1. No category

Georgian Math. J. Galley Proof, 14 pages
DOI 10.1515 / gmj-2013-0043
© de Gruyter 2013
Some integral inequalities of Simpson type for
GA-"-convex functions
Feng Qi and Bo-Yan Xi
Abstract. We introduce a new concept “GA-"-convex function” and establish some integral inequalities of Simpson type for GA-"-convex functions.
Keywords. Integral inequality of Simpson type, GA-"-convex function, integral identity.
2010 Mathematics Subject Classification. 26A51, 26D15, 41A55.
1
Introduction
Let us recall some definitions of several kinds of convex functions.
Definition 1.1. A function f W I R D . 1; 1/ ! R is said to be convex if
the inequality
f .tx C .1 t /y/ tf .x/ C .1 t/f .y/
holds for all x; y 2 I and t 2 Œ0; 1.
Definition 1.2. Let f W I RC D .0; 1/ ! R. If the inequality
f .x t y 1 t / tf .x/ C .1
t/f .y/
is valid for x; y 2 I and t 2 Œ0; 1, then f .x/ is said to be GA-convex on I .
Definition 1.3 ([7]). Let X be a real linear space, the set D X be convex, and
f W D ! R be a mapping. For some constant " 0, if
f .tx C .1
t /y/ tf .x/ C .1
t/f .y/ C "
holds for all x; y 2 D and t 2 Œ0; 1, then f .x/ is said to be "-convex on D.
This work was partially supported by the NNSF of China under Grant No. 11361038 and by the
Foundation of the Research Program of Science and Technology at the Universities of the Inner
Mongolia Autonomous Region under Grant No. NJZY13159, China.
Note 1:
Red parts
indicate
major
changes.
Please
check them
carefully.
2
F. Qi and B.-Y. Xi
It is well known that the classical inequalities for usually convex functions defined by Definition 1.1 are due to Hermite–Hadamard, which can be stated as the
following theorem.
Theorem 1.4. If f W I R ! R is a convex function on Œa; b with a; b 2 I and
a < b, then we have
Z b
a C b 1
f .a/ C f .b/
f
f .x/ dx :
2
b a a
2
In the literature, Simpson’s inequality may be referred to as Theorem 1.5 below.
Theorem 1.5 ([6]). If f W Œa; b ! R is a four times continuously differentiable
function on .a; b/ and its fourth derivative on .a; b/ is bounded by kf .4/ k D
supx2.a;b/ jf .4/ .x/j < 1, then
ˇZ b
a C b iˇ .b a/5 kf .4/ k
b a h f .a/ C f .b/
ˇ
ˇ
f .x/ dx
C 2f
:
ˇ
ˇ
3
2
2
2880
a
Let us reformulate some Hermite–Hadamard and Simpson type inequalities for
the above convex functions.
Theorem 1.6 ([5, Theorem 2.2]). Let a; b 2 I ı with a < b and f W I ı R ! R
be a differentiable mapping on I ı . If jf 0 .x/j is convex on Œa; b, then
Z b
ˇ f .a/ C f .b/
ˇ .b a/.jf 0 .a/j C jf 0 .b/j/
1
ˇ
ˇ
f .x/ dx ˇ :
ˇ
2
b a a
8
Theorem 1.7 ([12, Theorems 1 and 2]). Let a; b 2 I with a < b and f W I R ! R be differentiable on I ı . If jf 0 .x/jq is convex on Œa; b and q 1, then
Z b
ˇ f .a/ C f .b/
ˇ b a h jf 0 .a/jq C jf 0 .b/jq i1=q
1
ˇ
ˇ
f .x/ dx ˇ ˇ
2
b a a
4
2
and
ˇ a C b ˇ
ˇf
2
1
b
Z
a
a
b
ˇ b
ˇ
f .x/ dx ˇ a h jf 0 .a/jq C jf 0 .b/jq i1=q
:
4
2
Theorem 1.8 ([8, Theorems 2.3 and 2.4]). Let f W I R ! R be differentiable
on I ı , a; b 2 I ı with a < b, and p > 1. If jf 0 .x/jp=.p 1/ is convex on Œa; b,
then
ˇ 1 Z b
a C b ˇ b a 4 1=p °
ˇ
ˇ
jf 0 .a/jp=.p 1/
f .x/ dx f
ˇ
ˇ
b a a
2
16 p C 1
.p 1/=p .p 1/=p ±
C 3jf 0 .b/jp=.p 1/
C 3jf 0 .a/jp=.p 1/ C jf 0 .b/jp=.p 1/
Some integral inequalities of Simpson type
3
and
ˇ 1 Z b
ˇ
f .x/ dx
ˇ
b a a
f
a C b ˇ b a 4 1=p
ˇ
.jf 0 .a/j C jf 0 .b/j/:
ˇ
2
4
pC1
Theorem 1.9 ([14]). Let f W I R ! R be differentiable on I ı , a; b 2 I ı
with a < b, and f 0 2 L.Œa; b/, where L.Œa; b/ denotes the set of all Lebesgue
integrable functions on the interval Œa; b. If jf 0 .x/jq for q 1 is convex on Œa; b,
then
Z b
ˇ1h
ˇ b a h 2qC1 C 1 i1=q
a C b i
1
ˇ
ˇ
f .x/ dx ˇ ˇ f .a/ C f .b/ C 4f
6
2
b a a
12 3.q C 1/
h 3jf 0 .a/jq C jf 0 .b/jq 1=q jf 0 .a/jq C 3jf 0 .b/jq 1=q i
C
4
4
and
Z b
ˇ1h
ˇ 5.b a/
a C b i
1
ˇ
ˇ
f
.a/
C
f
.b/
C
4f
f
.x/
dx
ˇ
ˇ
6
2
b a a
72
h 61jf 0 .a/jq C 29jf 0 .b/jq 1=q 29jf 0 .a/jq C 61jf 0 .b/jq 1=q i
C
:
90
90
For more information on Hermite–Hadamard and Simpson type inequalities for
various convex functions, we refer the reader to the recently published articles
[1–4, 8–11, 13, 15–25] and the closely related references therein.
In this paper, we will introduce a new concept “GA-"-convex function” and
establish some integral inequalities of Simpson type for GA-"-convex functions.
2
A definition and a lemma
The so-called GA-"-convex functions can be defined as follows.
Definition 2.1. Let f W I RC ! R be a mapping and " 0 be a constant. If
the inequality
f .x t y 1 t / tf .x/ C .1 t/f .y/ C "
holds for all x; y 2 I and t 2 Œ0; 1, then f .x/ is said to be a GA-"-convex
function on I .
Remark 2.2. If f .x/ is increasing and "-convex on I RC , then it is GA-"convex on I . If f .x/ is decreasing and GA-"-convex on I RC , then it is
"-convex on I .
4
F. Qi and B.-Y. Xi
Example 2.3. Let f .x/ D x r for x 2 I D .0; 1 and r ¤ 0; 1. Then
f .x t y 1 t / D .x r /t .y r /1
D tf .x/ C .1
t
tx r C .1
t/y r
t /f .y/ tf .x/ C .1
t/f .y/ C "
for all x; y 2 I and t 2 Œ0; 1 and for any constant " 0. This implies that f .x/
is GA-convex and GA-"-convex on I . Furthermore,
(i) when r > 1 or r < 0,
a. the function f .x/ is convex on I ,
b. for any constant " 0, it is "-convex on I ;
(ii) when 0 < r < 1,
a. the function f .x/ is concave on I ,
b. for any constant " 1, it is "-convex on I .
Example 2.4. Let f .x/ D 21x for x 2 I D .0; 1.
It is easy to see that f .x/ is convex on I .
When putting x D 0:1, y D 0:9, and t D 0:5, since
f .0:10:5 0:91
0:5
/
0:5f .0:1/
.1
0:5/f .0:9/ > 0:07;
the function f .x/ is not GA-convex on I .
For " 12 , the function f .x/ is GA-"-convex on I .
To establish some new Simpson type inequalities for GA-"-convex functions,
we need the following lemma.
Lemma 2.5. Let f W I RC ! R be a differentiable function on I ı and a; b 2
I ı with a < b. If f 0 2 L.Œa; b/, then
p
Z b
f .a/ C 4f . ab/ C f .b/
1
f .x/
ln b ln a
dx D
6
ln b ln a a
x
4
Z 1
1 1 t=2 t=2 0 1 t=2 t=2
t
a
b f .a
b / at=2 b 1 t=2 f 0 .at=2 b 1 t=2 / dt:
3
0
Proof. Integrating by parts and letting x D a1 t=2 b t=2 for 0 t 1 lead to
Z
ln b ln a 1 1 1 t=2 t=2 0 1 t=2 t=2
t
a
b f .a
b / dt
2
3
0
Z 1
1 D
t
d f .a1 t=2 b t=2 /
3
0
Some integral inequalities of Simpson type
5
Z 1
ˇtD1
1
f .a1 t=2 b t=2 / dt
f .a1 t=2 b t=2 /ˇ t D0
3
0
Z 1
2 p
1
D f . ab/ C f .a/
f .a1 t=2 b t=2 / dt
3
3
0
Z pab
1
2 p
2
f .x/
D f .a/ C f . ab/
dx:
3
3
ln b ln a a
x
D t
Similarly, we have
ln b
2
D
D
D
D
3
Z 1
1 t=2 1 t=2 0 t=2 1 t=2
a b
f .a b
/ dt
3
0
Z 1
1 t
d f .at=2 b 1 t=2 /
3
0
Z 1
ˇ
1
t=2 1 t=2 ˇt D1
t
f .a b
/ tD0 C
f .at=2 b 1 t=2 / dt
3
0
Z 1
2 p
1
f . ab/
f .b/ C
f .at=2 b 1 t=2 / dt
3
3
0
Z b
1
f .x/
2 p
2
f . ab/
f .b/ C
dx:
p
3
3
ln b ln a
x
ab
ln a
t
Some new integral inequalities of Simpson type
Now we start out to establish some new integral inequalities of Simpson type for
GA-"-convex functions.
Theorem 3.1. Let f W I RC ! R be differentiable on I ı , a; b 2 I ı with
a < b, and f 0 2 L.Œa; b/. If jf 0 jq is GA-"-convex on Œa; b for some constant
" 0 and q 1, then
Z b
ˇ f .a/ C 4f .pab/ C f .b/
1
f .x/ ˇˇ ln b ln a
ˇ
dx ˇ ˇ
6
ln b ln a a
x
4
°
.q 1/=q
M1
.a; b/ .M1 .a; b/ M2 .a; b//jf 0 .a/jq C M2 .a; b/jf 0 .b/jq
1=q
.q
C "M1 .a; b/
C M1
C .M1 .b; a/
1/=q
.b; a/ M2 .b; a/jf 0 .a/jq
1=q ±
M2 .b; a//jf 0 .b/jq C "M1 .b; a/
;
6
F. Qi and B.-Y. Xi
where
M1 .u; v/ D
2
3.ln v
ln u/
5=6
u L.u1=6 ; v 1=6 / C u1=2 .2v 1=2
u1=2 /
2u1=2 v 1=6 L.u1=3 ; v 1=3 / ;
M2 .u; v/ D
1=6
2u1=2
4v L.u1=3 ; v 1=3 / C v 1=2 .ln v
2
3.ln v ln u/
2u1=3 L.u1=6 ; v 1=6 /
ln u/
5v 1=2 C 2u1=3 v 1=6 C u1=2 ;
and
Note 2:
´
L.u; v/ D
u v
ln u ln v ;
u;
u ¤ v;
uDv
()
is the logarithmic mean.
Proof. Since jf 0 jq is a GA-"-convex function on Œa; b, by Lemma 2.5 and by
Hölder’s integral inequality, we have
Z b
ˇ f .a/ C 4f .pab/ C f .b/
1
f .x/ ˇˇ
ˇ
dx ˇ
ˇ
6
ln b ln a a
x
Z
1 ˇˇ 1 t=2 t=2 0 1 t=2 t=2
ln b ln a 1 ˇˇ
b jf .a
b /j
ˇt
ˇ a
4
3
0
C at =2 b 1 t=2 jf 0 .at=2 b 1 t=2 /j dt
Z
ln b ln a ° 1 ˇˇ
1 ˇˇ 1 t=2 t=2 .q 1/=q
b dt
ˇt
ˇa
4
3
0
h Z 1ˇ
1 ˇˇ 1 t=2 t=2 0 1 t=2 t=2 q i1=q
ˇ
b jf .a
b /j dt
ˇt
ˇa
3
0
Z 1ˇ
1 ˇˇ t=2 1 t=2 .q 1/=q
ˇ
C
dt
ˇt
ˇa b
3
0
h Z 1ˇ
1 ˇˇ t=2 1 t=2 0 t=2 1 t=2 q i1=q ±
ˇ
jf .a b
/j dt
ˇt
ˇa b
3
0
Z
ln b ln a ° 1 ˇˇ
1 ˇˇ 1 t=2 t=2 .q 1/=q
b dt
ˇt
ˇa
4
3
0
Z 1ˇ
i 1=q
1 ˇˇ 1 t=2 t=2 h
t 0
t
ˇ
b
1
jf .a/jq C jf 0 .b/jq C " dt
ˇt
ˇa
3
2
2
0
We deleted
all formula
labels except
for this one.
Some integral inequalities of Simpson type
C
Z 1ˇ
ˇ
ˇt
0
Z 1ˇ
ˇ
ˇt
0
1 ˇˇ t=2 1
ˇa b
3
1 ˇˇ t=2 1
ˇa b
3
t=2
t=2
dt
ht
2
.q
1/=q
jf 0 .a/jq C 1
i 1=q ±
t 0
jf .b/jq C " dt
;
2
where
Z 1ˇ
ˇ
ˇt
1 ˇˇ 1 t=2 t=2
b dt
ˇa
3
0
Z 1=3 Z 1
1
1 1 t=2 t=2
1 t=2 t=2
D
t a
b dt C
t
a
b dt
3
3
0
1=3
p
2 .ln b ln a/.2 ab a/ 6.a1=2 b 1=2 2a5=6 b 1=6 C a/
D
3.ln b ln a/2
D M1 .a; b/;
Z 1ˇ
1 ˇˇ t =2 1 t=2
ˇ
dt D M1 .b; a/;
ˇt
ˇa b
3
0
Z 1ˇ
i
t 0
t
1 ˇˇ 1 t=2 t=2 h
ˇ
b
1
jf .a/jq C jf 0 .b/jq C " dt
ˇa
ˇt
3
2
2
0
D .M1 .a; b/
and
Z 1ˇ
ˇ
ˇt
0
1 ˇˇ t =2 1
ˇa b
3
M2 .a; b//jf 0 .a/jq C M2 .a; b/jf 0 .b/jq C "M1 .a; b/;
t=2
ht
2
7
jf 0 .a/jq C 1
D M2 .b; a/jf 0 .a/jq C .M1 .b; a/
i
t 0
jf .b/jq C " dt
2
M2 .b; a//jf 0 .b/jq C "M1 .b; a/:
The proof of Theorem 3.1 is thus complete.
Corollary 3.2. Under the conditions of Theorem 3.1, when q D 1, we have
Z b
ˇ f .a/ C 4f .pab/ C f .b/
f .x/ ˇˇ
1
ˇ
dx ˇ
ˇ
6
ln b ln a a
x
ln b ln a °
M1 .a; b/ M2 .a; b/ C M2 .b; a/ jf 0 .a/j C M2 .a; b/
4
±
C M1 .b; a/ M2 .b; a/ jf 0 .b/j C " M1 .a; b/ C M1 .b; a/ :
8
F. Qi and B.-Y. Xi
Theorem 3.3. Let f W I RC ! R be differentiable on I ı , a; b 2 I ı with
a < b, and f 0 2 L.Œa; b/. If jf 0 jq for q > 1 is GA-"-convex on Œa; b for some
constant " 0, then
Z b
ˇ f .a/ C 4f .pab/ C f .b/
1
f .x/ ˇˇ
ˇ
dx ˇ
ˇ
6
ln b ln a a
x
ln b ln a ° q 1 h 2 .2q 1/=.q 1/ 1 .2q 1/=.q 1/ i±1 1=q
C
4
2q 1 3
3
°h
aq=2 L.aq=2 ; b q=2 / M3 .aq=2 ; b q=2 / jf 0 .a/jq
i1=q
C M3 .aq=2 ; b q=2 /jf 0 .b/jq C "aq=2 L.aq=2 ; b q=2 /
h
C M3 .aq=2 ; b q=2 /jf 0 .a/jq C b q=2 L.aq=2 ; b q=2 /
i1=q ±
M3 .b q=2 ; aq=2 / jf 0 .b/jq C "b q=2 L.aq=2 ; b q=2 /
;
where L.u; v/ is defined by () and
M3 .u; v/ D
uŒv L.u; v/
2.ln v ln u/
for positive numbers u ¤ v.
Proof. By Lemma 2.5, Hölder’s inequality, and the GA-"-convexity of jf 0 jq on
Œa; b, we derive
Z b
ˇ f .a/ C 4f .pab/ C f .b/
1
f .x/ ˇˇ
ˇ
dx ˇ
ˇ
6
ln b ln a a
x
Z
ln b ln a 1 ˇˇ
1 ˇˇ 1 t=2 t=2 0 1 t=2 t=2
b jf .a
b /j
ˇt
ˇ a
4
3
0
C at =2 b 1 t=2 jf 0 .at=2 b 1 t=2 /j dt
Z
1 ˇˇq=.q 1/ 1 1=q
ln b ln a 1 ˇˇ
dt
ˇt
ˇ
4
3
0
°h Z 1
i1=q
aq.1 t=2/ b qt=2 jf 0 .a1 t=2 b t=2 /jq dt
C
hZ
0
1
0
aqt=2 b q.1
t=2/
jf 0 .at=2 b 1
t=2 q
/j dt
i1=q ±
9
Some integral inequalities of Simpson type
where
Z 1ˇ
ˇ
ˇt
Z
0
1
Z 1ˇ
ˇ
ˇt
1 ˇˇq=.q 1/ 1 1=q
dt
ˇ
4
3
0
°Z 1
i ±1=q
h
t
t 0
aq.1 t=2/ b qt=2 1
jf .a/jq C jf 0 .b/jq C " dt
2
2
0
°Z 1
i ±1=q ht
t
C
aqt=2 b q.1 t=2/ jf 0 .a/jq C 1
jf 0 .b/jq C " dt
;
2
2
0
ln b
ln a 1 ˇˇq=.q
ˇ
3
aq.1
1/
t =2/ qt=2
b
0
Z
1
aqt=2 b q.1
t=2/
0
dt D
q 1 h 2 .2q
2q 1 3
1/=.q 1/
C
1 .2q
1/=.q 1/ i
3
dt D aq=2 L.aq=2 ; b q=2 /;
dt D b q=2 L.aq=2 ; b q=2 /;
and
Z
1
aq.1
0
t =2/ qt=2
b
h
1
i
t
t 0
jf .a/jq C jf 0 .b/jq dt
2
2
°
aq=2
L.aq=2 ; b q=2 / C .b q=2 2aq=2 / jf 0 .a/jq
q.ln b ln a/
±
C b q=2 L.aq=2 ; b q=2 / jf 0 .b/jq
D aq=2 L.aq=2 ; b q=2 / M3 .aq=2 ; b q=2 / jf 0 .a/jq
D
C M3 .aq=2 ; b q=2 /jf 0 .b/jq ;
Z 1
ht
i
t 0
aqt=2 b q.1 t=2/ jf 0 .a/jq C 1
jf .b/jq dt
2
2
0
°
b q=2
D
L.aq=2 ; b q=2 / aq=2 jf 0 .a/jq
q.ln b ln a/
±
C .2b q=2 aq=2 / L.aq=2 ; b q=2 / jf 0 .b/jq
D M3 .b q=2 ; aq=2 /jf 0 .a/jq
C b q=2 L.aq=2 ; b q=2 /
This completes the proof of Theorem 3.3.
M3 .b q=2 ; aq=2 / jf 0 .b/jq :
;
10
F. Qi and B.-Y. Xi
Theorem 3.4. Let f W I RC ! R be differentiable on I ı , a; b 2 I ı with
a < b, and f 0 2 L.Œa; b/. If jf 0 jq for q > 1 is GA-"-convex on Œa; b for some
constant " 0, then
Z b
ˇ f .a/ C 4f .pab/ C f .b/
1
f .x/ ˇˇ ln b ln a
ˇ
dx ˇ ˇ
6
ln b ln a a
x
4
i
°
h
1=q 1 1=q
3 .qC2/
aq=.2.q 1// L.aq=.2.q 1// ; b q=.2.q 1// /
2.q C 1/.q C 2/
qC1
.2
.3q C 8/ C .6q C 11//jf 0 .a/jq C .2qC1 .3q C 4/ C 1/jf 0 .b/jq
1=q q=.2.q 1//
1 1=q
C 6".q C 2/.2qC1 C 1/
C b
L.aq=.2.q 1// ; b q=.2.q 1// /
.2qC1 .3q C 4/ C 1/jf 0 .a/jq C .2qC1 .3q C 8/ C .6q C 11//jf 0 .b/jq
1=q ±
C 6".q C 2/.2qC1 C 1/
;
where L.u; v/ is defined by ().
Proof. By Lemma 2.5, Hölder’s inequality, and the GA-"-convexity of jf 0 jq on
Œa; b, we obtain
Z b
ˇ f .a/ C 4f .pab/ C f .b/
1
f .x/ ˇˇ
ˇ
dx ˇ
ˇ
6
ln b ln a a
x
Z
1 ˇˇ 1 t=2 t=2 0 1 t=2 t=2
ln b ln a 1 ˇˇ
b jf .a
b /j
ˇt
ˇ a
4
3
0
C at=2 b 1 t=2 jf 0 .at=2 b 1 t=2 /j dt
Z
ln b ln a °h 1 q.1 t=2/=.q 1/ qt=.2.q 1// i1 1=q
a
b
dt
4
0
h Z 1ˇ
1 ˇˇq 0 1 t=2 t=2 q i1=q
ˇ
b /j dt
ˇt
ˇ jf .a
3
0
hZ 1
i1 1=q
C
aqt=.2.q 1// b q.1 t=2/=.q 1/ dt
0
h Z 1ˇ
ˇ
ˇt
ln b
1 ˇˇq 0 t=2 1
ˇ jf .a b
3
0
Z
ln a °h 1 q.1 t=2/=.q
a
4
h Z 1ˇ
ˇ
ˇt
0
t=2 q
/j dt
i1=q ±
1/ qt=.2.q 1//
b
dt
i1
1=q
0
1 ˇˇq 1
ˇ
3
i1=q
t 0
t
jf .a/jq C jf 0 .b/jq C " dt
2
2
11
Some integral inequalities of Simpson type
C
1
hZ
aqt=.2.q
0
1
1 ˇˇq t 0
jf .a/jq C 1
ˇ
3
2
aq.1
t =2/=.q 1/ qt=.2.q 1//
b
0
Z
1
dt
i1
1=q
i1=q ±
t 0
jf .b/jq C " dt
;
2
1 ˇˇq
3 .qC1/ qC1
.2
C 1/;
ˇ dt D
3
qC1
0
Z
b
0
h Z 1ˇ
ˇ
ˇt
where
Z 1ˇ
ˇ
ˇt
1// q.1 t=2/=.q 1/
aqt =2.q
1/ q.1 t=2/=.q 1/
b
0
dt D aq=.2.q
dt D b q=.2.q
1//
1//
L.aq=.2.q
L.aq=.2.q
1//
1//
; b q=.2.q
; b q=.2.q
1//
1//
/;
/;
1
qC1
1 ˇˇq
3 .qC2/
t ˇˇ
.3q C 4/ C 1 ;
2
ˇt
ˇ dt D
3
2.q C 1/.q C 2/
0 2
Z 1
ˇ
ˇ
qC1
t ˇ
1 ˇq
3 .qC2/
.3q C 8/ C .6q C 11/ :
1
2
ˇt
ˇ dt D
2
3
2.q C 1/.q C 2/
0
Z
The proof is complete.
Theorem 3.5. Let f W Œa; b RC ! R0 be GA-"-convex on Œa; b and f 2
L.Œa; b/. Then
1
ln b
Z
ln a
a
b
f .x/
f .a/ C f .b/
dx C"
x
2
and
Z
a
b
f .x/ dx ŒL.a; b/
af .a/ C Œb
L.a; b/f .b/ C ".b
a/;
where L.u; v/ is defined by ()
Proof. Letting x D a1 t b t for 0 t 1 gives
1
ln b
Z
b
ln a a
Z 1
.1
0
f .x/
dx D
x
Z
1
f .a1 t b t / dt
0
f .a/ C f .b/
t/f .a/ C tf .b/ C " dt D
C"
2
12
F. Qi and B.-Y. Xi
and
Z
a
b
Z
f .x/ dx D .ln b
ln a/
.ln b
ln a/
1
0
Z 1
D ŒL.a; b/
a1 t b t f .a1 t b t / dt
a1 t b t .1
0
af .a/ C Œb
t/f .a/ C tf .b/ C " dt
L.a; b/f .b/ C ".b
a/:
Theorem 3.5 is thus proved.
Acknowledgments. The authors wish to express their thanks to the anonymous
referee for his/her careful corrections to and valuable comments on the original
version of this paper.
Bibliography
[1] M. Alomari, M. Darus and U. S. Kirmaci, Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and to special means, Comput. Math. Appl. 59 (2010), no. 1, 225–232.
[2] R.-F. Bai, F. Qi and B.-Y. Xi, Hermite–Hadamard type inequalities for the m- and
.˛; m/-logarithmically convex functions, Filomat 26 (2013), no. 1, 1–7.
[3] S.-P. Bai, S.-H. Wang and F. Qi, Some Hermite–Hadamard type inequalities for
n-time differentiable .˛; m/-convex functions, J. Inequal. Appl. 2012 (2012), DOI
10.1186/1029-242X-2012-267.
[4] L. Chun and F. Qi, Integral inequalities of Hermite–Hadamard type for functions
whose 3rd derivatives are s-convex, Appl. Math. 3 (2012), no. 11, 1680–1685.
[5] S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings
and applications to special means of real numbers and to trapezoidal formula, Appl.
Math. Lett. 11 (1998), no. 5, 91–95.
[6] S. S. Dragomir, R. P. Agarwal and P. Cerone, On Simpson’s inequality and applications, J. Inequal. Appl. 5 (2000), no. 6, 533–579.
[7] D. H. Hyers and S. M. Ulam, Approximately convex functions, Proc. Amer. Math.
Soc. 3 (1952), 821–828.
[8] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special
means of real numbers and to midpoint formula, Appl. Math. Comput. 147 (2004),
no. 1, 137–146.
[9] F. Mirzapour and A. Morassaei, On approximately convex functions, Iran. Int. J. Sci.
2 (2001), no. 1, 83–95.
Some integral inequalities of Simpson type
13
[10] C. T. Ng and K. Nikodem, On approximately convex functions, Proc. Amer. Math.
Soc. 118 (1993), no. 1, 103–108.
[11] Z. Páles, On approximately convex functions, Proc. Amer. Math. Soc. 131 (2003),
no. 1, 243–252.
[12] C. E. M. Pearce and J. Peˇcari´c, Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett. 13 (2000), no. 2,
51–55.
[13] F. Qi, Z.-L. Wei and Q. Yang, Generalizations and refinements of Hermite–
Hadamard’s inequality, Rocky Mountain J. Math. 35 (2005), no. 1, 235–251.
[14] M. Z. Sarikaya, E. Set and M. E. Özdemir, On new inequalities of Simpson’s type
for convex functions, RGMIA Res. Rep. Coll. 13 (2010), no. 2, article 2.
[15] Y. Shuang, H.-P. Yin and F. Qi, Hermite–Hadamard type integral inequalities for
geometric-arithmetically s-convex functions, Analysis (Munich) 33 (2013), no. 2,
197–208.
[16] S.-H. Wang, B.-Y. Xi and F. Qi, On Hermite–Hadamard type inequalities for .˛; m/convex functions, Int. J. Open Probl. Comput. Sci. Math. 5 (2012), no. 4, 47–56.
[17] S.-H. Wang, B.-Y. Xi and F. Qi, Some new inequalities of Hermite–Hadamard
type for n-time differentiable functions which are m-convex, Analysis (Munich) 32
(2012), no. 3, 247–262.
[18] B.-Y. Xi, R.-F. Bai and F. Qi, Hermite–Hadamard type inequalities for the m- and
.˛; m/-geometrically convex functions, Aequationes Math. 84 (2012), no. 3, 261–
269.
[19] B.-Y. Xi, J. Hua and F. Qi, Hermite–Hadamard type inequalities for extended sconvex functions on the co-ordinates in a rectangle, J. Appl. Anal., to appear.
[20] B.-Y. Xi and F. Qi, Some integral inequalities of Hermite–Hadamard type for convex
functions with applications to means, J. Funct. Spaces Appl. 2012, article ID 980438.
[21] B.-Y. Xi and F. Qi, Hermite–Hadamard type inequalities for functions whose derivatives are of convexities, Nonlinear Funct. Anal. Appl. 18 (2013), no. 2, 163–176.
[22] B.-Y. Xi and F. Qi, Some Hermite–Hadamard type inequalities for differentiable
convex functions and applications, Hacet. J. Math. Stat. 42 (2013), no. 3, 243–257.
[23] B.-Y. Xi and F. Qi, Some inequalities of Hermite–Hadamard type for h-convex functions, Adv. Inequal. Appl. 2 (2013), no. 1, 1–15.
[24] B.-Y. Xi, S.-H. Wang and F. Qi, Some inequalities of Hermite–Hadamard type for
functions whose 3rd derivatives are P -convex, Appl. Math. 3 (2012), no. 12, 1898–
1902.
Note 3:
[25] T.-Y. Zhang, A.-P. Ji and F. Qi, Some inequalities of Hermite–Hadamard type for
GA-convex functions with applications to means, Matematiche (Catania) 68 (2013),
no. 1, 229–239.
Please note
that we
only state
one e-mail
address per
author.
14
F. Qi and B.-Y. Xi
Received April 4, 2013; revised June 23, 2013; accepted July 9, 2013.
Author information
Feng Qi, College of Mathematics, Inner Mongolia University for Nationalities,
Tongliao City, Inner Mongolia Autonomous Region, 028043, China.
E-mail: [email protected]
Bo-Yan Xi, College of Mathematics, Inner Mongolia University for Nationalities,
Tongliao City, Inner Mongolia Autonomous Region, 028043, China.
E-mail: [email protected]