# Generalized Steffensen Type Inequalities Involving Convex Functions

Authors: Josip Pečarić; Ksenija Smoljak Kalamir
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

In this paper generalized Steffensen type inequalities related to the class of functions that are “convex at pointc” are derived and as a consequence inequalities involving the class of convex functions are obtained. Moreover, linear functionals from the difference of the right- and left-hand side of the obtained generalized inequalities are constructed and new families of exponentially convex functions related to constructed functionals are derived.


---

## Body

## 1. Introduction

The well-known Steffensen inequality [1] reads as follows.Theorem 1.
Suppose thatf is nonincreasing and g is integrable on [
a
,
b
] with 0
≤
g
≤
1 and λ
=
∫
a
b
‍
g
(
t
)
d
t. Then one has
(1)
∫
b
-
λ
b
f
(
t
)
d
t
≤
∫
a
b
f
(
t
)
g
(
t
)
d
t
≤
∫
a
a
+
λ
f
(
t
)
d
t
.
The inequalities are reversed for f is nondecreasing.Since its appearance in 1918 numerous papers have been devoted to generalizations and refinements of Steffensen’s inequality. In [2] Pečarić proved the following generalization.Theorem 2.
Leth be a positive integrable function on [
a
,
b
] and f an integrable function such that f
/
h is nondecreasing on [
a
,
b
]. If g is a real-valued integrable function on [
a
,
b
] such that 0
≤
g
≤
1, then
(2)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≥
∫
a
a
+
λ
‍
f
(
t
)
d
t
holds, where λ is the solution of the equation
(3)
∫
a
a
+
λ
‍
h
(
t
)
d
t
=
∫
a
b
‍
h
(
t
)
g
(
t
)
d
t
.
If f
/
h is a nonincreasing function, then the reverse inequality in (2) holds.By substitutionsg
(
x
)
→
1
-
g
(
x
) and λ
→
b
-
a
-
λ, Theorem 2 becomes as follows.Theorem 3.
Let the conditions of Theorem2 be fulfilled. Then
(4)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≤
∫
b
-
λ
b
‍
f
(
t
)
d
t
holds, where λ is the solution of the equation
(5)
∫
b
-
λ
b
‍
h
(
t
)
d
t
=
∫
a
b
‍
h
(
t
)
g
(
t
)
d
t
.
If f
/
h is a nonincreasing function, then the reverse inequality in (4) holds.In 2000 Mercer [3] gave a generalization that contains various already known generalizations, one of which is the aforementioned generalization given by Pečarić. In 2007 Wu and Srivastava [4] noted that Mercer’s result is incorrect and they have corrected it and gave a refinement of Steffensen’s inequality. Liu [5] also noted that Mercer’s result is incorrect as stated.Motivated by refinement of Steffensen’s inequality given in [4], Pečarić et al. [6] obtained the following refined version of results given in Theorems 2 and 3.Corollary 4.
Leth be a positive integrable function on [
a
,
b
] and f, g integrable functions on [
a
,
b
] such that f
/
h is nonincreasing and 0
≤
g
≤
1. Then
(6)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≤
∫
a
a
+
λ
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
a
+
λ
)
h
(
a
+
λ
)
]
h
(
t
)
[
1
-
g
(
t
)
]
)
d
t
≤
∫
a
a
+
λ
‍
f
(
t
)
d
t
,
where λ is given by (3).
Iff
/
h is a nondecreasing function, then the reverse inequality in (6) holds.Corollary 5.
Leth be a positive integrable function on [
a
,
b
] and f
,
g integrable functions on [
a
,
b
] such that f
/
h is nonincreasing and 0
≤
g
≤
1. Then
(7)
∫
b
-
λ
b
‍
f
(
t
)
d
t
≤
∫
b
-
λ
b
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
b
-
λ
)
h
(
b
-
λ
)
]
h
(
t
)
[
1
-
g
(
t
)
]
)
d
t
≤
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
,
where λ is given by (5).
Iff
/
h is a nondecreasing function, then the reverse inequality in (7) holds.In this paper we obtain generalized Steffensen type inequalities, related to the aforementioned generalizations and refinements of Steffensen’s inequality, for the class of functions that are convex at pointc. Moreover, we construct linear functionals from the difference of the right- and left-hand side of obtained generalized inequalities and derive new families of exponentially convex functions related to constructed functionals.


## 2. Main Results

Let us begin by introducing a class of functions that extends the class of convex functions.Definition 6.
Leth
:
[
a
,
b
]
→
R be a positive function, f
:
[
a
,
b
]
→
R a function, and c
∈
(
a
,
b
). We say that f
/
h belongs to the class M
1
c
[
a
,
b
] (M
2
c
[
a
,
b
]) if there exists a constant A such that the function F
(
x
)
/
h
(
x
)
=
(
f
(
x
)
/
h
(
x
)
)
-
A
x is nonincreasing (nondecreasing) on [
a
,
c
] and nondecreasing (nonincreasing) on [
c
,
b
].As noted in [7] we can describe the property from Definition 6 as “convexity at point c.” In [7] Pečarić and Smoljak also proved that there is a connection between the class of functions M
1
c
[
a
,
b
] and the class of convex functions. This connection is given in the following theorem.Theorem 7.
The functionf
/
h is convex (concave) on [
a
,
b
] if and only if f
/
h
∈
M
1
c
[
a
,
b
]
(
f
/
h
∈
M
2
c
[
a
,
b
]
) for every c
∈
(
a
,
b
).Applying the generalizations of Steffensen’s inequality given in the Introduction to functions that are convex at pointc we obtain the following results.Theorem 8.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
1. Let λ
1 be the solution of the equation
(8)
∫
a
a
+
λ
1
‍
h
(
t
)
d
t
=
∫
a
c
‍
h
(
t
)
g
(
t
)
d
t
and let λ
2 be the solution of the equation
(9)
∫
b
-
λ
2
b
‍
h
(
t
)
d
t
=
∫
c
b
‍
h
(
t
)
g
(
t
)
d
t
.
If f
/
h
∈
M
1
c
[
a
,
b
] and
(10)
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
=
∫
a
a
+
λ
1
‍
t
h
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
t
h
(
t
)
d
t
,
then
(11)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≤
∫
a
a
+
λ
1
‍
f
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
f
(
t
)
d
t
.
If f
/
h
∈
M
2
c
[
a
,
b
] and (10) holds, the inequality in (11) is reversed.Proof.
Letf
/
h
∈
M
1
c
[
a
,
b
] and let F
(
x
)
=
f
(
x
)
-
A
x
h
(
x
), where A is the constant from Definition 6. Since F
/
h
:
[
a
,
c
]
→
R is nonincreasing, from Theorem 2 we obtain
(12)
0
≤
∫
a
a
+
λ
1
‍
F
(
t
)
d
t
-
∫
a
c
‍
F
(
t
)
g
(
t
)
d
t
=
∫
a
a
+
λ
1
‍
f
(
t
)
d
t
-
∫
a
c
‍
f
(
t
)
g
(
t
)
d
t
-
A
(
∫
a
a
+
λ
1
‍
t
h
(
t
)
d
t
-
∫
a
c
‍
t
h
(
t
)
g
(
t
)
d
t
)
.
Since F
/
h
:
[
c
,
b
]
→
R is nondecreasing, from Theorem 3 we obtain
(13)
0
≥
∫
c
b
‍
F
(
t
)
g
(
t
)
d
t
-
∫
b
-
λ
2
b
‍
F
(
t
)
d
t
=
∫
c
b
‍
f
(
t
)
g
(
t
)
d
t
-
∫
b
-
λ
2
b
‍
f
(
t
)
d
t
-
A
(
∫
c
b
‍
t
h
(
t
)
g
(
t
)
d
t
-
∫
b
-
λ
2
b
‍
t
h
(
t
)
d
t
)
.
Now from (12) and (13) we obtain
(14)
∫
a
a
+
λ
1
‍
f
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
f
(
t
)
d
t
-
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≥
A
(
∫
a
a
+
λ
1
‍
t
h
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
t
h
(
t
)
d
t
-
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
)
.
Hence, if (10) is satisfied, then (11) holds.
It is similar forf
/
h
∈
M
2
c
[
a
,
b
].Remark 9.
From the proof we deduce that condition (10) can be weakened. So, for f
/
h
∈
M
1
c
[
a
,
b
] inequality (11) still holds if (10) is replaced by the weaker condition
(15)
A
(
∫
a
a
+
λ
1
‍
t
h
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
t
h
(
t
)
d
t
-
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
)
≥
0
,
where A is the constant from Definition 6. For f
/
h
∈
M
2
c
[
a
,
b
] the reverse inequality in (11) holds if (10) is replaced by (15) with the reverse inequality.
Moreover, condition (15) can be further weakened if the function f
/
h is monotonic.
First, let us show that forf
/
h
∈
M
1
c
[
a
,
b
] we have
(16)
(
f
h
)
-
′
(
c
)
≤
A
≤
(
f
h
)
+
′
(
c
)
.
Since F
/
h is nonincreasing on [
a
,
c
] and nondecreasing on [
c
,
b
] for all distinct points x
1
,
x
2
∈
[
a
,
c
] and y
1
,
y
2
∈
[
c
,
b
] we have
(17)
[
x
1
,
x
2
;
F
h
]
=
[
x
1
,
x
2
;
f
h
]
-
A
≤
0
≤
[
y
1
,
y
2
;
f
h
]
-
A
=
[
y
1
,
y
2
;
F
h
]
.
Therefore, if (
f
/
h
)
-
′
(
c
) and (
f
/
h
)
+
′
(
c
) exist, letting x
i
↗
c and y
i
↘
c, i
=
1,2 we get (16). Similarly, for f
/
h
∈
M
2
c
[
a
,
b
] we have (16) with the reverse inequality.
Hence, if the functionf
/
h
∈
M
1
c
[
a
,
b
] is nondecreasing or f
/
h
∈
M
2
c
[
a
,
b
] is nonincreasing, from (15) we obtain that (10) can be weakened to
(18)
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
≤
∫
a
a
+
λ
1
‍
t
h
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
t
h
(
t
)
d
t
.
Further, if f
/
h
∈
M
1
c
[
a
,
b
] is nonincreasing or f
/
h
∈
M
2
c
[
a
,
b
] is nondecreasing, (10) can be weakened to (18) with the reverse inequality.Theorem 10.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
1. Let λ
1 be the solution of the equation
(19)
∫
c
-
λ
1
c
‍
h
(
t
)
d
t
=
∫
a
c
‍
h
(
t
)
g
(
t
)
d
t
and let λ
2 be the solution of the equation
(20)
∫
c
c
+
λ
2
‍
h
(
t
)
d
t
=
∫
c
b
‍
h
(
t
)
g
(
t
)
d
t
.
If f
/
h
∈
M
1
c
[
a
,
b
] and
(21)
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
=
∫
c
-
λ
1
c
+
λ
2
‍
t
h
(
t
)
d
t
,
then
(22)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≥
∫
c
-
λ
1
c
+
λ
2
‍
f
(
t
)
d
t
.
If f
/
h
∈
M
2
c
[
a
,
b
] and (21) holds, the inequality in (22) is reversed.Proof.
Letf
/
h
∈
M
1
c
[
a
,
b
] and let F
(
x
)
=
f
(
x
)
-
A
x
h
(
x
), where A is the constant from Definition 6. F
/
h
:
[
a
,
c
]
→
R is nonincreasing, so from Theorem 3 we obtain
(23)
0
≤
∫
a
c
‍
f
(
t
)
g
(
t
)
d
t
-
∫
c
-
λ
1
c
‍
f
(
t
)
d
t
-
A
(
∫
a
c
‍
t
h
(
t
)
g
(
t
)
d
t
-
∫
c
-
λ
1
c
‍
t
h
(
t
)
d
t
)
.
F
/
h
:
[
c
,
b
]
→
R is nondecreasing, so from Theorem 2 we obtain
(24)
0
≥
∫
c
c
+
λ
2
‍
f
(
t
)
d
t
-
∫
c
b
‍
f
(
t
)
g
(
t
)
d
t
-
A
(
∫
c
c
+
λ
2
‍
t
h
(
t
)
d
t
-
∫
c
b
‍
t
h
(
t
)
g
(
t
)
d
t
)
.
Hence, from (23) and (24) we conclude
(25)
∫
a
b
f
(
t
)
g
(
t
)
d
t
-
∫
c
-
λ
1
c
+
λ
2
‍
f
(
t
)
d
t
≥
A
(
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
-
∫
c
-
λ
1
c
+
λ
2
‍
t
h
(
t
)
d
t
)
.
Hence, if ∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
=
∫
c
-
λ
1
c
+
λ
2
‍
t
h
(
t
)
d
t, then (22) holds.
It is similar forf
/
h
∈
M
2
c
[
a
,
b
].Remark 11.
Forf
/
h
∈
M
1
c
[
a
,
b
] inequality (22) still holds if condition (21) is replaced by the weaker condition
(26)
A
(
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
-
∫
c
-
λ
1
c
+
λ
2
‍
t
h
(
t
)
d
t
)
≥
0
,
where A is the constant from Definition 6. Also, for f
/
h
∈
M
2
c
[
a
,
b
] the reverse inequality in (22) holds if (21) is replaced by (26) with the reverse inequality.
Additionally, condition (26) can be further weakened if the function f
/
h is monotonic. Similarly, as in Remark 9, if the function f
/
h
∈
M
1
c
[
a
,
b
] is nondecreasing or f
/
h
∈
M
2
c
[
a
,
b
] is nonincreasing, from (26) we obtain that (21) can be weakened to
(27)
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
≥
∫
c
-
λ
1
c
+
λ
2
‍
t
h
(
t
)
d
t
.
Further, if f
/
h
∈
M
1
c
[
a
,
b
] is nonincreasing or f
/
h
∈
M
2
c
[
a
,
b
] is nondecreasing, (21) can be weakened to (27) with the reverse inequality.As a consequence of Theorems8 and 10 we obtain generalized Steffensen type inequalities that involve convex functions.Corollary 12.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
1. Let λ
1 be the solution of (8) and λ
2 the solution of (9). If f
/
h
:
[
a
,
b
]
→
R is convex and (10) holds, then the inequality in (11) holds.
Iff
/
h
:
[
a
,
b
]
→
R is concave, the inequality in (11) is reversed.Proof.
Sincef
/
h is convex, from Theorem 7, we have that f
/
h
∈
M
1
c
[
a
,
b
] for every c
∈
(
a
,
b
). Hence, we can apply Theorem 8.Corollary 13.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
1. Let λ
1 be the solution of (19) and λ
2 the solution of (20). If f
/
h
:
[
a
,
b
]
→
R is convex and (21) holds, then the inequality in (22) holds.
Iff
/
h
:
[
a
,
b
]
→
R is concave, the inequality in (22) is reversed.Proof.
The proof is similar to that of Corollary12 applying Theorem 10.Remark 14.
Similarly, as in Remarks9 and 11, we obtain that conditions (10) and (21) in Corollaries 12 and 13 can be weakened if, additionally, the function f
/
h is monotonic.Remark 15.
Forh
≡
1 in Theorems 8 and 10 and Corollaries 12 and 13 we obtain the results given in [7].In [6, Theorems 2.4 and 2.5] Pečarić et al. gave a corrected version of Mercer’s result [3, Theorem 2] which follows from Theorems 2 and 3. In the following theorems we obtain generalizations of these results for functions from the class M
1
c
[
a
,
b
].Theorem 16.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
h. Let λ
1 be the solution of the equation ∫
a
a
+
λ
1
‍
h
(
t
)
d
t
=
∫
a
c
‍
g
(
t
)
d
t and λ
2 the solution of the equation ∫
b
-
λ
2
b
‍
h
(
t
)
d
t
=
∫
c
b
‍
g
(
t
)
d
t. If f
∈
M
1
c
[
a
,
b
] and
(28)
∫
a
b
‍
t
g
(
t
)
d
t
=
∫
a
a
+
λ
1
‍
t
h
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
t
h
(
t
)
d
t
,
then
(29)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≤
∫
a
a
+
λ
1
‍
f
(
t
)
h
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
f
(
t
)
h
(
t
)
d
t
.
If f
∈
M
2
c
[
a
,
b
] and (28) holds, the inequality in (29) is reversed.Proof.
Takeg
↦
g
/
h and f
↦
f
h in Theorem 8.Theorem 17.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
h. Let λ
1 be the solution of the equation ∫
c
-
λ
1
c
‍
h
(
t
)
d
t
=
∫
a
c
‍
g
(
t
)
d
t and λ
2 the solution of the equation ∫
c
c
+
λ
2
‍
h
(
t
)
d
t
=
∫
c
b
‍
g
(
t
)
d
t. If f
∈
M
1
c
[
a
,
b
] and
(30)
∫
a
b
‍
t
g
(
t
)
d
t
=
∫
c
-
λ
1
c
+
λ
2
‍
t
h
(
t
)
d
t
,
then
(31)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≥
∫
c
-
λ
1
c
+
λ
2
‍
f
(
t
)
h
(
t
)
d
t
.
If f
∈
M
2
c
[
a
,
b
] and (30) holds, the inequality in (31) is reversed.Proof.
Takeg
↦
g
/
h and f
↦
f
h in Theorem 10.In [6, Theorem 2.6] Pečarić et al. showed that Mercer’s generalization [3, Theorem 3] is equivalent to Theorem 2. Further, in [6, Theorem 2.7] they obtained analogue theorem equivalent to Theorem 3. Motivated by the mentioned generalizations in the following theorems we obtain generalizations for functions from class M
1
c
[
a
,
b
].Theorem 18.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
,
k
:
[
a
,
b
]
→
R be integrable functions such that 0
≤
g
≤
k. Let λ
1 be the solution of the equation
(32)
∫
a
a
+
λ
1
‍
k
(
t
)
h
(
t
)
d
t
=
∫
a
c
‍
h
(
t
)
g
(
t
)
d
t
and let λ
2 be the solution of the equation
(33)
∫
b
-
λ
2
b
‍
k
(
t
)
h
(
t
)
d
t
=
∫
c
b
‍
h
(
t
)
g
(
t
)
d
t
.
If f
/
h
∈
M
1
c
[
a
,
b
] and
(34)
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
=
∫
a
a
+
λ
1
‍
t
k
(
t
)
h
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
t
k
(
t
)
h
(
t
)
d
t
,
then
(35)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≤
∫
a
a
+
λ
1
‍
f
(
t
)
k
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
f
(
t
)
k
(
t
)
d
t
.
If f
/
h
∈
M
2
c
[
a
,
b
] and (34) holds, the inequality in (35) is reversed.Proof.
Takeh
↦
k
h, g
↦
g
/
k and f
↦
f
k in Theorem 8.Theorem 19.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
,
k
:
[
a
,
b
]
→
R be integrable functions such that 0
≤
g
≤
k. Let λ
1 be the solution of the equation
(36)
∫
c
-
λ
1
c
‍
k
(
t
)
h
(
t
)
d
t
=
∫
a
c
‍
h
(
t
)
g
(
t
)
d
t
and let λ
2 be the solution of the equation
(37)
∫
c
c
+
λ
2
‍
k
(
t
)
h
(
t
)
d
t
=
∫
c
b
‍
h
(
t
)
g
(
t
)
d
t
.
If f
/
h
∈
M
1
c
[
a
,
b
] and
(38)
∫
a
b
‍
t
h
(
t
)
g
(
t
)
d
t
=
∫
c
-
λ
1
c
+
λ
2
‍
t
k
(
t
)
h
(
t
)
d
t
,
then
(39)
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
≥
∫
c
-
λ
1
c
+
λ
2
‍
f
(
t
)
k
(
t
)
d
t
.
If f
/
h
∈
M
2
c
[
a
,
b
] and (38) holds, the inequality in (39) is reversed.Proof.
Takeh
↦
k
h, g
↦
g
/
k and f
↦
f
k in Theorem 10.Remark 20.
Takingk
≡
1 in Theorems 18 and 19 we obtain Theorems 8 and 10, respectively.In the following theorems we obtain refined version of results given in Theorems8 and 10.Theorem 21.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
1. Let λ
1 be the solution of (8) and λ
2 the solution of (9). If f
/
h
∈
M
1
c
[
a
,
b
] and
(40)
∫
a
b
t
h
(
t
)
g
(
t
)
d
t
=
∫
a
a
+
λ
1
‍
(
t
h
(
t
)
-
[
t
-
a
-
λ
1
]
h
(
t
)
[
1
-
g
(
t
)
]
)
d
t
+
∫
b
-
λ
2
b
‍
(
t
h
(
t
)
-
[
t
-
b
+
λ
2
]
h
(
t
)
[
1
-
g
(
t
)
]
)
d
t
,
then
(41)
∫
a
b
f
(
t
)
g
(
t
)
d
t
≤
∫
a
a
+
λ
1
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
a
+
λ
1
)
h
(
a
+
λ
1
)
]
K
K
K
K
K
K
K
×
h
(
t
)
[
1
-
g
(
t
)
]
f
(
t
)
h
(
t
)
)
d
t
+
∫
b
-
λ
2
b
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
b
-
λ
2
)
h
(
b
-
λ
2
)
]
K
K
K
K
K
K
K
K
×
h
(
t
)
[
1
-
g
(
t
)
]
f
(
b
-
λ
2
)
h
(
b
-
λ
2
)
)
d
t
.
If f
/
h
∈
M
2
c
[
a
,
b
] and (40) holds, the inequality in (41) is reversed.Proof.
The proof is similar to that of Theorem8 applying Corollary 4 for F
/
h
:
[
a
,
c
]
→
R nonincreasing and Corollary 5 for F
/
h
:
[
c
,
b
]
→
R nondecreasing.Theorem 22.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
1. Let λ
1 be the solution of (19) and λ
2 the solution of (20). If f
/
h
∈
M
1
c
[
a
,
b
] and
(42)
∫
a
b
t
h
(
t
)
g
(
t
)
d
t
=
∫
c
-
λ
1
c
‍
(
t
h
(
t
)
-
[
t
-
c
+
λ
1
]
h
(
t
)
[
1
-
g
(
t
)
]
)
d
t
+
∫
c
c
+
λ
2
‍
(
t
h
(
t
)
-
[
t
-
c
-
λ
2
]
h
(
t
)
[
1
-
g
(
t
)
]
)
d
t
,
then
(43)
∫
a
b
f
(
t
)
g
(
t
)
d
t
≥
∫
c
-
λ
1
c
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
c
-
λ
1
)
h
(
c
-
λ
1
)
]
K
K
K
K
K
l
l
K
×
h
(
t
)
[
1
-
g
(
t
)
]
f
(
c
-
λ
1
)
h
(
c
-
λ
1
)
)
d
t
+
∫
c
c
+
λ
2
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
c
+
λ
2
)
h
(
c
+
λ
2
)
]
K
K
K
K
K
K
K
K
×
h
(
t
)
[
1
-
g
(
t
)
]
f
(
t
)
h
(
t
)
)
d
t
.
If f
/
h
∈
M
2
c
[
a
,
b
] and (42) holds, the inequality in (43) is reversed.Proof.
The proof is similar to that of Theorem10 applying Corollary 5 for F
/
h
:
[
a
,
c
]
→
R nonincreasing and Corollary 4 for F
/
h
:
[
c
,
b
]
→
R nondecreasing.Motivated by sharpened and generalized versions of Theorems2 and 3 obtained by Pečarić at al. in [6, Corollaries 2.4 and 2.5] we obtain the following results.Theorem 23.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
,
ψ
:
[
a
,
b
]
→
R be integrable functions such that 0
≤
ψ
≤
g
≤
1
-
ψ. Let λ
1 be the solution of (8) and λ
2 the solution of (9). If f
/
h
∈
M
1
c
[
a
,
b
] and
(44)
∫
a
b
t
h
(
t
)
g
(
t
)
d
t
=
∫
a
a
+
λ
1
‍
t
h
(
t
)
d
t
-
∫
a
c
‍
|
t
-
a
-
λ
1
|
h
(
t
)
ψ
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
t
h
(
t
)
d
t
+
∫
c
b
‍
|
t
-
b
+
λ
2
|
h
(
t
)
ψ
(
t
)
d
t
,
then
(45)
∫
a
b
f
(
t
)
g
(
t
)
d
t
≤
∫
a
a
+
λ
1
‍
f
(
t
)
d
t
-
∫
a
c
‍
|
f
(
t
)
h
(
t
)
-
f
(
a
+
λ
1
)
h
(
a
+
λ
1
)
|
h
(
t
)
ψ
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
f
(
t
)
d
t
+
∫
c
b
‍
|
f
(
t
)
h
(
t
)
-
f
(
b
-
λ
2
)
h
(
b
-
λ
2
)
|
h
(
t
)
ψ
(
t
)
d
t
.
If f
/
h
∈
M
2
c
[
a
,
b
] and (44) holds, the inequality in (45) is reversed.Proof.
The proof is similar to that of Theorem8 applying [6, Corollary 2.3] for F
/
h
:
[
a
,
c
]
→
R nonincreasing and [6, Corollary 2.4] for F
/
h
:
[
c
,
b
]
→
R nondecreasing.Theorem 24.
Leth
:
[
a
,
b
]
→
R be a positive integrable function, let f
:
[
a
,
b
]
→
R be an integrable function, and let c
∈
(
a
,
b
). Let g
,
ψ
:
[
a
,
b
]
→
R be integrable functions such that 0
≤
ψ
≤
g
≤
1
-
ψ. Let λ
1 be the solution of (19) and λ
2 the solution of (20). If f
/
h
∈
M
1
c
[
a
,
b
] and
(46)
∫
a
b
t
h
(
t
)
g
(
t
)
d
t
=
∫
c
-
λ
1
c
+
λ
2
‍
t
h
(
t
)
d
t
-
∫
a
c
‍
|
t
-
c
+
λ
1
|
h
(
t
)
ψ
(
t
)
d
t
+
∫
c
b
‍
|
t
-
c
-
λ
2
|
h
(
t
)
ψ
(
t
)
d
t
,
then
(47)
∫
a
b
f
(
t
)
g
(
t
)
d
t
≥
∫
c
-
λ
1
c
+
λ
2
‍
f
(
t
)
d
t
+
∫
a
c
‍
|
f
(
t
)
h
(
t
)
-
f
(
c
-
λ
1
)
h
(
c
-
λ
1
)
|
h
(
t
)
ψ
(
t
)
d
t
-
∫
c
b
‍
|
f
(
t
)
h
(
t
)
-
f
(
c
+
λ
2
)
h
(
c
+
λ
2
)
|
h
(
t
)
ψ
(
t
)
d
t
.
If f
/
h
∈
M
2
c
[
a
,
b
] and (46) holds, the inequality in (47) is reversed.Proof.
The proof is similar to that of Theorem10 applying [6, Corollary 2.4] for F
/
h
:
[
a
,
c
]
→
R nonincreasing and [6, Corollary 2.3] for F
/
h
:
[
c
,
b
]
→
R nondecreasing.Remark 25.
Generalized Steffensen type inequalities obtained in Theorems18–24 also hold if the function f
/
h is convex (concave). This follows from Theorem 7; that is, if f
/
h is a convex function, then f
/
h
∈
M
1
c
[
a
,
b
] for every c
∈
(
a
,
b
).Remark 26.
Similarly, as in Remarks9 and 11, we obtain that conditions (28), (30), (34), (38), (40), (42), (44), and (46) can be weakened, but here we omit the details.


## 3. Mean Value Theorems

Generalizations of Steffensen type inequalities given by (11), (22), (35), (39), (41), and (43) are linear in f. Hence, we can define the following linear functionals:
(48)
L
1
(
f
)
=
∫
a
a
+
λ
1
‍
f
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
f
(
t
)
d
t
-
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
,
(49)
L
2
(
f
)
=
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
-
∫
c
-
λ
1
c
+
λ
2
‍
f
(
t
)
d
t
,
(50)
L
3
(
f
)
=
∫
a
a
+
λ
1
‍
f
(
t
)
k
(
t
)
d
t
+
∫
b
-
λ
2
b
‍
f
(
t
)
k
(
t
)
d
t
-
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
,
(51)
L
4
(
f
)
=
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
-
∫
c
-
λ
1
c
+
λ
2
‍
f
(
t
)
k
(
t
)
d
t
,
(52)
L
5
(
f
)
=
∫
a
a
+
λ
1
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
a
+
λ
1
)
h
(
a
+
λ
1
)
]
k
k
k
k
k
k
k
k
k
k
l
k
k
×
h
(
t
)
[
1
-
g
(
t
)
]
f
(
t
)
h
(
t
)
)
d
t
+
∫
b
-
λ
2
b
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
b
-
λ
2
)
h
(
b
-
λ
2
)
]
k
k
k
k
k
k
k
k
l
×
h
(
t
)
[
1
-
g
(
t
)
]
f
(
t
)
h
(
t
)
)
d
t
-
∫
a
b
‍
f
(
t
)
g
(
t
)
d
t
,
(53)
L
6
(
f
)
=
∫
a
b
f
(
t
)
g
(
t
)
d
t
‍
-
∫
c
-
λ
1
c
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
c
-
λ
1
)
h
(
c
-
λ
1
)
]
k
k
k
k
k
k
k
k
k
×
h
(
t
)
[
1
-
g
(
t
)
]
f
(
t
)
h
(
t
)
)
d
t
-
∫
c
c
+
λ
2
‍
(
f
(
t
)
-
[
f
(
t
)
h
(
t
)
-
f
(
c
+
λ
2
)
h
(
c
+
λ
2
)
]
k
k
k
k
k
k
k
l
k
k
×
h
(
t
)
[
1
-
g
(
t
)
]
f
(
t
)
h
(
t
)
)
d
t
.Under the assumptions of Theorems8, 10, 18, 19, 21, and 22 we have that L
i
(
f
)
≥
0, i
=
1
,
…
,
6, for f
/
h
∈
M
1
c
[
a
,
b
]. Further, Corollaries 12 and 13 and Remark 25 assure that L
i
(
f
)
≥
0, i
=
1
,
…
,
6, for any convex function f
/
h.Let us begin by showing a Lagrange type mean value theorem for the functionalL
1.Theorem 27.
Leth
:
[
a
,
b
]
→
R be a positive integrable function and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
1. Let λ
1 be the solution of (8) and λ
2 the solution of (9). Let f
:
[
a
,
b
]
→
R be such that f
/
h
∈
C
2
[
a
,
b
]. If (10) holds, then there exists ξ
∈
[
a
,
b
] such that
(54)
L
1
(
f
)
=
1
2
(
f
(
ξ
)
h
(
ξ
)
)
′
′
×
[
∫
a
b
‍
t
2
h
(
t
)
g
(
t
)
d
t
-
∫
a
a
+
λ
1
‍
t
2
h
(
t
)
d
t
kk
-
∫
b
-
λ
2
b
‍
t
2
h
(
t
)
d
t
]
,
where L
1 is defined by (48).Proof.
Sincef
/
h
∈
C
2
[
a
,
b
], there exist
(55)
m
=
min
⁡
x
∈
[
a
,
b
]
(
f
(
x
)
h
(
x
)
)
′
′
,
M
=
max
⁡
x
∈
[
a
,
b
]
(
f
(
x
)
h
(
x
)
)
′
′
.
Let
(56)
Φ
1
(
x
)
=
f
(
x
)
-
m
2
x
2
h
(
x
)
,
Φ
2
(
x
)
=
M
2
x
2
h
(
x
)
-
f
(
x
)
.
The functions Φ
1
/
h and Φ
2
/
h are convex since Φ
i
/
h
≥
0, i
=
1,2. Hence, L
1
(
Φ
i
)
≥
0, i
=
1,2, and we obtain
(57)
m
2
L
1
(
f
~
)
≤
L
1
(
f
)
≤
M
2
L
1
(
f
~
)
,
where f
~
(
x
)
=
x
2
h
(
x
). Since f
~
/
h is convex we have L
1
(
f
~
)
≥
0.
IfL
1
(
f
~
)
=
0, then (57) implies L
1
(
f
)
=
0 and (54) holds for every ξ
∈
[
a
,
b
]. Otherwise, multiplying (57) by 2
/
L
1
(
f
~
) we obtain
(58)
m
≤
2
L
1
(
f
)
L
1
(
f
~
)
≤
M
,
so continuinity of (
f
/
h
)
′′ ensures the existence of ξ
∈
[
a
,
b
] satisfying (54).We continue with a Cauchy type mean value theorem for the functionalL
1.Theorem 28.
Leth
:
[
a
,
b
]
→
R be a positive integrable function and let c
∈
(
a
,
b
). Let g
:
[
a
,
b
]
→
R be an integrable function such that 0
≤
g
≤
1. Let λ
1 be the solution of (8) and λ
2 the solution of (9). Let the functions f and F be such that f
/
h
,
F
/
h
∈
C
2
[
a
,
b
]. If (10) holds and L
1
(
F
)
≠
0, then there exists ξ
∈
[
a
,
b
] such that
(59)
L
1
(
f
)
L
1
(
F
)
=
(
f
/
h
)
′′
(
ξ
)
(
F
/
h
)
′′
(
ξ
)
,
where L
1 is defined by (48).Proof.
DefineΨ
(
x
)
=
L
1
(
F
)
f
(
x
)
-
L
1
(
f
)
F
(
x
). Due to linearity of L
1 we have L
1
(
Ψ
)
=
0. Now by Theorem 27 there exist ξ
,
ξ
1
∈
[
a
,
b
] such that
(60)
0
=
L
1
(
Ψ
)
=
1
2
(
Ψ
(
ξ
)
h
(
ξ
)
)
′′
L
1
(
f
~
)
0
≠
L
1
(
F
)
=
1
2
(
F
(
ξ
1
)
h
(
ξ
1
)
)
′′
L
1
(
f
~
)
,
where f
~
(
x
)
=
x
2
h
(
x
). Therefore, L
1
(
f
~
)
≠
0 and
(61)
0
=
(
Ψ
(
ξ
)
h
(
ξ
)
)
′′
=
L
1
(
F
)
(
f
(
ξ
)
h
(
ξ
)
)
′′
-
L
1
(
f
)
(
F
(
ξ
)
h
(
ξ
)
)
′′
,
which gives the claim of the theorem.Remark 29.
As in Theorem27 we can obtain Lagrange type mean value theorems for the functionals L
i, i
=
2
,
…
,
6. Similarly, Cauchy type mean value theorems can be obtained for the functionals L
i, i
=
2
,
…
,
6. Hence, we can obtain that there exist ξ
i
∈
[
a
,
b
], i
=
2
,
…
,
6, such that
(62)
L
i
(
f
)
L
i
(
F
)
=
(
f
/
h
)
′′
(
ξ
i
)
(
F
/
h
)
′′
(
ξ
i
)
,
i
=
2
,
…
,
6
.


## 4. Exponential Convexity

We begin by recalling some definitions and results on exponential convexity; see [8, 9].Definition 30.
A functionψ
:
I
→
R is n-exponentially convex in the Jensen sense on I if
(63)
∑
i
,
j
=
1
n
‍
ξ
i
ξ
j
ψ
(
x
i
+
x
j
2
)
≥
0
holds for all choices ξ
1
,
…
,
ξ
n
∈
R and all choices x
1
,
…
,
x
n
∈
I.
A functionψ
:
I
→
R is n
-exponentially convex on I if it is n-exponentially convex in the Jensen sense and continuous on I.Remark 31.
It is clear from the definition that1-exponentially convex functions in the Jensen sense are nonnegative functions.
Also,n-exponentially convex functions in the Jensen sense are k-exponentially convex in the Jensen sense for all k
≤
n and k
∈
N.Definition 32.
A functionψ
:
I
→
R is exponentially convex in the Jensen sense on I if it is n-exponentially convex in the Jensen sense on I for every n
∈
N.
A functionψ
:
I
→
R is exponentially convex on I if it is exponentially convex in the Jensen sense and continuous on I.Remark 33.
A functionψ
:
I
→
R is log-convex in the Jensen sense, that is,
(64)
ψ
(
x
+
y
2
)
2
≤
ψ
(
x
)
ψ
(
y
)
,
∀
x
,
y
∈
I
,
if and only if
(65)
α
2
ψ
(
x
)
+
2
α
β
ψ
(
x
+
y
2
)
+
β
2
ψ
(
y
)
≥
0
holds for all α
,
β
∈
R and x
,
y
∈
I, that is, if and only if ψ is 2-exponentially convex in the Jensen sense. By induction from (64) we have
(66)
ψ
(
1
2
k
x
+
(
1
-
1
2
k
)
y
)
≤
ψ
(
x
)
1
/
2
k
ψ
(
y
)
1
-
(
1
/
2
k
)
.
Therefore, if ψ is continuous and ψ
(
x
)
=
0 for some x
∈
I, then from the last inequality and nonnegativity of ψ (see Remark 31) we get
(67)
ψ
(
y
)
=
lim
⁡
k
→
∞
ψ
(
1
2
k
x
+
(
1
-
1
2
k
)
y
)
=
0
∀
y
∈
I
.
Hence, either 2-exponentially convex function is identically equal to zero or it is strictly positive and log-convex.We also use the following well known results for convex functions.Lemma 34.
A functionψ
:
I
→
R is convex if and only if the inequality
(68)
(
x
3
-
x
2
)
ψ
(
x
1
)
+
(
x
1
-
x
3
)
ψ
(
x
2
)
+
(
x
2
-
x
1
)
ψ
(
x
3
)
≥
0
holds for all x
1
,
x
2
,
x
3
∈
I such that x
1
<
x
2
<
x
3.Proposition 35.
Iff is a convex function on I and if x
1
≤
y
1, x
2
≤
y
2, x
1
≠
x
2, y
1
≠
y
2, then the following inequality holds:
(69)
f
(
x
2
)
-
f
(
x
1
)
x
2
-
x
1
≤
f
(
y
2
)
-
f
(
y
1
)
y
2
-
y
1
.
If the function f is concave, the inequality is reversed.Definition 36.
Letf be a function defined on [
a
,
b
]. The nth order divided difference of f at distinct points x
0
,
x
1
,
…
,
x
n in [
a
,
b
] is defined recursively by
(70)
[
x
j
;
f
]
=
f
(
x
j
)
,
j
=
0
,
…
,
n
,
[
x
0
,
x
1
,
…
,
x
n
;
f
]
=
[
x
1
,
…
,
x
n
;
f
]
-
[
x
0
,
…
,
x
n
-
1
;
f
]
x
n
-
x
0
.Remark 37.
The value[
x
0
,
x
1
,
…
,
x
n
;
f
] is independent of the order of the points x
0
,
…
,
x
n. Previous definition can be extended to include the case in which some or all of the points coincide by assuming that x
0
≤
⋯
≤
x
k and letting
(71)
[
x
,
…
,
x
︸
;
f
j
+
1
times
]
=
f
(
j
)
(
x
)
j
!
,
provided that f
(
j
)
(
x
) exists.Next, we construct exponentially convex functions using the previously defined functionalsL
i, i
=
1
,
…
,
6. In the sequel the notation log
⁡ denotes the natural logarithm function and I, J denote intervals in R.Theorem 38.
LetΩ
=
{
f
p
/
h
:
I
→
R
∣
p
∈
J
} be a family of functions such that for all mutually different points x
0
,
x
1
,
x
2
∈
I the mapping p
↦
[
x
0
,
x
1
,
x
2
;
f
p
/
h
] is n-exponentially convex in the Jensen sense on J. Let L
i, i
=
1
,
…
,
6, be linear functionals defined by (48)–(53). Then the mapping p
↦
L
i
(
f
p
) is n-exponentially convex in the Jensen sense on J.
If the mappingp
↦
L
i
(
f
p
) is continuous on J, then it is n-exponentially convex on J.Proof.
Forξ
j
∈
R and p
j
∈
J, j
=
1
,
…
,
n, we define the function
(72)
Φ
(
x
)
=
∑
j
,
k
=
1
n
‍
ξ
j
ξ
k
f
(
p
j
+
p
k
)
/
2
(
x
)
.
Since the mapping p
↦
[
x
0
,
x
1
,
x
2
;
f
p
/
h
] is n-exponentially convex in the Jensen sense we have
(73)
[
x
0
,
x
1
,
x
2
;
Φ
h
]
=
∑
j
,
k
=
1
n
‍
ξ
j
ξ
k
[
x
0
,
x
1
,
x
2
;
f
(
p
j
+
p
k
)
/
2
h
]
≥
0
.
So Φ
/
h is a convex function and
(74)
0
≤
L
i
(
Φ
)
=
∑
j
,
k
=
1
n
‍
ξ
j
ξ
k
L
i
(
f
(
p
j
+
p
k
)
/
2
)
,
i
=
1
,
…
,
6
.
Therefore, the mapping p
↦
L
i
(
f
p
) is n-exponentially convex on J in the Jensen sense.
If the mappingp
↦
L
i
(
f
p
) is also continuous on J, then p
↦
L
i
(
f
p
) is n-exponentially convex by definition.If the assumptions of Theorem38 hold for all n
∈
N, then we have the following corollary.Corollary 39.
LetΩ
=
{
f
p
/
h
:
I
→
R
∣
p
∈
J
} be a family of functions such that for all mutually different points x
0
,
x
1
,
x
2
∈
I the mapping p
↦
[
x
0
,
x
1
,
x
2
;
f
p
/
h
] is exponentially convex in the Jensen sense on J. Let L
i, i
=
1
,
…
,
6, be linear functionals defined by (48)–(53). Then the mapping p
↦
L
i
(
f
p
) is exponentially convex in the Jensen sense on J.
If the mappingp
↦
L
i
(
f
p
) is continuous on J, then it is exponentially convex on J.Corollary 40.
LetΩ
=
{
f
p
/
h
:
I
→
R
∣
p
∈
J
} be a family of functions such that for all mutually different points x
0
,
x
1
,
x
2
∈
I the mapping p
↦
[
x
0
,
x
1
,
x
2
;
f
p
/
h
] is 2-exponentially convex in the Jensen sense on J. Let L
i, i
=
1
,
…
,
6, be linear functionals defined by (48)–(53). Then the following statements hold. (i)
If the mappingp
↦
L
i
(
f
p
) is continuous on J, then, for r
,
s
,
t
∈
J, such that r
<
s
<
t, we have
(75)
[
L
i
(
f
s
)
]
t
-
r
≤
[
L
i
(
f
r
)
]
t
-
s
[
L
i
(
f
t
)
]
s
-
r
,
i
=
1
,
…
,
6
.
(ii)
If the mappingp
↦
L
i
(
f
p
) is positive and differentiable on J, then for all p
,
q
,
u
,
v
∈
J such that p
≤
u and q
≤
v we have
(76)
μ
p
,
q
(
L
i
,
Ω
)
≤
μ
u
,
v
(
L
i
,
Ω
)
,where
(77)
μ
p
,
q
(
L
i
,
Ω
)
=
{
(
L
i
(
f
p
)
L
i
(
f
q
)
)
1
/
(
p
-
q
)
,
p
≠
q
,
exp
⁡
(
(
d
/
d
p
)
L
i
(
f
p
)
L
i
(
f
p
)
)
,
p
=
q
.Proof.
(i) By Theorem38 the mapping p
↦
L
i
(
f
p
) is 2-exponentially convex. Hence, by Remark 33, either this mapping is identically equal to zero, in which case inequality (75) holds trivially with zeros on both sides, or it is strictly positive and log-convex. Therefore, for r
,
s
,
t
∈
J such that r
<
s
<
t Lemma 34 gives
(78)
(
t
-
s
)
log
⁡
L
i
(
f
r
)

+

(
r
-
t
)
log
⁡
L
i
(
f
s
)
+
(
s
-
r
)
log
⁡
L
i
(
f
t
)
≥
0
,
which is equivalent to inequality (75).
(ii) By (i) we have that the mappingp
↦
L
i
(
f
p
) is log-convex on J; that is, the function p
↦
log
⁡
L
i
(
f
p
) is convex on J. Applying Proposition 35 with p
≤
u, q
≤
v, p
≠
q, u
≠
v, we obtain
(79)
log
⁡
L
i
(
f
p
)
-
log
⁡
L
i
(
f
q
)
p
-
q
≤
log
⁡
L
i
(
f
u
)
-
log
⁡
L
i
(
f
v
)
u
-
v
;
that is
(80)
μ
p
,
q
(
L
i
,
Ω
)
≤
μ
u
,
v
(
L
i
,
Ω
)
.
The limit casesp
=
q and u
=
v are obtained by taking the limits p
→
q and u
→
v.Remark 41.
The results stated in Theorem38 and Corollaries 39 and 40 still hold when some or all of the points x
0
,
x
1
,
x
2
∈
I coincide. The proofs are obtained by recalling Remark 37 and a suitable characterization of convexity.
We continue with an example of a family of functions which satisfies the previous conditions.
Leth be a positive integrable function and let
(81)
Υ
=
{
f
p
h
:
R
⟶
(
0
,
∞
)
∣
p
∈
R
}
be a family of functions where f
p is defined by
(82)
f
p
(
x
)
=
{
1
p
2
e
p
x
h
(
x
)
,
p
≠
0
,
1
2
2
1
2
x
2
h
(
x
)
,
p
=
0
.
We have (
d
2
/
d
x
2
)
(
f
p
(
x
)
/
h
(
x
)
)
=
e
p
x
>
0, so f
p
/
h is convex on R for every p
∈
R and p
↦
(
d
2
/
d
x
2
)
(
f
p
(
x
)
/
h
(
x
)
) is exponentially convex by definition. Using analogous arguing as in the proof of Theorem 38 we have that p
↦
[
t
0
,
t
1
,
t
2
;
f
p
/
h
] is exponentially convex (and so exponentially convex in the Jensen sense). We see that the family Υ satisfies the assumptions of Corollary 39, so mappings p
↦
L
i
(
f
p
) are exponentially convex in the Jensen sense. It is easy to verify that these mappings are continuous, so they are exponentially convex.For this family of functionsμ
p
,
q
(
L
i
,
Υ
), i
=
1
,
…
,
6, from (77) becomes
(83)
μ
p
,
q
(
L
i
,
Υ
)
=
{
(
L
i
(
f
p
)
L
i
(
f
q
)
)
1
/
(
p
-
q
)
,
p
≠
q
,
exp
⁡
(
L
i
(
id
·
f
p
)
L
i
(
f
p
)
-
2
p
)
,
p
=
q
≠
0
,
exp
⁡
(
L
i
(
id
·
f
0
)
3
L
i
(
f
0
)
)
,
p
=
q
=
0
.Explicitly forμ
p
,
q
(
L
1
,
Υ
) we have the following:(i)
forp
≠
q,
(84)
μ
p
,
q
(
L
1
,
Υ
)
=
(
q
2
p
2
(
∫
a
b
‍
e
p
t
h
(
t
)
g
(
t
)
d
t
-
∫
a
a
+
λ
1
‍
e
p
t
h
(
t
)
d
t
kkkkkklklk
-
∫
b
-
λ
2
b
‍
e
p
t
h
(
t
)
d
t
)
k
k
k
k
k
×
(
∫
a
b
‍
e
q
t
h
(
t
)
g
(
t
)
d
t
-
∫
a
a
+
λ
1
‍
e
q
t
h
(
t
)
d
t
kkkkkklk
-
∫
b
-
λ
2
b
‍
e
q
t
h
(
t
)
d
t
)
-
1
)
1
/
(
p
-
q
)
;
(ii)
forp
=
q
≠
0,
(85)
μ
p
,
p
(
L
1
,
Υ
)
=
exp
⁡
(
(
∫
a
b
t
e
p
t
h
(
t
)
g
(
t
)
d
t
-
∫
a
a
+
λ
1
‍
t
e
p
t
h
(
t
)
d
t
‍
kkkkkkkkk
-
∫
b
-
λ
2
b
‍
t
e
p
t
h
(
t
)
d
t
)
k
k
k
k
k
×
(
∫
a
b
‍
e
p
t
h
(
t
)
g
(
t
)
d
t
-
∫
a
a
+
λ
1
‍
e
p
t
h
(
t
)
d
t
kkkkklkk
-
∫
b
-
λ
2
b
‍
e
p
t
h
(
t
)
d
t
)
-
1
-
2
p
)
;
(iii)
forp
=
q
=
0,
(86)
μ
0,0
(
L
1
,
Υ
)
=
exp
⁡
(
1
3
(
∫
a
b
t
3
h
(
t
)
g
(
t
)
d
t
-
∫
a
a
+
λ
1
‍
t
3
h
(
t
)
d
t
‍
kkkkkkkklklk
-
∫
b
-
λ
2
b
‍
t
3
h
(
t
)
d
t
)
k
k
k
k
k
k
k
k
×
(
∫
a
b
‍
t
2
h
(
t
)
g
(
t
)
d
t
-
∫
a
a
+
λ
1
‍
t
2
h
(
t
)
d
t
kkkkkkkkklk
-
∫
a
-
λ
2
b
‍
t
2
h
(
t
)
d
t
)
-
1
)
.Theorem28 applied on functions f
p
/
h
,
f
q
/
h
∈
Υ implies that
(87)
M
p
,
q
(
L
1
,
Υ
)
=
log
⁡
μ
p
,
q
(
L
1
,
Υ
)
satisfies a
≤
M
p
,
q
(
L
1
,
Υ
)
≤
b. Hence M
p
,
q is a monotonic mean by (76).



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*Source: 428030-2014-08-05.xml*