and only if, the function f satisfies the inequality(1)
f
a
+
b
2
≤
1
b
-
a
∫
a
b
f
x
d
x
≤
f
a
+
f
b
2
,which is known as the Hermite-Hadamard inequality. For the applications in various fields of pure and applied sciences, we refer the reader to [6, 7, 11–17].The concepts of the convex sets and convex functions have been extended and generalized in several directions by using innovative ideas and techniques. Hanson [13] introduced the invex functions in mathematical programming. Note that the invex functions are not convex functions. Ben-Israel and Mond [18] introduced the concept of invex sets and peinvex functions. They have shown that the differentiable preinvex functions are invex functions, but the inverse may not be true. It is well known that the preinvex functions may not be convex functions (see [19–21]). For example, the function f
(
x
)
=
-
|
x
| is not a convex function, but it is a preinvex function with respect to η, where(2)
η
v
,
u
=
v
-
u
,
if
v
≤
0
,
u
≤
0
,
v
≥
0
,
u
≥
0
,
u
-
v
,
o
t
h
e
r
w
i
s
e
.Pitea and Postolache [16, 22, 23] investigated the properties of quasi-invexity in theoretical mechanics and nonlinear optimization. This shows that the preinvexity and its variant generalizations play an important and significant role in the developments of various fields of pure and applied sciences. Noor [19] has shown that the following inequality(3)
f
2
a
+
η
b
,
a
2
≤
1
η
b
,
a
∫
a
a
+
η
b
,
a
f
x
d
x
≤
f
a
+
f
b
2holds for preinvex function f. For more integral inequalities of type (3), see [20].In recent years, harmonic convex functions, which can be viewed as an important and significant extension of the convex functions, are being used to develop several iterative methods for solving nonlinear equations. Anderson et al. [1] and İşcan [5] have investigated various properties of harmonic convex functions. It has been shown that if f is a harmonic convex function, then the following inequality holds:(4)
f
2
a
b
a
+
b
≤
a
b
b
-
a
∫
a
b
f
x
x
2
d
x
≤
f
a
+
f
b
2
,which is also called Hermite-Hadamard inequality for harmonic convex function.Later on, Noor et al. [24] introduced a new unified class of convex functions, the harmonic h-preinvex function, which includes two classes of convex functions (harmonic preinvex functions and harmonic s-preinvex functions) as special cases.Recently, the functions ofp-preinvexity and harmonic p-convexity have attracted the interest of many researchers. Motivated by the idea of Noor et al. [19, 20, 24, 25], in an earlier investigation, we have introduced a more general unified class of convex functions; this class of convex functions is called harmonic (
p
,
h
)-preinvex functions. One can easily show that harmonic (
p
,
h
)-preinvex functions include harmonic preinvex, harmonic p-convex, and preinvex functions. Following this way, the aim of this paper is to introduce a new generalized class of harmonically preinvex functions which is called harmonically (
p
,
h
,
m
)-preinvex functions; also we will discuss the properties of these classes of functions. Our results include several previous results as special cases. We hope that the interested readers may discover new and innovative applications of these harmonic (
p
,
h
,
m
)-preinvex functions.


## 2. Preliminaries

We begin with recalling some basic concepts and notions in the convex analysis. For more details, we refer the reader to [24, 26–28] and the references therein.Definition 1.
A setK in R
n is said to be a convex set, if(5)
1
-
t
x
+
t
y
∈
K
,
∀
x
,
y
∈
K
,
t
∈
0,1
.Definition 2.
A functionf on the convex set K is said to be a convex function, if(6)
f
1
-
t
x
+
t
y
≤
1
-
t
f
x
+
t
f
y
,
∀
x
,
y
∈
K
,
t
∈
0,1
.Definition 3.
A setK
η
⊆
R is said to be invex set with respect to the bifunction η
(
·
,
·
), if(7)
x
+
t
η
y
,
x
∈
K
η
,
∀
x
,
y
∈
K
η
,
t
∈
0,1
.Definition 4.
The functionf on the invex set K
η is said to be preinvex with respect to the bifunction η
(
·
,
·
), if(8)
f
x
+
t
η
y
,
x
≤
1
-
t
f
x
+
t
f
y
∀
x
,
y
∈
K
η
,
t
∈
0,1
.
The functionf is said to be preconcave if -
f is preinvex.The invex setK
η is also called η-connected set. Note that if η
(
b
,
a
)
=
b
-
a, then invex set becomes the convex set. Clearly, every convex set is an invex set but converse is not true in general.Definition 5.
LetK
η be an invex set in R, and let h
:
[
0,1
]
→
R be a nonnegative function. Then, a function f
:
R
→
R is said to be h-preinvex function with respect to the bifunction η
(
·
,
·
), if(9)
f
x
+
t
η
y
,
x
≤
h
1
-
t
f
x
+
h
t
f
y
∀
x
,
y
∈
K
η
,
t
∈
0,1
.Definition 6.
A setK
h
⊆
R
/
{
0
}
→
R is said to be a harmonically p-convex set, if(10)
x
y
t
y
p
+
1
-
t
x
p
1
/
p
∈
K
h
,
∀
x
,
y
∈
K
h
,
t
∈
0,1
.Definition 7.
Leth
:
[
0,1
]
→
R be a nonnegative function. A function f
:
K
h
⊆
R
/
{
0
}
→
R is said to be harmonically (
p
,
h
)-convex function, if(11)
f
x
y
t
y
p
+
1
-
t
x
p
1
/
p
≤
h
t
f
x
+
h
1
-
t
f
y
∀
x
,
y
∈
K
h
,
t
∈
0,1
.Definition 8.
A setI
=
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
} is said to be a harmonic invex set with respect to the bifunction η
(
·
,
·
), if(12)
x
x
+
η
y
,
x
1
-
t
x
+
t
x
+
η
y
,
x
∈
I
,
∀
x
,
y
∈
I
,
t
∈
0,1
.Definition 9.
Leth
:
[
0,1
]
→
R be a nonnegative function. A function f
:
I
→
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
}
→
R is said to be harmonically h-preinvex function with respect to an arbitrary bifunction η
(
·
,
·
), if(13)
f
x
x
+
η
y
,
x
1
-
t
x
+
t
x
+
η
y
,
x
≤
h
t
f
x
+
h
1
-
t
f
y
,
∀
x
,
y
∈
I
,
t
∈
0,1
.Definition 10.
Letp
∈
R
/
{
0
}. The function f on the p-invex set A
η
,
p is said to be p-preinvex function with respect to η, where p
∈
R
/
{
0
}, if(14)
f
1
-
t
x
p
+
t
x
+
η
y
,
x
p
1
/
p
≤
t
f
x
+
1
-
t
f
y
,for all x
,
y
∈
A
η
,
p and t
∈
[
0,1
].Definition 11.
The functionf
:
[
0
,
b
]
→
R, b
>
0, is said to be m-convex, where m
∈
[
0,1
], if we have(15)
f
t
x
+
1
-
t
m
y
≤
t
f
x
+
1
-
t
m
f
yfor all x
,
y
∈
[
0
,
b
] and t
∈
[
0,1
]. We say that f is m-concave if -
f is m-convex.


## 3. Definitions and Properties of Harmonically(
p
,
h
,
m
)-Preinvex Functions

In this section, we introduce some definitions and properties related to harmonically(
p
,
h
,
m
)-preinvex functions, which is a component part of our main results.Definition 12.
A functionf
:
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
}
→
R is said to be harmonically p-preinvex functions with respect to the bifunction η
(
·
,
·
) where p
∈
R
/
{
0
} if(16)
f
x
x
+
η
y
,
x
1
-
t
x
p
+
t
x
+
η
y
,
x
p
1
/
p
=
f
t
x
p
+
1
-
t
x
+
η
y
,
x
p
-
1
/
p
≤
t
f
x
+
1
-
t
f
yfor all x
,
y
∈
[
a
,
a
+
η
(
b
,
a
)
] and t
∈
[
0,1
].Remark 13.
Notice that forp
=
1, harmonic p-preinvexity reduces to harmonic preinvexity; for η
(
b
,
a
)
=
b
-
a, harmonic p-preinvexity reduces to harmonic p-convexity; for p
=
-
1, harmonic p-preinvexity reduces to preinvexity.Definition 14.
Two functionsf
,
g are said to be similarly ordered (f is g-monotone) on interval I, if and only if,(17)
f
x
-
f
y
g
x
-
g
y
≥
0
,
∀
x
,
y
∈
I
.Proposition 15.
Iff
,
g
:
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
}
→
R
+ are two similarly ordered harmonic p-preinvex functions, then the product f
g is also a harmonic p-preinvex function.Proof.
Sincef
,
g are two positive harmonic p-preinvex functions, by using Definition 12, one has(18)
f
x
x
+
η
y
,
x
1
-
t
x
p
+
t
x
+
η
y
,
x
p
1
/
p
g
x
x
+
η
y
,
x
1
-
t
x
p
+
t
x
+
η
y
,
x
p
1
/
p
≤
t
f
x
+
1
-
t
f
y
t
g
x
+
1
-
t
g
y
=
t
2
f
x
g
x
+
t
1
-
t
f
x
g
y
+
f
y
g
x
+
1
-
t
2
f
y
g
y
=
t
f
x
g
x
+
1
-
t
f
y
g
y
+
t
2
f
x
g
x
+
t
1
-
t
f
x
g
y
+
f
y
g
x
+
1
-
t
2
f
y
g
y
-
t
f
x
g
x
-
1
-
t
f
y
g
y
=
t
f
x
g
x
+
1
-
t
f
y
g
y
-
t
1
-
t
f
x
g
x
+
f
y
g
y
-
f
x
g
y
-
f
y
g
x
=
t
f
x
g
x
+
1
-
t
f
y
g
y
-
t
1
-
t
f
x
-
f
y
g
x
-
g
y
≤
t
f
x
g
x
+
1
-
t
f
y
g
y
,where, we have used the fact that two harmonic p-preinvex functions are similarly ordered functions. This shows that product of two similarly ordered harmonic p-preinvex functions is a harmonic p-preinvex function.Definition 16.
Leth
:
[
0,1
]
→
R be a nonnegative function. A function f
:
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
}
→
R is said to be harmonically (
p
,
h
,
m
)-preinvex functions with respect to the bifunction η
(
·
,
·
) or belongs to the class h
p
x
(
p
,
h
,
m
), where p
∈
R
/
{
0
}, if(19)
f
x
x
+
η
m
y
,
x
1
-
t
x
p
+
t
x
+
η
m
y
,
x
p
1
/
p
=
f
t
x
p
+
1
-
t
x
+
η
m
y
,
x
p
-
1
/
p
≤
h
t
f
x
+
m
h
1
-
t
f
yfor all x
,
y
∈
[
a
,
a
+
η
(
b
,
a
)
], t
∈
[
0,1
], and m
∈
(
0,1
].
Leth
:
[
0,1
]
→
R be a nonnegative function. A function f
:
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
}
→
R is said to be harmonically (
p
,
h
,
m
)-preconcave functions with respect to the bifunction η
(
·
,
·
) or belongs to the class h
p
v
(
p
,
h
,
m
), where p
∈
R
/
{
0
}, if(20)
f
x
x
+
η
m
y
,
x
1
-
t
x
p
+
t
x
+
η
m
y
,
x
p
1
/
p
=
f
t
x
p
+
1
-
t
x
+
η
m
y
,
x
p
-
1
/
p
≥
h
t
f
x
+
m
h
1
-
t
f
yfor all x
,
y
∈
[
a
,
a
+
η
(
b
,
a
)
], t
∈
[
0,1
], and m
∈
(
0,1
].Remark 17.
Note that form
=
1 and h
(
t
)
=
t harmonically (
p
,
h
,
m
)-preinvex functions reduce to the harmonically p-preinvex functions; for m
=
1 and p
=
1 harmonically (
p
,
h
,
m
)-preinvex functions reduce to the harmonically h-preinvex functions; for m
=
1 and η
(
m
y
,
x
)
=
m
y
-
x harmonically (
p
,
h
,
m
)-preinvex functions reduce to the harmonically (
p
,
h
)-convex functions; for p
=
-
1
,
h
(
t
)
=
t and η
(
m
y
,
x
)
=
m
y
-
x, harmonically (
p
,
h
,
m
)-preinvex functions reduce to the m-convex functions.Now, we discuss some interesting results of harmonically(
p
,
h
,
m
)-preinvex (preconcave) functions, which includes linearity, product, composition properties, and the order property of h and p.Proposition 18.
Iff
,
g
∈
h
p
x
(
p
,
h
,
m
) and λ
>
0, then f
+
g
,
λ
f
∈
h
p
x
(
p
,
h
,
m
). Similarly, if f
,
g
∈
h
p
v
(
p
,
h
,
m
) and λ
>
0, then f
+
g, λ
f
∈
h
p
v
(
p
,
h
,
m
).Proof.
The proof follows immediately from the definitions of the classesh
p
x
(
p
,
h
,
m
) and h
p
v
(
p
,
h
,
m
).Proposition 19.
(a) Letf
∈
h
p
x
(
p
1
,
h
,
m
) and 0
<
a
<
a
+
η
(
b
,
a
). If f is monotone decreasing (monotone increasing) and p
1
≥
p
2 (p
1
≤
p
2), then f
∈
h
p
x
(
p
2
,
h
,
m
).
(b) Letf
∈
h
p
v
(
p
1
,
h
,
m
) and 0
<
a
<
a
+
η
(
b
,
a
). If f is monotone decreasing (monotone increasing) and p
2
≥
p
1 (p
1
≥
p
2), then f
∈
h
p
v
(
p
2
,
h
,
m
).Proof.
Let us recall a well-known result on the monotonicity of weighted mean (see [29]).
Let(21)
M
n
p
x
;
λ
=
λ
1
x
1
p
+
λ
2
x
2
p
+
⋯
+
λ
n
x
n
p
λ
1
+
λ
2
+
⋯
+
λ
n
1
/
p
,where λ
i
>
0, x
i
>
0 (i
=
1,2
,
…
,
n) and(22)
M
n
0
x
;
λ
=
x
1
λ
1
x
2
λ
2
⋯
x
n
λ
n
1
/
λ
1
+
λ
2
+
⋯
+
λ
n
.
In [29], it is proved that M
n
[
p
]
(
x
;
λ
) is a strictly increasing function for p
∈
(
-
∞
,
+
∞
).
Let(23)
g
p
=
x
x
+
η
m
y
,
x
t
x
+
η
m
y
,
x
p
+
1
-
t
x
p
1
/
p
.
By using the monotonicity of weighted mean forn
=
2, we conclude that g
(
p
) is a strictly decreasing function for p
∈
(
-
∞
,
+
∞
). Thus, for p
1
≥
p
2 and x
,
x
+
η
(
m
y
,
x
)
∈
[
a
,
a
+
η
(
b
,
a
)
], we have g
(
p
2
)
≥
g
(
p
1
)
. Since f is monotone decreasing and f
∈
h
p
x
(
p
1
,
h
,
m
), we then obtain(24)
f
g
p
2
≤
f
g
p
1
≤
h
t
f
x
+
m
h
1
-
t
f
y
.
Therefore, we getf
∈
h
p
x
(
p
2
,
h
,
m
). The result of (b) follows by the similar arguments as discussed above.Proposition 20.
Letf and g be nonnegative functions, and let h
2
≤
h
1 in their domains [
0,1
]. If f
∈
h
p
x
(
p
,
h
2
,
m
), then f
∈
h
p
x
(
p
,
h
1
,
m
). Similarly, if f
∈
h
p
v
(
p
,
h
1
,
m
), then f
∈
h
p
v
(
p
,
h
2
,
m
).Proof.
Sincef
∈
h
p
x
(
p
,
h
2
,
m
) form the definition of the classes h
p
x
(
p
,
h
,
m
) and the constraint conditions for f, g, h
1, and h
2, we have(25)
f
t
x
p
+
1
-
t
x
+
η
m
y
,
x
p
-
1
/
p
≤
h
2
t
f
x
+
m
h
2
1
-
t
f
y
≤
h
1
t
f
x
+
m
h
1
1
-
t
f
y
,which implies that f
∈
h
p
x
(
p
,
h
1
,
m
). The result in the second part of Proposition 20 can be deduced in a similar way as above.Proposition 21.
(a) Letf
,
g be similarly ordered positive functions on [
a
,
a
+
η
(
b
,
a
)
], that is,(26)
f
x
-
f
y
g
x
-
g
y
≥
0for all x
,
y
∈
[
a
,
a
+
η
(
b
,
a
)
]. If f
∈
h
p
x
(
p
,
h
1
,
m
), g
∈
h
p
x
(
p
,
h
2
,
m
), and h
(
α
)
+
m
h
(
1
-
α
)
≤
c for all α
∈
[
0,1
], where h
(
t
)
=
m
a
x
(
h
1
(
t
)
,
h
2
(
t
)
) and c is a fixed positive number, then the product f
g belongs to h
p
x
(
p
,
c
h
,
m
).
(b) Letf and g be opposite ordered positive functions, that is,(27)
f
x
-
f
y
g
x
-
g
y
≤
0for all x
,
y
∈
[
a
,
a
+
η
(
b
,
a
)
]. If f
∈
h
p
v
(
p
,
h
1
,
m
), g
∈
h
p
v
(
p
,
h
2
,
m
), and h
(
α
)
+
m
h
(
1
-
α
)
≥
c for all α
∈
[
0,1
], where h
(
t
)
=
m
i
n
(
h
1
(
t
)
,
h
2
(
t
)
) and c is a fixed positive number, then the product f
g belongs to h
p
v
(
p
,
c
h
,
m
).Proof.
We only give a proof of the first part and the result of the second part of this proposition followed by a similar argument. Sincef and g are similarly ordered, we have(28)
f
x
g
x
+
f
y
g
y
≥
f
x
g
y
+
f
y
g
x
.Let α and β be positive numbers such that α
+
β
=
1. We obtain(29)
f
x
x
+
η
m
y
,
x
α
x
+
η
m
y
,
x
p
+
β
x
p
1
/
p
g
x
x
+
η
m
y
,
x
α
x
+
η
m
y
,
x
p
+
β
x
p
1
/
p
≤
h
1
α
f
x
+
m
h
1
β
f
y
h
2
α
g
x
+
m
h
2
β
g
y
≤
h
2
α
f
x
g
x
+
m
h
α
h
β
f
x
g
y
+
m
h
α
h
β
f
y
g
x
+
m
2
h
2
β
f
y
g
y
≤
h
2
α
f
x
g
x
+
m
h
α
h
β
f
x
g
x
+
m
h
α
h
β
f
y
g
y
+
m
2
h
2
β
f
y
g
y
=
h
α
+
m
h
β
h
α
f
x
g
x
+
m
h
β
f
y
g
y
≤
c
h
α
f
x
g
x
+
c
m
h
β
f
y
g
y
.
This completes the proof of Proposition21.Proposition 22.
Letf
:
I
1
→
(
0
,
∞
) and g
:
I
2
→
(
0
,
∞
) be functions with g
(
I
2
)
⊆
I
1.
(a) If the functionf is m-convex and increasing (decreasing) and g
∈
h
p
x
(
p
,
h
,
m
)  (
g
∈
h
p
v
(
p
,
h
,
m
)
) with h
(
α
)
+
h
(
1
-
α
)
=
1 for α
∈
[
0,1
], then f
∘
g belongs to h
p
x
(
p
,
h
,
m
).
(b) If the functionf is m-concave and increasing (decreasing) and g
∈
h
p
v
(
p
,
h
,
m
)
(
g
∈
h
p
x
(
p
,
h
,
m
)
) with h
(
α
)
+
h
(
1
-
α
)
=
1 for α
∈
[
0,1
], then f
∘
g belongs to h
p
v
(
p
,
h
,
m
).Proof.
In view ofg
∈
h
p
x
(
p
,
h
,
m
) and f is an increasing function, we have(30)
f
∘
g
x
x
+
η
m
y
,
x
α
x
+
η
m
y
,
x
p
+
1
-
α
x
p
1
/
p
≤
f
h
α
g
x
+
m
h
1
-
α
g
y
.
Since,h
(
α
)
+
h
(
1
-
α
)
=
1 and f is m-convex, we obtain(31)
f
h
α
g
x
+
m
h
1
-
α
g
y
≤
h
α
f
∘
g
x
+
m
h
1
-
α
f
∘
g
y
,which leads to(32)
f
∘
g
x
x
+
η
m
y
,
x
α
x
+
η
m
y
,
x
p
+
1
-
α
x
p
1
/
p
≤
h
α
f
∘
g
x
+
m
h
1
-
α
f
∘
g
y
.
We conclude thatf
∘
g
∈
h
p
x
(
p
,
h
,
m
). The rest of the proof of Proposition 22 can be dealt with in the same way as described above.


## 4. Integral Inequalities for Harmonically(
p
,
h
,
m
)-Preinvex Functions

We illustrate the applications of the definitions and properties of harmonically(
p
,
h
,
m
)-preinvex functions stated in foregoing section; we will establish some integral inequalities via harmonically (
p
,
h
,
m
)-preinvex functions. Our results include, as special cases, some related inequalities for harmonic p-preinvex functions, harmonic h-preinvex functions, and m-convex functions.Theorem 23.
Letf
:
[
a
,
a
+
η
(
m
b
,
a
)
]
→
(
0
,
∞
) be an integrable harmonic (
p
,
h
,
m
)-preinvex function, and let h be a concave function on [
0,1
]
. Then we have the following inequality:(33)
f
a
a
+
η
m
b
,
a
a
p
+
a
+
η
m
b
,
a
p
-
x
p
1
/
p
≤
2
f
a
+
m
f
b
h
1
2
-
f
a
a
+
η
m
b
,
a
x
.Proof.
Letx
=
[
(
1
-
t
)
a
p
+
t
(
a
+
η
(
m
b
,
a
)
)
p
]
1
/
p and t
∈
[
0,1
]
. Using the definition of harmonic (
p
,
h
,
m
)-preinvex function along with a simple computation gives (34)
f
a
a
+
η
m
b
,
a
a
p
+
a
+
η
m
b
,
a
p
-
x
p
1
/
p
=
f
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
≤
h
1
-
t
f
a
+
m
h
t
f
b
=
f
a
+
m
f
b
h
1
-
t
+
h
t
-
h
t
f
a
+
m
h
1
-
t
f
b
≤
f
a
+
m
f
b
h
1
-
t
+
h
t
-
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
=
f
a
+
m
f
b
h
1
-
t
+
h
t
-
f
a
a
+
η
m
b
,
a
x
≤
2
f
a
+
m
f
b
h
1
2
-
f
a
a
+
η
m
b
,
a
x
,where the last inequality follows from the assumption that h is a concave function on [
0,1
]
. This completes the proof of Theorem 23.Choosingm
=
1, h
(
t
)
=
t in Theorem 23, we get the following.Corollary 24.
Letf
:
[
a
,
a
+
η
(
b
,
a
)
]
→
(
0
,
∞
) be an integrable harmonic p-preinvex function. Then we have the inequality(35)
f
a
a
+
η
b
,
a
a
p
+
a
+
η
b
,
a
p
-
x
p
1
/
p
≤
f
a
+
f
b
-
f
a
a
+
a
+
η
b
,
a
x
.Theorem 25.
Letf
:
[
a
,
a
+
η
(
m
b
,
a
)
]
⊆
R
/
{
0
}
→
R be a harmonic (
p
,
h
,
m
)-preinvex function with 0
<
a
<
a
+
η
(
m
b
,
a
). If f, h are integrable, then(36)
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
f
x
x
p
+
1
d
x
≤
f
a
+
m
f
b
∫
0
1
h
t
d
t
.Proof.
Sincef is harmonic (
p
,
h
,
m
)-preinvex function, we have(37)
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
≤
h
t
f
a
+
m
h
1
-
t
f
b
.
Integrating both sides of the above inequality with respect tot over [
0,1
], we obtain (38)
∫
0
1
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
d
t
≤
∫
0
1
h
t
f
a
+
m
h
1
-
t
f
b
d
t
=
f
a
∫
0
1
h
t
d
t
+
m
f
b
∫
0
1
h
1
-
t
d
t
=
f
a
∫
0
1
h
t
d
t
+
m
f
b
∫
0
1
h
t
d
t
=
f
a
+
m
f
b
∫
0
1
h
t
d
t
.On the other hand, by setting(39)
x
=
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
,
t
∈
0,1
,then the left-hand side of integral expression can be simplified to(40)
∫
0
1
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
d
t
=
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
f
x
x
p
+
1
d
x
.
Thus(41)
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
f
x
x
p
+
1
d
x
≤
f
a
+
m
f
b
∫
0
1
h
t
d
t
.
The proof of Theorem25 is complete.Takingm
=
1, h
(
t
)
=
t in Theorem 25, we obtain the following.Corollary 26.
Letf
:
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
}
→
R be a harmonic p-preinvex function with 0
<
a
<
a
+
η
(
b
,
a
). If f is integrable, then(42)
p
a
a
+
η
b
,
a
p
a
+
η
b
,
a
p
-
a
p
∫
a
a
+
η
b
,
a
f
x
x
p
+
1
d
x
≤
f
a
+
f
b
2
.Theorem 27.
Letf, g
:
[
a
,
a
+
η
(
m
b
,
a
)
]
⊆
R
/
{
0
}
→
(
0
,
∞
) be harmonic (
p
,
h
,
m
)-preinvex functions with 0
<
a
<
a
+
η
(
m
b
,
a
)
. If f
,
g
,
h are integrable, then the following inequality holds:(43)
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
f
x
g
x
x
p
+
1
d
x
≤
M
a
,
b
,
m
∫
0
1
h
2
x
d
x
+
N
a
,
b
,
m
∫
0
1
h
x
h
1
-
x
d
x
,where M
a
,
b
,
m
=
f
a
g
a
+
m
2
f
b
g
b and N
a
,
b
,
m
=
m
f
a
g
b
+
f
b
g
a.Proof.
Sincef
,
g are harmonic (
p
,
h
,
m
)-preinvex functions, it follows that(44)
∫
0
1
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
g
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
d
t
≤
∫
0
1
h
t
f
a
+
m
h
1
-
t
f
b
h
t
g
a
+
m
h
1
-
t
g
b
d
t
=
∫
0
1
h
t
2
f
a
g
a
+
m
2
h
1
-
t
2
f
b
g
b
d
t
+
∫
0
1
m
h
t
h
1
-
t
f
a
g
b
+
f
b
g
a
d
t
=
f
a
g
a
∫
0
1
h
t
2
d
t
+
m
2
f
b
g
b
∫
0
1
h
1
-
t
2
d
t
+
m
f
a
g
b
+
f
b
g
a
∫
0
1
h
t
h
1
-
t
d
t
=
f
a
g
a
+
m
2
f
b
g
b
∫
0
1
h
t
2
d
t
+
m
f
a
g
b
+
f
b
g
a
∫
0
1
h
t
h
1
-
t
d
t
=
M
a
,
b
,
m
∫
0
1
h
2
x
d
x
+
N
a
,
b
,
m
∫
0
1
h
x
h
1
-
x
d
x
.
On the other hand, by using a substitution,(45)
x
=
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
,
t
∈
0,1
,and then a straightforward calculation gives(46)
∫
0
1
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
g
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
d
t
=
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
f
x
g
x
x
p
+
1
d
x
.
The proof of Theorem27 is completed.Puttingm
=
1, h
(
t
)
=
t into Theorem 27, we have the following.Corollary 28.
Letf
,
g
:
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
}
→
(
0
,
∞
) be harmonic p-preinvex functions with 0
<
a
<
a
+
η
(
b
,
a
)
. If f
,
g are integrable, then the following inequality holds:(47)
p
a
a
+
η
b
,
a
p
a
+
η
b
,
a
p
-
a
p
∫
a
a
+
η
b
,
a
f
x
g
x
x
p
+
1
d
x
≤
1
3
M
a
,
b
+
1
6
N
a
,
b
,where M
(
a
,
b
)
=
f
(
a
)
g
(
a
)
+
f
(
b
)
g
(
b
) and N
(
a
,
b
)
=
f
(
a
)
g
(
b
)
+
f
(
b
)
g
(
a
).Theorem 29.
Letf
:
[
a
,
a
+
η
(
m
b
,
a
)
]
⊆
R
/
{
0
}
→
R be harmonic (
p
,
h
,
m
)-preinvex functions with 0
<
a
<
a
+
η
(
m
b
,
a
)
. If f
,
h
,
w are integrable, then the following inequality holds:(48)
∫
a
a
+
η
m
b
,
a
f
x
w
x
x
p
+
1
d
x
≤
M
2
∫
a
a
+
η
m
b
,
a
w
x
x
p
+
1
d
x
,where M
=
sup
t
∈
[
0,1
]
⁡
(
f
(
a
)
+
m
f
(
b
)
)
(
h
(
t
)
+
h
(
1
-
t
)
), and w
(
x
) is nonnegative and satisfies(49)
w
x
=
w
a
x
a
+
η
m
b
,
a
a
p
+
a
+
η
m
b
,
a
p
x
p
-
a
a
+
η
m
b
,
a
p
1
/
p
.Proof.
Note thatf is harmonic (
p
,
h
,
m
)-preinvex function; we have(50)
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
+
f
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
≤
h
t
f
a
+
m
h
1
-
t
f
b
+
h
1
-
t
f
a
+
m
h
t
f
b
=
f
a
h
t
+
h
1
-
t
+
m
f
b
h
t
+
h
1
-
t
=
f
a
+
m
f
b
h
t
+
h
1
-
t
.
Using a substitution(51)
x
=
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
,
t
∈
0,1
,we get (52)
∫
0
1
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
w
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
d
t
=
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
f
x
w
x
x
p
+
1
d
x
.
Similarly, we have(53)
∫
0
1
f
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
w
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
d
t
=
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
f
x
w
x
x
p
+
1
d
x
.
From the properties of expressionw
(
x
), it is easy to find that(54)
w
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
=
w
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
.
By utilizing the above inequality and identities, we deduce that(55)
2
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
f
x
w
x
x
p
+
1
d
x
=
∫
0
1
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
w
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
d
t
+
∫
0
1
f
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
w
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
d
t
=
∫
0
1
f
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
+
∫
0
1
f
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
·
w
a
a
+
η
m
b
,
a
1
-
t
a
p
+
t
a
+
η
m
b
,
a
p
1
/
p
d
t
≤
∫
0
1
f
a
+
m
f
b
h
t
+
h
1
-
t
w
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
d
t
≤
M
∫
0
1
w
a
a
+
η
m
b
,
a
t
a
p
+
1
-
t
a
+
η
m
b
,
a
p
1
/
p
d
t
=
M
p
a
a
+
η
m
b
,
a
p
a
+
η
m
b
,
a
p
-
a
p
∫
a
a
+
η
m
b
,
a
w
x
x
p
+
1
d
x
,where M
=
sup
t
∈
[
0,1
]
⁡
(
f
(
a
)
+
m
f
(
b
)
)
(
h
(
t
)
+
h
(
1
-
t
)
)
. The proof of Theorem 29 is completed.Puttingm
=
1, h
(
t
)
=
t into Theorem 29, we have the following.Corollary 30.
Letf
:
[
a
,
a
+
η
(
b
,
a
)
]
⊆
R
/
{
0
}
→
R be harmonic p-preinvex functions with 0
<
a
<
a
+
η
(
b
,
a
)
. If f
,
w are integrable, then we have the following inequality:(56)
∫
a
a
+
η
b
,
a
f
x
w
x
x
p
+
1
d
x
≤
f
a
+
f
b
2
∫
a
a
+
η
b
,
a
w
x
x
p
+
1
d
x
,where w
(
x
) is nonnegative and satisfies(57)
w
x
=
w
a
x
a
+
η
b
,
a
a
p
+
a
+
η
b
,
a
p
x
p
-
a
a
+
η
b
,
a
p
1
/
p
.



---

*Source: 2148529-2017-01-31.xml*