Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex
We establish some new Hermite–Hadamard-type inequalities for functions whose first derivatives are of convexity and apply these inequalities to construct inequalities for special means.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507908363517952 |
|---|---|
| author | Qi, Feng Xi, Bo-Yan Zhang, Tian-Yu Ци, Фен Сі, Бо-Ян Чжан, Т.-Ю |
| author_facet | Qi, Feng Xi, Bo-Yan Zhang, Tian-Yu Ци, Фен Сі, Бо-Ян Чжан, Т.-Ю |
| author_sort | Qi, Feng |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:48:26Z |
| description | We establish some new Hermite–Hadamard-type inequalities for functions whose first derivatives are of convexity and apply these inequalities to construct inequalities for special means. |
| first_indexed | 2026-03-24T02:16:47Z |
| format | Article |
| fulltext |
UDC 517.5
Feng Qi (Inst. Math., Henan Polytechnic Univ. and College Sci., Tianjin Polytechnic Univ., China),
Tian-Yu Zhang, Bo-Yan Xi (College Math., Inner Mongolia Univ. Nat., China)
HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES
FOR FUNCTIONS WHOSE FIRST DERIVATIVES ARE OF CONVEXITY*
IНТЕГРАЛЬНI НЕРIВНОСТI ТИПУ ЕРМIТА – АДАМАРА,
ПЕРШI ПОХIДНI ЯКИХ МАЮТЬ ОПУКЛIСТЬ
We establish some new Hermite – Hadamard-type inequalities for functions whose first derivatives are of convexity and
apply these inequalities to construct inequalities for special means.
Встановлено деякi новi нерiвностi типу Ермiта – Aдамара для функцiй, похiднi яких мають опуклiсть. Цi нерiвностi
застосовано при побудовi нерiвностей для спецiальних середнiх.
1. Introduction. Throughout this paper, we will use I and I◦ to denote an interval on the real line
R and the interior of I respectively.
In [4], the following Hermite – Hadamard type inequalities for continuously differentiable convex
functions were proved.
Theorem 1.1 ([4], Theorem 2.2). Let f : I◦ ⊆ R→ R be a continuously differentiable mapping
on I◦ and a, b ∈ I◦ with a < b. If |f ′| is convex on [a, b], then∣∣∣∣∣∣f(a) + f(b)
2
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ (b− a)(|f ′(a)|+ |f ′(b)|)
8
.
Theorem 1.2 ([4], Theorem 2.3). Let f : I◦ ⊆ R→ R be a continuously differentiable mapping
on I◦, a, b ∈ I◦ with a < b, and p > 1. If the new mapping |f ′|p/(p−1) is convex on [a, b], then∣∣∣∣∣∣f(a) + f(b)
2
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ b− a
2(p+ 1)1/p
[
|f ′(a)|p/(p−1) + |f ′(b)|p/(p−1)
2
](p−1)/p
.
In [9], the above inequalities were generalized as follows.
Theorem 1.3 ([9], Theorems 1 and 2). Let f : I ⊆ R→ R be continuously differentiable on I◦,
a, b ∈ I with a < b, and q ≥ 1. If |f ′|q is convex on [a, b], then∣∣∣∣∣∣f(a) + f(b)
2
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ b− a
4
[
|f ′(a)|q + |f ′(b)|q
2
]1/q
and ∣∣∣∣∣∣f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ b− a
4
[
|f ′(a)|q + |f ′(b)|q
2
]1/q
.
* This paper was partially supported by the National Natural Science Foundation under Grant No. 11361038 of China
and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous
Region under Grant No. NJZY14191, China.
c© FENG QI, TIAN-YU ZHANG, BO-YAN XI, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4 555
556 FENG QI, TIAN-YU ZHANG, BO-YAN XI
In [7], the above inequalities were further generalized as follows.
Theorem 1.4 ([7], Theorems 2.3 and 2.4). Let f : I ⊆ R→ R be continuously differentiable on
I◦, a, b ∈ I with a < b, and p > 1. If |f ′|p/(p−1) is convex on [a, b], then∣∣∣∣∣∣ 1
b− a
b∫
a
f(x)dx− f
(
a+ b
2
)∣∣∣∣∣∣ ≤ b− a
16
(
4
p+ 1
)1/p
×
×
{[
|f ′(a)|p/(p−1) + 3|f ′(b)|p/(p−1)
](p−1)/p
+
[
3|f ′(a)|p/(p−1) + |f ′(b)|p/(p−1)
](p−1)/p}
and ∣∣∣∣∣∣ 1
b− a
b∫
a
f(x)dx− f
(
a+ b
2
)∣∣∣∣∣∣ ≤ b− a
4
(
4
p+ 1
)1/p
(|f ′(a)|+ |f ′(b)|).
In [5], an inequality similar to the above ones was given as follows.
Theorem 1.5 ([5], Theorem 3). Let f : [a, b]→ R be an absolutely continuous mapping on [a, b]
whose derivative belongs to Lp([a, b]). Then∣∣∣∣∣∣13
[
f(a) + f(b)
2
+ 2f
(
a+ b
2
)]
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ 1
6
[
2q+1 + 1
3(q + 1)
]1/q
(b− a)1/q‖f ′‖p,
where
1
p
+
1
q
= 1 and p > 1.
Recently, the following inequalities were obtained in [10, 11].
Theorem 1.6 [10]. Let I ⊆ R be an open interval, with a, b ∈ I and a < b, and f : I → R be
twice continuously differentiable mapping such that f ′′ is integrable. If 0 ≤ λ ≤ 1 and |f ′′| is a
convex function on [a, b], then∣∣∣∣∣∣(λ− 1)f
(
a+ b
2
)
− λf(a) + f(b)
2
+
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤
(b− a)2
24
{[
λ4 + (1 + λ)(1− λ)3 + 5λ− 3
4
]
|f ′′(a)|+
+
[
λ4 + (2− λ)λ3 + 1− 3λ
4
]
|f ′′(b)|
}
, 0 ≤ λ ≤ 1
2
,
(b− a)2
48
(3λ− 1)
(
|f ′′(a)|+ |f ′′(b)|
)
,
1
2
≤ λ ≤ 1.
Theorem 1.7 [11]. Let f : I ⊆ R→ R be continuously differentiable on I◦, a, b ∈ I with a < b,
and f ′ ∈ L([a, b]). If |f ′|q is convex for q ≥ 1 on [a, b], then∣∣∣∣∣∣16
[
f(a) + f(b) + 4f
(
a+ b
2
)]
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ b− a
12
[
2q+1 + 1
3(q + 1)
]1/q [(
3|f ′(a)|q + |f ′(b)|q
4
)1/q
+
(
|f ′(a)|q + 3|f ′(b)|q
4
)1/q
]
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES FOR FUNCTIONS . . . 557
and ∣∣∣∣∣∣16
[
f(a) + f(b) + 4f
(
a+ b
2
)]
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ 5(b− a)
72
[(
61|f ′(a)|q + 29|f ′(b)|q
90
)1/q
+
(
29|f ′(a)|q + 61|f ′(b)|q
90
)1/q
]
.
A function f : I ⊆ R→ [0,∞) is said to be s-convex if the inequality
f(αx+ βy) ≤ αsf(x) + βsf(y)
holds for all x, y ∈ I, α, β ∈ [0, 1] with α+ β = 1, and some fixed s ∈ (0, 1].
In [1], some inequalities of Hermite – Hadamard type for s-convex functions were established as
follows.
Theorem 1.8 ([1], Theorems 2.3 and 2.4). Let f : I ⊆ R → R be a continuously differentiable
mapping on I◦ such that f ′ ∈ L([a, b]), where a, b ∈ I with a < b.
If |f ′|p/(p−1) is s-convex on [a, b] for p > 1 and some fixed s ∈ (0, 1], then∣∣∣∣∣∣f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ b− a
4
(
1
p+ 1
)1/p( 1
s+ 1
)2/q
×
×
{[(
21−s + s+ 1
)
|f ′(a)|q + 21−s|f ′(b)|q
]1/q
+
[
21−s|f ′(a)|q +
(
21−s + s+ 1
)
|f ′(b)|q
]1/q}
,
where p is the conjugate of q, that is,
1
p
+
1
q
= 1.
If |f ′|q is s-convex on [a, b] for q ≥ 1 and some fixed s ∈ (0, 1], then∣∣∣∣∣∣f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ b− a
8
[
2
(s+ 1)(s+ 2)
]1/q
×
×
{[(
21−s + 1
)
|f ′(a)|q + 21−s|f ′(b)|q
]1/q
+
[(
21−s + 1
)
|f ′(b)|q + 21−s|f ′(a)|q
]1/q}
.
Some inequalities of Hermite – Hadamard type were also obtained in [2, 3, 6, 8, 12, 14 – 17, 19]
and plenty of references therein.
In this paper we will establish some new Hermite – Hadamard type integral inequalities for
functions whose first derivatives are of convexity and apply them to derive some inequalities of
special means.
2. Lemmas. For establishing our new integral inequalities of Hermite – Hadamard type, we need
the following lemmas.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
558 FENG QI, TIAN-YU ZHANG, BO-YAN XI
Lemma 2.1. Let I be an interval and f : I → R be continuously differentiable on I◦, with
a, b ∈ I and a < b, and λ, µ ∈ R. If f ′ ∈ L([a, b]), then
(1− µ)f(a) + λf(b) + (µ− λ)f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx =
= (b− a)
1/2∫
0
(λ− t)f ′(ta+ (1− t)b)dt+
1∫
1/2
(µ− t)f ′(ta+ (1− t)b)dt
.
Proof. This follows from standard integration by parts.
Lemma 2.2 [18]. Let f : I ⊆ R→ R be continuously differentiable on I◦, a, b ∈ I with a < b.
If f ′ ∈ L([a, b]) and λ, µ ∈ R, then
λf(a) + µf(b)
2
+
2− λ− µ
2
f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx =
=
b− a
4
1∫
0
[
(1− λ− t)f ′
(
ta+ (1− t)a+ b
2
)
+ (µ− t)f ′
(
t
a+ b
2
+ (1− t)b
)]
dt.
Proof. This may be derived via standard integration by parts.
Remark 2.1. Lemmas 2.1 and 2.2 are equivalent to each other.
3. New integral inequalities of Hermite – Hadamard type. Now we are in a position to
establish some new integral inequalities of Hermite – Hadamard type for functions whose derivatives
are of convexity.
Theorem 3.1. Let f : I ⊆ R→ R be a continuously differentiable function on I◦, a, b ∈ I with
a < b and 0 ≤ λ ≤ 1
2
≤ µ ≤ 1. If |f ′| is convex on [a, b], then
∣∣∣∣∣∣(1− µ)f(a) + λf(b) + (µ− λ)f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ b− a
24
[(
10− 3λ+
+8λ3 − 15µ+ 8µ3
)
|f ′(a)|+
(
8− 9λ+ 24λ2 − 8λ3 − 21µ+ 24µ2 − 8µ3
)
|f ′(b)|
]
. (3.1)
Proof. By Lemma 2.1 and the convexity of |f ′| on [a, b], we have∣∣∣∣∣∣(1− µ)f(a) + λf(b) + (µ− λ)f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ (b− a)
1/2∫
0
|λ− t||f ′(ta+ (1− t)b)|dt+
1∫
1/2
|µ− t||f ′(ta+ (1− t)b)|dt
≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES FOR FUNCTIONS . . . 559
≤ (b− a)
1/2∫
0
|λ− t|
(
t|f ′(a)|+ (1− t)|f ′(b)|
)
dt+
1∫
1/2
|µ− t|
(
t|f ′(a)|+ (1− t)|f ′(b)|
)
dt
.
Substituting equations
1/2∫
0
|λ− t|
(
t|f ′(a)|+ (1− t)|f ′(b)|
)
dt=
1
24
[(
1− 3λ+ 8λ3
)
|f ′(a)|+
(
2− 9λ+24λ2−8λ3
)
|f ′(b)|
]
and
1∫
1/2
|µ− t|
(
t|f ′(a)|+ (1− t)|f ′(b)|
)
dt =
=
1
24
[(
9− 15µ+ 8µ3
)
|f ′(a)|+
(
6− 21µ+ 24µ2 − 8µ3
)
|f ′(b)|
]
into the above inequality leads to (3.1).
Theorem 3.1 is proved.
Theorem 3.2. Let f : I ⊆ R→ R be a continuously differentiable function on I◦, a, b ∈ I with
a < b and 0 ≤ λ ≤ 1
2
≤ µ ≤ 1. If |f ′|q for q > 1 is convex on [a, b] and q ≥ p > 0, then∣∣∣∣∣∣(1− µ)f(a) + λf(b) + (µ− λ)f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ (b− a)
(
q − 1
2q − p− 1
)1−1/q [ 1
(p+ 1)(p+ 2)
]1/q
×
×
[(
1
2
− λ
)(2q−p−1)/(q−1)
+ λ(2q−p−1)/(q−1)
]1−1/q
×
×
([
1
2
(p+ 1 + 2λ)
(
1
2
− λ
)p+1
+ λp+2
]
|f ′(a)|q+
+
[
1
2
(p+ 3− 2λ)
(
1
2
− λ
)p+1
+ (p+ 2− λ)λp+1
]
|f ′(b)|q
)1/q
+
+
[(
µ− 1
2
)(2q−p−1)/(q−1)
+ (1− µ)(2q−p−1)/(q−1)
]1−1/q
×
×
([
1
2
(p+ 1 + 2µ)
(
µ− 1
2
)p+1
+ (p+ 1 + µ)(1− µ)p+1
]
|f ′(a)|q+
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
560 FENG QI, TIAN-YU ZHANG, BO-YAN XI
+
[
1
2
(p+ 3− 2µ)
(
µ− 1
2
)p+1
+ (1− µ)p+2
]
|f ′(b)|q
)1/q
.
Proof. By Lemma 2.1, the convexity of |f ′|q on [a, b], and Hölder’s integral inequality, we obtain∣∣∣∣∣∣(1− µ)f(a) + λf(b) + (µ− λ)f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ (b− a)
1/2∫
0
|λ− t||f ′(ta+ (1− t)b)|dt+
1∫
1/2
|µ− t||f ′(ta+ (1− t)b)|dt
≤
≤ (b− a)
1/2∫
0
|λ− t|(q−p)/(q−1)dt
1−1/q 1/2∫
0
|λ− t|p|f ′(ta+ (1− t)b)|qdt
1/q
+
+
1∫
1/2
|µ− t|(q−p)/(q−1)dt
1−1/q 1∫
1/2
|µ− t|p|f ′(ta+ (1− t)b)|qdt
1/q
≤
≤ (b− a)
1/2∫
0
|λ− t|(q−p)/(q−1)dt
1−1/q 1/2∫
0
|λ− t|p
(
t|f ′(a)|q + (1− t)|f ′(b)|q
)
dt
1/q
+
+
1∫
1/2
|µ− t|(q−p)/(q−1)dt
1−1/q 1∫
1/2
|µ− t|p
(
t|f ′(a)|q + (1− t)|f ′(b)|q
)
dt
1/q
. (3.2)
Furthermore, a straightforward computation gives
1/2∫
0
|λ− t|(q−p)/(q−1)dt = q − 1
2q − p− 1
[(
1
2
− λ
)(2q−p−1)/(q−1)
+ λ(2q−p−1)/(q−1)
]
, (3.3)
1∫
1/2
|µ− t|(q−p)/(q−1)dt = q − 1
2q − p− 1
[(
µ− 1
2
)(2q−p−1)/(q−1)
+ (1− µ)(2q−p−1)/(q−1)
]
,
1/2∫
0
|λ− t|p
(
t|f ′(a)|q + (1− t)|f ′(b)|q
)
dt =
1
(p+ 1)(p+ 2)
{[
1
2
(p+ 1 + 2λ)
(
1
2
− λ
)p+1
+
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES FOR FUNCTIONS . . . 561
+ λp+2
]
|f ′(a)|q +
[
1
2
(p+ 3− 2λ)
(
1
2
− λ
)p+1
+ (p+ 2− λ)λp+1
]
|f ′(b)|q
}
,
and
1∫
1/2
|µ− t|p
(
t|f ′(a)|q + (1− t)|f ′(b)|q
)
dt =
1
(p+ 1)(p+ 2)
{[
1
2
(p+ 1 + 2µ)
(
µ− 1
2
)p+1
+
+ (p+ 1 + µ)(1− µ)p+1
]
|f ′(a)|q +
[
1
2
(p+ 3− 2µ)
(
µ− 1
2
)p+1
+ (1− µ)p+2
]
|f ′(b)|q
}
.
(3.4)
Substituting the equations from (3.3) to (3.4) into (3.2) results in the inequality (3.1).
Theorem 3.2 is proved.
Corollary 3.1. Let f : I ⊆ R→ R be a continuously differentiable function on I◦, a, b ∈ I with
a < b and 0 ≤ λ ≤ 1
2
≤ µ ≤ 1. If |f ′|q for q ≥ 1 is convex on [a, b], then
∣∣∣∣∣∣(1− µ)f(a) + λf(b) + (µ− λ)f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤ b− a
2
(
1
3
)1/q
×
×
[(
1
2
− λ
)2
+ λ2
]1−1/q ([
(1 + λ)
(
1
2
− λ
)2
+ λ3
]
|f ′(a)|q +
[
(2− λ)
(
1
2
− λ
)2
+
+ (3− λ)λ2
]
|f ′(b)|q
)1/q
+
[(
µ− 1
2
)2
+ (1− µ)2
]1−1/q ([
(1 + µ)
(
µ− 1
2
)2
+
+ (2 + µ)(1− µ)2
]
|f ′(a)|q +
[
(2− µ)
(
µ− 1
2
)2
+ (1− µ)3
]
|f ′(b)|q
)1/q
and ∣∣∣∣∣∣(1− µ)f(a) + λf(b) + (µ− λ)f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ b− a
2
[
2
(q + 1)(q + 2)
]1/q{([1
2
(q + 1 + 2λ)
(
1
2
− λ
)q+1
+ λq+2
]
|f ′(a)|q+
+
[
1
2
(q + 3− 2λ)
(
1
2
− λ
)q+1
+ (q + 2− λ)λq+1
]
|f ′(b)|q
)1/q
+
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
562 FENG QI, TIAN-YU ZHANG, BO-YAN XI
+
([
1
2
(q + 1 + 2µ)
(
µ− 1
2
)q+1
+ (q + 1 + µ)(1− µ)q+1
]
|f ′(a)|q+
+
[
1
2
(q + 3− 2µ)
(
µ− 1
2
)q+1
+ (1− µ)q+2
]
|f ′(b)|q
)1/q
.
Proof. This follows from Theorem 3.1 and setting p = 1 and p = q in Theorem 3.2.
Corollary 3.2. Let f : I ⊆ R→ R be a continuously differentiable function on I◦, a, b ∈ I with
a < b, m > 0, and m ≥ 2` ≥ 0. If |f ′|q for q > 1 is convex on [a, b], and q ≥ p > 0, then∣∣∣∣∣∣ `m [f(a) + f(b)] +
m− 2`
m
f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ b− a
4m2
(
q − 1
2q − p− 1
)1−1/q ( 1
2m(p+ 1)(p+ 2)
)1/q
×
×[(2`)(2q−p−1)/(q−1) + (m− 2`)(2q−p−1)/(q−1)]1−1/q×
×
{(
[(2`)p+2 + (mp+m+ 2`)(m− 2`)p+1]|f ′(a)|q+
+[(2mp+ 4m− 2`)(2`)p+1 + (mp+ 3m− 2`)(m− 2`)p+1]|f ′(b)|q
)1/q
+
+
(
[(2mp+ 4m− 2`)(2`)p+1 + (mp+ 3m− 2`)(m− 2`)p+1]|f ′(a)|q+
+[(2`)p+2 + (mp+m+ 2`)(m− 2`)p+1]|f ′(b)|q
)1/q}
.
Proof. This follows from letting λ = 1− µ =
`
m
in Theorem 3.2.
Corollary 3.3. Let f : I ⊆ R→ R be a continuously differentiable function on I◦, a, b ∈ I with
a < b, m > 0, and m ≥ 2` ≥ 0. If |f ′|q for q ≥ 1 is convex on [a, b], then∣∣∣∣∣∣ `m [f(a) + f(b)] +
m− 2`
m
f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ b− a
8m2
(
1
3m
)1/q
[(m− 2`)2 + (2`)2]1−1/q
{(
[(m+ `)(m− 2`)2 + 4`3]|f ′(a)|q+
+[(2m− `)(m− 2`)2 + 4(3m− `)`2]|f ′(b)|q
)1/q
+
(
[4(3m− `)`2+
+(2m− `)(m− 2`)2]|f ′(a)|q + [(m+ `)(m− 2`)2 + 4`3]|f ′(b)|q
)1/q}
and
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES FOR FUNCTIONS . . . 563∣∣∣∣∣∣ `m [f(a) + f(b)] +
m− 2`
m
f
(
a+ b
2
)
− 1
b− a
b∫
a
f(x)dx
∣∣∣∣∣∣ ≤
≤ b− a
4m
[
1
2m2(q + 1)(q + 2)
]1/q {(
[(2`)q+2 + (mq +m+ 2`)(m− 2`)q+1]|f ′(a)|q+
+[(2mq + 4m− 2`)(2`)q+1 + (mq + 3m− 2`)(m− 2`)q+1]|f ′(b)|q
)1/q
+
+
(
[(2mq + 4m− 2`)(2`)q+1 + (mq + 3m− 2`)(m− 2`)q+1]|f ′(a)|q+
+[(2`)q+2 + (mq +m+ 2`)(m− 2`)q+1]|f ′(b)|q
)1/q}
.
Proof. This follows from setting λ = 1− µ =
`
m
in Corollary 3.1.
4. Applications to special means. For two positive numbers a > 0 and b > 0, define
A(a, b) =
a+ b
2
, G(a, b) =
√
ab , H(a, b) =
2ab
a+ b
,
I(a, b) =
1
e
(
bb
aa
)1/(b−a)
, a 6= b,
a, a = b,
L(a, b) =
b− a
ln b− ln a
, a 6= b,
a, a = b,
and
Ls(a, b) =
[
bs+1 − as+1
(s+ 1)(b− a)
]1/s
, s 6= 0,−1 and a 6= b,
L(a, b), s = −1 and a 6= b,
I(a, b), s = 0 and a 6= b,
a, a = b.
It is well known that A, G, H, L = L−1, I = L0, and Ls are respectively called the arithmetic,
geometric, harmonic, logarithmic, exponential, and generalized logarithmic means of two positive
number a and b.
Theorem 4.1. Let b > a > 0, q > 1, q ≥ p > 0, m > 0, m ≥ 2` ≥ 0, and s ∈ R.
(1) If either s > 1 and (s− 1)q ≥ 1 or s < 1 and s 6= 0, then∣∣∣∣∣2`A
(
as, bs
)
+ (m− 2`)[A(a, b)]s
m
− [Ls(a, b)]
s
∣∣∣∣∣ ≤ b− a
4m2
|s|
(
q − 1
2q − p− 1
)1−1/q
×
×
[
1
2m(p+ 1)(p+ 2)
]1/q [
(2`)(2q−p−1)/(q−1) + (m− 2`)(2q−p−1)/(q−1)
]1−1/q×
×
{[(
(2`)p+2 + (mp+m+ 2`)(m− 2`)p+1
)
a(s−1)q+
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
564 FENG QI, TIAN-YU ZHANG, BO-YAN XI
+
(
(2mp+ 4m− 2`)(2`)p+1 + (mp+ 3m− 2`)(m− 2`)p+1
)
b(s−1)q
]1/q
+
+
[(
(2mp+ 4m− 2`)(2`)p+1 + (mp+ 3m− 2`)(m− 2`)p+1
)
a(s−1)q+
+
(
(2`)p+2 + (mp+m+ 2`)(m− 2`)p+1
)
b(s−1)q
]1/q}
.
(2) If s = −1, then∣∣∣∣ 1m
[
2`
H(a, b)
+
m− 2`
A(a, b)
]
− 1
L(a, b)
∣∣∣∣ ≤ b− a
4m2
(
q − 1
2q − p− 1
)1−1/q [ 1
2m(p+ 1)(p+ 2)
]1/q
×
×
[
(2`)(2q−p−1)/(q−1) + (m− 2`)(2q−p−1)/(q−1)
]1−1/q×
×
{[
(2`)p+2 + (mp+m+ 2`)(m− 2`)p+1
a2q
+
+
(2mp+ 4m− 2`)(2`)p+1 + (mp+ 3m− 2`)(m− 2`)p+1
b2q
]1/q
+
+
[
(2mp+ 4m− 2`)(2`)p+1 + (mp+ 3m− 2`)(m− 2`)p+1
a2q
+
+
(2`)p+2 + (mp+m+ 2`)(m− 2`)p+1
b2q
]1/q}
.
Proof. Set f(x) = xs for x > 0 and s 6= 0, 1. Then it is easy to obtain that
f ′(x) = sxs−1, |f ′(x)|q = |s|qx(s−1)q,
(
|f ′(x)|q
)′′
= (s− 1)q[(s− 1)q − 1]|s|qx(s−1)q−2.
Hence, when s > 1 and (s − 1)q ≥ 1, or when s < 1 and s 6= 0, the function |f ′|q is convex on
[a, b]. From Corollary 3.2, Theorem 4.1 follows.
By the argument similar to Theorem 4.1, we further obtain the following conclusions.
Theorem 4.2. Let b > a > 0, q ≥ 1, m > 0, m ≥ 2` ≥ 0, and s ∈ R.
(1) If either s > 1 and (s− 1)q ≥ 1 or s < 1 and s 6= 0, then∣∣∣∣2`A(as, bs) + (m− 2`)[A(a, b)]s
m
− [Ls(a, b)]
s
∣∣∣∣ ≤ b− a
8m2
(
1
3m
)1/q [
4`2 + (m− 2`)2
]1−1/q|s|×
×
{[(
4`3 + (m+ `)(m− 2`)2
)
a(s−1)q +
(
4(3m− `)`2 + (2m− `)(m− 2`)2
)
b(s−1)q
]1/q
+
+
[(
4(3m− `)`2 + (2m− `)(m− 2`)2
)
a(s−1)q +
(
4`3 + (m+ `)(m− 2`)2
)
b(s−1)q
]1/q}
and ∣∣∣∣2`A(as, bs) + (m− 2`)[A(a, b)]s
m
− [Ls(a, b)]
s
∣∣∣∣ ≤ b− a
4m
|s|
[
1
2m2(q + 1)(q + 2)
]1/q
×
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES FOR FUNCTIONS . . . 565
×
{[(
(2`)q+2 + (mq +m+ 2`)(m− 2`)q+1
)
a(s−1)q+
+
(
(2mq + 4m− 2`)(2`)q+1 + (mq + 3m− 2`)(m− 2`)q+1
)
b(s−1)q
]1/q
+
+
[(
(2mq + 4m− 2`)(2`)q+2 + (mq + 3m− 2`)(m− 2`)q+1
)
a(s−1)q+
+
(
(2`)q+1 + (mq +m+ 2`)(m− 2`)q+1
)
b(s−1)q
]1/q}
.
(2) If s = −1, then∣∣∣∣ 1m
[
2`
H(a, b)
+
m− 2`
A(a, b)
]
− 1
L(a, b)
∣∣∣∣ ≤ b− a
8m2
(
1
3m
)1/q [
4`2 + (m− 2`)2
]1−1/q×
×
{[
4`3 + (m+ `)(m− 2`)2
a2q
+
4(3m− `)`2 + (2m− `)(m− 2`)2
b2q
]1/q
+
+
[
4(3m− `)`2 + (2m− `)(m− 2`)2
a2q
+
4`3 + (m+ `)(m− 2`)2
b2q
]1/q}
and ∣∣∣∣ 1m
[
2`
H(a, b)
+
m− 2`
A(a, b)
]
− 1
L(a, b)
∣∣∣∣ ≤ b− a
4m
[
1
2m2(q + 1)(q + 2)
]1/q
×
×
{[
(2`)q+2 + (mq +m+ 2`)(m− 2`)q+1
a2q
+
+
(mq + 3m− 2`)(m− 2`)q+1 + (2mq + 4m− 2`)(2`)q+1
b2q
]1/q
+
+
[
(2mq + 4m− 2`)(2`)q+1 + (mq + 3m− 2`)(m− 2`)q+1
a2q
+
+
(2`)q+2 + (mq +m+ 2`)(m− 2`)q+1
b2q
]1/q}
.
In particular, we have∣∣∣∣2`A(as, bs) + (m− 2`)[A(a, b)]s
m
− [Ls(a, b)]
s
∣∣∣∣ ≤ b− a
4m2
|s|
[
4`2 + (m− 2`)2
]
A
(
as−1, bs−1
)
and ∣∣∣∣ 1m
[
2`
H(a, b)
+
m− 2`
A(a, b)
]
− 1
L(a, b)
∣∣∣∣ ≤ b− a
4m2
4`2 + (m− 2`)2
H(a2, b2)
.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
566 FENG QI, TIAN-YU ZHANG, BO-YAN XI
Theorem 4.3. Let b > a > 0, q > 1, q ≥ p > 0, m > 0, and m ≥ 2` ≥ 0. Then∣∣∣∣2` lnG(a, b) + (m− 2`) lnA(a, b)
m
− ln I(a, b)
∣∣∣∣ ≤ b− a
4m2
(
q − 1
2q − p− 1
)1−1/q
×
×
[
1
2m(p+ 1)(p+ 2)
]1/q [
(m− 2`)(2q−p−1)/(q−1) + (2`)(2q−p−1)/(q−1)
]1−1/q×
×
{[
(2`)p+2 + (mp+m+ 2`)(m− 2`)p+1
aq
+
+
(mp+ 3m− 2`)(m− 2`)p+1 + (2mp+ 4m− 2`)(2`)p+1
bq
]1/q
+
+
[
(2mp+ 4m− 2`)(2`)p+1 + (mp+ 3m− 2`)(m− 2`)p+1
aq
+
+
(2`)p+2 + (mp+m+ 2`)(m− 2`)p+1
bq
]1/q}
.
Proof. This follows from taking f(x) = lnx for x > 0 in Corollary 3.2.
By the similar argument to Theorem 4.3, we can obtain the following inequalities.
Theorem 4.4. Let b > a > 0, q ≥ 1, m > 0, and m ≥ 2` ≥ 0. Then∣∣∣∣2` lnG(a, b) + (m− 2`) lnA(a, b)
m
− ln I(a, b)
∣∣∣∣ ≤ b− a
4m
[
1
2m2(q + 1)(q + 2)
]1/q
×
×
{[
(2`)q+2 + (mq +m+ 2`)(m− 2`)q+1
aq
+
+
(mq + 3m− 2`)(m− 2`)q+1 + (2mq + 4m− 2`)(2`)q+1
bq
]1/q
+
+
[
(2mq + 4m− 2`)(2`)q+1 + (mq + 3m− 2`)(m− 2`)q+1
aq
+
+
(2`)q+2 + (mq +m+ 2`)(m− 2`)q+1
bq
]1/q}
and∣∣∣∣2` lnG(a, b) + (m− 2`) lnA(a, b)
m
− ln I(a, b)
∣∣∣∣ ≤ b− a
8m2
(
1
3m
)1/q [
(m− 2`)2 + (2`)2
]1−1/q×
×
{[
4`3 + (m+ `)(m− 2`)2
aq
+
(2m− `)(m− 2`)2 + 4(3m− `)`2
bq
]1/q
+
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
HERMITE – HADAMARD TYPE INTEGRAL INEQUALITIES FOR FUNCTIONS . . . 567
+
[
4(3m− `)`2 + (2m− `)(m− 2`)2
aq
+
4`3 + (m+ `)(m− 2`)2
bq
]1/q}
.
In particular, we have∣∣∣∣2` lnG(a, b) + (m− 2`) lnA(a, b)
m
− ln I(a, b)
∣∣∣∣ ≤ b− a
4m2
4`2 + (m− 2`)2
H(a, b)
.
Remark 4.1. This paper is a simplified version of the preprint [13].
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Received 27.11.12,
after revision — 29.08.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 4
|
| id | umjimathkievua-article-2002 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:16:47Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c4/dd2657af9653fcc1d050895509797ac4.pdf |
| spelling | umjimathkievua-article-20022019-12-05T09:48:26Z Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex Інтегральні нерівності типу Ерміта - Адамара, перші похідні яких мають опуклість Qi, Feng Xi, Bo-Yan Zhang, Tian-Yu Ци, Фен Сі, Бо-Ян Чжан, Т.-Ю We establish some new Hermite–Hadamard-type inequalities for functions whose first derivatives are of convexity and apply these inequalities to construct inequalities for special means. Встановлено дєякі нові нєрівності типу Ерміта - Адамара для Функцій, похідні яких мають опуклість. Ці нєрівності застосовано при побудові нерівностей для спеціальних середніх. Institute of Mathematics, NAS of Ukraine 2015-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2002 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 4 (2015); 555-567 Український математичний журнал; Том 67 № 4 (2015); 555-567 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2002/1022 https://umj.imath.kiev.ua/index.php/umj/article/view/2002/1023 Copyright (c) 2015 Qi Feng; Xi Bo-Yan; Zhang Tian-Yu |
| spellingShingle | Qi, Feng Xi, Bo-Yan Zhang, Tian-Yu Ци, Фен Сі, Бо-Ян Чжан, Т.-Ю Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex |
| title | Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex |
| title_alt | Інтегральні нерівності типу Ерміта - Адамара, перші похідні яких мають опуклість |
| title_full | Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex |
| title_fullStr | Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex |
| title_full_unstemmed | Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex |
| title_short | Hermite–Hadamard-Type Integral Inequalities for Functions Whose First Derivatives are Convex |
| title_sort | hermite–hadamard-type integral inequalities for functions whose first derivatives are convex |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2002 |
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