Some integral inequalities for α-, m-, (α,m)-logarithmically convex functions

0.14 MB·English

by  Mevlut Tunc

SOME INTEGRAL INEQUALITIES FOR α-, m-,
(α,m)-LOGARITHMICALLY CONVEX FUNCTIONS
3
1 MEVLU¨TTUNC¸(cid:3) ANDEBRUYU¨KSEL▽
0
2
Abstract. In this paper, we establish some new Hadamard type inequali-
n ties usingelementary well knowninequalities forfunctions whoseinequalities
a absolutevaluesareα-,m-,(α,m)-logarithmicallyconvex.
J

8
2

1. INTRODUCTION
]
A
Let f : I ⊆ R → R be a convex mapping defined on the interval I of real
C
numbers and a,b∈I, with a<b. The following double inequalities:
.
h a+b b f(a)+f(b)
t f ≤ f(x)dx≤
a 2 2
m (cid:18) (cid:19) Za
hold. This double inequality is known in the literature as the Hermite-Hadamard
[ inequality for convex functions (see [1]-[8]).

1 In this section, we will present definitions and some results used in this paper.
v
Definition 1. Let I be an interval in R. Then f : I →R, ∅=6 I ⊆R is said to be
7
5 convex if
0
(1.1) f(tx+(1−t)y)≤tf(x)+(1−t)f(y).
7
. for all x,y ∈I and t∈[0,1].
1
0
In [1], the concepts of α-, m- and (α,m)-logarithmically convex functions were
3
introduced as follows.
1
v: Definition 2. [1] A function f : [0,b] → (0,∞) is said to be m-logarithmically
i convex if the inequality
X
(1.2) f(tx+m(1−t)y)≤[f(x)]t[f(y)]m(1−t)
r
a
holds for all x,y ∈[0,b], m∈(0,1], and t∈[0,1].
Obviously,if putting m=1 in Definition 2, then f is just the ordinarylogarith-
mically convex on [0,b].
Definition 3. [8] A function f : [0,b] → (0,∞) is said to be α-logarithmically
convex if
(1.3) f(tx+(1−t)y)≤[f(x)]tα[f(y)](1−tα)
Date:January10,2013.
2000 Mathematics Subject Classification. 26A15,26A51,26D10.
Key words and phrases. α-, m-, (α,m)-logarithmically convex, Hadamard’s inequality,
H¨older’sinequality,powermeaninequality,Cauchy’sinequality.
(cid:3)CorrespondingAuthor.
Thispaperisinfinalformandnoversionofitwillbesubmittedforpublicationelsewhere.
1
2 MEVLU¨TTUNC¸(cid:3) ANDEBRUYU¨KSEL▽
holds for all x,y ∈[0,b], α∈(0,1] and t∈[0,1].
Clearly, when taking α = 1 in Definition 3, then f becomes the ordinary loga-
rithmically convex on [0,b].
Definition 4. [1] A function f :[0,b]→(0,∞) is said to be (α,m)-logarithmically
convex if
(1.4) f(tx+m(1−t)y)≤[f(x)]tα[f(y)]m(1−tα)
holds for all x,y ∈[0,b], (α,m)∈(0,1]×(0,1], and t∈[0,1].
Clearly, when taking α = 1 in Definition 4, then f becomes the standard m-
logarithmically convex function on [0,b], and, when taking m = 1 in Definition 4,
then f becomes the α-logarithmically convex function on [0,b].
2. NEW HADAMARD-TYPE INEQUALITIES
In order to prove our main theorems, we need the following lemma [2].
Lemma 1. [2] Let f : I ⊂ R → R be a differentiable mapping on I◦, a,b ∈ I◦
with a < b. If f′ ∈ L[a,b], then the following equality holds:
f(a)+f(b) 1 b
(2.1) − f(x)dx
2 b−a
Za
b−a 1 1
′ ′
= [f (ta+(1−t)b)−f (sa+(1−s)b)](s−t)dtds.
2
Z0 Z0
Asimple proofofthis equalitycanbe alsodoneintegratingbypartsinthe right
hand side (see [2]).
ThenexttheoremsgivesanewresultoftheupperHermite-Hadamardinequality
for α-, m-, (α,m)-logarithmically convex functions.
Theorem 1. Let I ⊃ [0,∞) be an open interval and let f : I → (0,∞) be a
differentiable function on I such that f′ ∈L(a,b) for 0≤a<b<∞. If |f′(x)| is
(α,m)-logarithmically convex on 0, b for (α,m)∈(0,1]2, then
m
(cid:2) (cid:3)
f(a)+f(b) 1 b
(2.2) − f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12) (b−a) f′ b m, (cid:12)(cid:12) η =1
≤ (cid:12)( (b3−2a)(cid:12)f′(cid:0)mmb(cid:1)(cid:12)m −α2ln2ηα−32lαn3lnηη(cid:12)+2ηα−2, η <1
(cid:12) (cid:12)
(cid:12) (cid:0) (cid:1)(cid:12)
where η =|f′(a)|/ f′ b m(cid:12) . (cid:12)
m
(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
SOME INTEGRAL INEQUALITIES 3
Proof. By Lemma 1 and since |f′| is an (α,m)-logarithmically convex on 0, b ,
m
then we have
(cid:2) (cid:3)
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12)b−a 1 1 ′ (cid:12)(cid:12) ′
≤ (cid:12) |(f (ta+(1−t)b))−(cid:12) (f (sa+(1−s)b))||s−t|dtds
2
Z0 Z0
b−a 1 1 ′ tα ′ b m(1−tα)
≤ |s−t||f (a)| f dtds
2 m
"Z0 Z0 (cid:12) (cid:18) (cid:19)(cid:12) #
(cid:12) (cid:12)
b−a 1 1 ′ s(cid:12)(cid:12)α ′ b(cid:12)(cid:12) m(1−sα)
+ |s−t||f (a)| f dtds
2 m
"Z0 Z0 (cid:12) (cid:18) (cid:19)(cid:12) #
(cid:12) (cid:12)
(cid:12) (cid:12)
If 0<k≤1, 0<m,n≤1 (cid:12) (cid:12)
(2.3) kmn ≤kmn.
When η =1, by (2.3), we get
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12)b−a ′ b m 1 1 (cid:12)(cid:12) 1 1
≤ (cid:12) f |s−t|d(cid:12)tds+ |s−t|dtds
2 m
(cid:12) (cid:18) (cid:19)(cid:12) (cid:20)Z0 Z0 Z0 Z0 (cid:21)
(cid:12) (cid:12)m
b−a(cid:12) ′ b (cid:12)
= (cid:12)f (cid:12)
3 m
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
When η <1, by (2.3)(cid:12), we get(cid:12)
(cid:12) (cid:12)
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
≤ (cid:12)(cid:12)(cid:12)b−a f′ b m 1 1|s−t|η(cid:12)(cid:12)(cid:12)αtdtds+ 1 1|s−t|ηαsdtds
2 m
(cid:12) (cid:18) (cid:19)(cid:12) (cid:20)Z0 Z0 Z0 Z0 (cid:21)
b−a(cid:12)(cid:12) ′ b (cid:12)(cid:12)m −α2ln2η−2αlnη+4ηα+α2ηαln2η−2αηαlnη−4
= (cid:12)f (cid:12)
2 m 2α3ln3η
(cid:12) (cid:18) (cid:19)(cid:12) (cid:20)
−α(cid:12)(cid:12)lnη+2η(cid:12)(cid:12)α−αηαlnη−2
+ (cid:12) (cid:12)
2α2ln2η
(cid:21)
which completes the proof. (cid:3)
Corollary 1. Let I ⊃ [0,∞) be an open interval and let f : I → (0,∞) be a
differentiable function on I such that f′ ∈L(a,b) for 0≤a<b<∞. If |f′(x)| is
m-logarithmically convex on 0, b for m∈(0,1] , then
m
f(a)+f(b) 1 b (cid:2) (cid:3) (b−a) f′ b m, η =1
(cid:12)(cid:12) 2 − b−aZa f(x)dx(cid:12)(cid:12)≤( (b3−2a)(cid:12)(cid:12)f′(cid:0)mmb(cid:1)(cid:12)(cid:12)m −ln2η−l2n3lnηη+2η−2, η <1
(cid:12) (cid:12)
(cid:12)where η is same as Theorem 1. (cid:12) (cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
4 MEVLU¨TTUNC¸(cid:3) ANDEBRUYU¨KSEL▽
Corollary 2. Let I ⊃ [0,∞) be an open interval and let f : I → (0,∞) be a
differentiable function on I such that f′ ∈ L(a,b) for 0 ≤ a < b < ∞. If |f′(x)|is
α-logarithmically convex on [0,b] for α∈(0,1] , then
f(a)+f(b) 1 b (b−a)|f′(b)|, η =1
(cid:12)(cid:12) 2 − b−aZa f(x)dx(cid:12)(cid:12)≤( (b3−2a)|f′(b)|4ηα−4α2lnαη3−ln23αη2ln2η−4, η <1
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where η =|f′(a)|/|f′(b)|.
Theorem 2. Let I ⊃ [0,∞) be an open interval and let f : I → (0,∞) be a
differentiable function on I such that f′ ∈ L(a,b) for 0 ≤ a < b < ∞. If |f′(x)|q
is an (α,m)-logarithmically convex on 0, b for (α,m)∈(0,1]2 and p,q >1 with
m
1 + 1 =1, then
p q (cid:2) (cid:3)
f(a)+f(b) 1 b
(2.4) − f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12) (cid:12)1
(cid:12)(cid:12) (b−a) f′ b m 2 (cid:12)(cid:12)p , η =1
m (p+1)(p+2)
≤
 1 1
 (b−a(cid:12)(cid:12)) f(cid:0)′ (cid:1)mb(cid:12)(cid:12) (cid:16)m (p+1)2(p+(cid:17)2) p × ηln(αηq(,ααqq,)α−q)1 q , η <1
 (cid:12)(cid:12) (cid:0) (cid:1)(cid:12)(cid:12) (cid:16) (cid:17) (cid:16) (cid:17)
where η(α,α) is same as Theorem 1
Proof. Since|f′|q isan(α,m)-logarithmicallyconvexon 0, b ,fromLemma1and
m
the well known Ho¨lder inequality, we have
(cid:2) (cid:3)
(2.5)
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12)b−a 1 1 ′ (cid:12)(cid:12) ′
≤ (cid:12) |(f (ta+(1−t)b))−(cid:12) (f (sa+(1−s)b))||s−t|dtds
2
Z0 Z0
b−a 1 1 ′ tα ′ b m(1−tα)
≤ |s−t||f (a)| f dtds
2 m
Z0 Z0 (cid:12) (cid:18) (cid:19)(cid:12)
b−a 1 1 ′ s(cid:12)(cid:12)α ′ b(cid:12)(cid:12) m(1−sα)
+ |s−t||f (a)|(cid:12) f (cid:12) dtds
2 m
Z0 Z0 (cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)1
b−a b m 1 1 (cid:12) (cid:12)p
≤ f′ |s−(cid:12)t|pdtds (cid:12)
2 m
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19)
(cid:12) (cid:12) 1 1
(cid:12) 1 1 (cid:12) q 1 1 q
× (cid:12) ηqt(cid:12)αdtds + ηqsαdtds
"(cid:18)Z0 Z0 (cid:19) (cid:18)Z0 Z0 (cid:19) #
SOME INTEGRAL INEQUALITIES 5
If η =1, by (2.3), we obtain
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12) (cid:12) 1
(cid:12) b m 1 1 (cid:12) p
≤ ((cid:12)b−a) f′ |s−t(cid:12)|pdtds
m
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19)
(cid:12) (cid:12)m 1
= (b−a)(cid:12)(cid:12)f′ b (cid:12)(cid:12) 2 p
m (p+1)(p+2)
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
If η <1, by (2.3), we obtain (cid:12) (cid:12)
f(a)+f(b) 1 b
(2.6) − f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12) (cid:12) 1
(cid:12)b−a b m 1 1 (cid:12) p
≤ (cid:12) f′ |s−t|p(cid:12)dtds
2 m
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19)
(cid:12) (cid:12) 1 1
(cid:12) 1 1 (cid:12) q 1 1 q
× (cid:12) ηqt(cid:12)αdtds + ηqsαdtds
"(cid:18)Z0 Z0 (cid:19) (cid:18)Z0 Z0 (cid:19) #
m 1 1
′ b 2 p η(αq,αq)−1 q
= (b−a) f ×
m (p+1)(p+2) lnη(αq,αq)
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:12) (cid:12)
which completes th(cid:12)e proof. (cid:12) (cid:3)
(cid:12) (cid:12)
Corollary 3. Let I ⊃ [0,∞) be an open interval and let f : I → (0,∞) be a
differentiable function on I such that f′ ∈L(a,b) for 0≤a<b<∞. If |f′(x)|qis
an m-logarithmically convex on 0, b for m∈(0,1] and p=q =2, then
m
(cid:2) (cid:3)
f(a)+f(b) 1 b (b−a) f′ mb m 16, η =1
− f(x)dx ≤ 1
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)  (b−a(cid:12)(cid:12)) f(cid:0)′ m(cid:1)b(cid:12)(cid:12) mq 16 ηln(2η,(22),−2)1 2 , η <1
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:0) (cid:1)(cid:12) q (cid:16) (cid:17)
C(cid:12) orollary 4. Let I ⊃ [0,∞) be(cid:12)anopen interva(cid:12)l and l(cid:12)et f : I → (0,∞) be a
differentiable function on I such that f′ ∈ L(a,b) for 0 ≤ a < b < ∞. If |f′(x)|is
α-logarithmically convex on [0,b] for α∈(0,1] , then
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12) 1 (cid:12)
(cid:12)(cid:12) (b−a)|f′(b)| 2 p , (cid:12)(cid:12) η =1
(p+1)(p+2)
≤
 1 1
 (b−a)|f′(b(cid:16))| (p+1)2(p+2(cid:17)) p ηln(αηq(,ααqq,)α−q)1 q , η <1
(cid:16) (cid:17) (cid:16) (cid:17)
where η =|f′(a)|/|f′(b)|.
Theorem 3. Let I ⊃ [0,∞) be an open interval and let f : I → (0,∞) be a
differentiable function on I such that f′ ∈L(a,b) for 0≤a<b<∞. If |f′(x)|q is
6 MEVLU¨TTUNC¸(cid:3) ANDEBRUYU¨KSEL▽
(α,m)-logarithmically convex on 0, b for (α,m)∈(0,1]2, and then
m
(cid:2) (cid:3)
(2.7)
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12) b−a f′ b m, (cid:12)(cid:12) η =1
≤ (cid:12) b−3a(cid:12) 1 (cid:0)1m−1q(cid:1)(cid:12)f′ b m 2ϕ−2(cid:12)− ϕ+1 − 1−ϕ 1q + ϕ−1 − ϕ+1 q1 , η <1
 2 (cid:12) 3 (cid:12) m [lnϕ]3 [lnϕ]2 2lnϕ [lnϕ]2 2lnϕ
(cid:0) (cid:1) (cid:12) (cid:0) (cid:1)(cid:12) (cid:26)h i h i (cid:27)
 (cid:12) (cid:12)
where η(α,α) is same as Theorem 1, and we take η(αq,αq)=ϕ .
Proof. Since |f′|q is an (α,m)-logarithmically convex on 0, b , for q ≥ 1, from
m
Lemma 1 and the well known power mean integral inequality, we get
(cid:2) (cid:3)
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12)b−a 1 1 ′ (cid:12)(cid:12) ′
≤ (cid:12) |(f (ta+(1−t)b))−(cid:12) (f (sa+(1−s)b))||s−t|dtds
2
Z0 Z0
b−a 1 1 1−q1 1 1 q1
′ q
≤ |s−t|dtds |s−t||f (ta+(1−t)b)| dtds
2
(cid:18)Z0 Z0 (cid:19) (cid:18)Z0 Z0 (cid:19)
b−a 1 1 1−1q 1 1 q1
′ q
+ |s−t|dtds |s−t||f (sa+(1−s)b)| dtds
2
(cid:18)Z0 Z0 (cid:19) (cid:18)Z0 Z0 (cid:19)
b−a b m 1 1 1−q1 1 1 q1
≤ f′ |s−t|dtds |s−t|ηqtαdtds
2 m
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19) (cid:18)Z0 Z0 (cid:19)
b−a(cid:12)(cid:12) b(cid:12)(cid:12) m 1 1 1−q1 1 1 q1
+ (cid:12) f′ (cid:12) |s−t|dtds |s−t|ηqsαdtds
2 m
(cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19) (cid:18)Z0 Z0 (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
When η =1, by (2.3), we obtain
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
≤ (cid:12)(cid:12)(cid:12)b−a 1 1−q1 f′ b m 1 (cid:12)(cid:12)(cid:12)1|s−t|dtds q1
2 3 m
(cid:18) (cid:19) (cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19)
+b−a 1 1−(cid:12)(cid:12)(cid:12)q1 f′ b (cid:12)(cid:12)(cid:12) m 1 1|s−t|dtds q1
2 3 m
(cid:18) (cid:19) (cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19)
m(cid:12) (cid:12)
b−a ′ b (cid:12) (cid:12)
= f (cid:12) (cid:12)
3 m
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
SOME INTEGRAL INEQUALITIES 7
When η <1, by (2.3), we obtain
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
≤ (cid:12)(cid:12)(cid:12)b−a 1 1−q1 f′ b m 1 (cid:12)(cid:12)(cid:12)1|s−t|ηαqtdtds q1
2 3 m
(cid:18) (cid:19) (cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19)
+b−a 1 1−(cid:12)(cid:12)(cid:12)1q f′ b (cid:12)(cid:12)(cid:12) m 1 1|s−t|ηαqsdtds q1
2 3 m
(cid:18) (cid:19) (cid:12) (cid:18) (cid:19)(cid:12) (cid:18)Z0 Z0 (cid:19)
= b−a 1 1−q1 f(cid:12)(cid:12)(cid:12)′ b m(cid:12)(cid:12)(cid:12)
2 3 m
(cid:18) (cid:19) (cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12) 1
2η(αq,αq(cid:12)(cid:12))−2 (cid:12)(cid:12) η(αq,αq)+1 1−η(αq,αq) q
× − −
"[ln(η(αq,αq))]3 [ln(η(αq,αq))]2 2ln(η(αq,αq))#

1
 η(αq,αq)−1 η(αq,αq)+1 q
+ −
"[ln(η(αq,αq))]2 2ln(η(αq,αq))# 

which completes the proof. (cid:3)

Corollary 5. Let I ⊃ [0,∞) be an open interval and let f : I → (0,∞) be a
differentiable function on I such that f′ ∈L(a,b) for 0≤a<b<∞. If |f′(x)|qis
m-logarithmically convex on 0, b for m∈(0,1], then
m
f(a)+f(b) 1 (cid:2) b (cid:3)
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12) b−a f′ b m, (cid:12)(cid:12) η =1
≤ (cid:12) (b−2a) 13 1−3q1 (cid:12)f′(cid:0)mmb(cid:1)(cid:12)m (cid:12)
(cid:12) (cid:12) 1 1
 (cid:0)×(cid:1) [2lnη(cid:12)(cid:12)(ηq(,qq(cid:0),)q−)]23(cid:1)(cid:12)(cid:12)− [lηn(ηq,(qq),+q)1]2 − 21l−nηη((qq,,qq)) q + [lηn(ηq,(qq),−q)1]2 − 2ηl(nq,ηq()q+,q1) q ,η <1
(cid:26)h i h i (cid:27)
Corollary 6. Let I ⊃ [0,∞) be an open interval and let f : I → (0,∞) be a
differentiable function on I such that f′ ∈L(a,b) for 0≤a<b<∞. If |f′(x)| is
α-logarithmically convex on [0,b] for α∈(0,1] , then
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12) (b−a) 1 |f′(b)|, (cid:12)(cid:12) η =1
(cid:12) (b−a) 1 1−(cid:0)31q(cid:1)|f′(b)| 2η(αq,(cid:12)αq)−2 − η(αq,αq)+1 − 1−η(αq,αq) 1q
≤ 2 3 [ln(η(αq,αq))]3 [ln(η(αq,αq))]2 2ln(η(αq,αq))
 (cid:0) (cid:1) + (cid:18)ηh(αq,αq)−1 − η(αq,αq)+1 1q i , η <1
[ln(η(αq,αq))]2 2ln(η(αq,αq))
where η =|f′(a)|/|f′(b)|.h i (cid:19)
Theorem 4. Let f :I ⊂R →R be differentiable on I◦, a,b∈I, with a<b and
+ +
f′ ∈L([a,b]). If |f′| is an (α,m)-logarithmically convex 0, b for (α,m)∈(0,1]2
m
(cid:2) (cid:3)
8 MEVLU¨TTUNC¸(cid:3) ANDEBRUYU¨KSEL▽
and µ ,µ ,τ ,τ >0 with µ +τ =1 and µ +τ =1, then
1 2 1 2 1 1 2 2
(2.8)
f(a)+f(b) 1 b
− f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12) (b−a) f′ b m 2µ31 (cid:12)(cid:12)+ 2µ32 +τ +τ , η =1
(cid:12) 2 m (2µ1+1)(µ1+1)(cid:12) (2µ2+1)(µ2+1) 1 2
 (b−a)(cid:12)f′(cid:0) b(cid:1)(cid:12)mh 2µ31 + 2µ32 i
≤  2 (cid:12)+(cid:12)(cid:12) τ(cid:0)1mη(cid:16)(cid:1)(cid:12)τ(cid:12)(cid:12)α1,τnα1((cid:17)2−µ11++1)(τµ21η+(cid:16)1)τα2,τα2(2(cid:17)µ−21+1)(,µ2+1) η <1
 lnη(cid:16)τα1,τα1(cid:17) lnη(cid:16)τα2,τα2(cid:17)(cid:27)
where η(α,α) is same as Theorem 1.
Proof. Since|f′|q isan(α,m)-logarithmicallyconvexon 0, b ,fromLemma1,we
m
have
(cid:2) (cid:3)
f(a)+f(b) 1 b
(2.9) − f(x)dx
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)
(cid:12)(cid:12)(b−a) 1 1 ′ (cid:12)(cid:12) ′
≤ (cid:12) |(f (ta+(1−t)b)(cid:12))−(f (sa+(1−s)b))||s−t|dtds
2
Z0 Z0
(b−a) 1 1 ′ tα ′ b m(1−tα)
≤ |s−t||f (a)| f dtds
2 m
"Z0 Z0 (cid:12) (cid:18) (cid:19)(cid:12) #
(cid:12) (cid:12)
(b−a) 1 1 ′ s(cid:12)(cid:12)α ′ b(cid:12)(cid:12) m(1−sα)
+ |s−t||f (a)| f dtds
2 m
"Z0 Z0 (cid:12) (cid:18) (cid:19)(cid:12) #
(cid:12) (cid:12)
(b−a) b m 1 1 (cid:12) (cid:12) 1 1
= f′ |s−t|(cid:12)ηtαdtds+(cid:12) |s−t|ηsαdtds
2 m
(cid:12) (cid:18) (cid:19)(cid:12) (cid:20)Z0 Z0 Z0 Z0 (cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
for all t∈[0,1]. U(cid:12)sing the w(cid:12)ell knowninequality rt≤µrµ1 +τtτ1, on the right side
of (2.9), we get
1 1 1 1
(2.10) |s−t|ηtαdtds+ |s−t|ηsαdtds
Z0 Z0 Z0 Z0
1 1 1 1 1 tα
≤ µ1 |s−t|µ1 dtds+τ1 ητ1dtds
Z0 Z0 Z0 Z0
1 1 1 1 1 sα
+µ2 |s−t|µ2 dtds+τ2 ητ2dtds
Z0 Z0 Z0 Z0
When η =1, by (2.3), we get
1 1 1 1
(2.11) |s−t|ηtαdtds+ |s−t|ηsαdtds
Z0 Z0 Z0 Z0
2µ3 2µ3
≤ 1 + 2 +τ +τ
1 2
(2µ +1)(µ +1) (2µ +1)(µ +1)
1 1 2 2
SOME INTEGRAL INEQUALITIES 9
When η <1, by (2.3), we get
(2.12)
1 1 1 1
|s−t|ηtαdtds+ |s−t|ηsαdtds
Z0 Z0 Z0 Z0
1 1 1 1 1 tα 1 1 1 1 1 sα
≤ µ1 |s−t|µ1 dtds+τ1 ητ1dtds+µ2 |s−t|µ2 dtds+τ2 ητ2dtds
Z0 Z0 Z0 Z0 Z0 Z0 Z0 Z0
1 1 1 1 1 1 1 1
1 1 αt αs
≤ µ1 |s−t|µ1 dtds+µ2 |s−t|µ2 dtds+τ1 ητ1dtds+τ2 ητ2dtds
Z0 Z0 Z0 Z0 Z0 Z0 Z0 Z0
= 2µ31 + 2µ32 +τ η τα1,τα1 −1 +τ η τα2,τα2 −1
(2µ1+1)(µ1+1) (2µ2+1)(µ2+1) 1 ln(cid:16)η α,(cid:17)α 2 ln(cid:16)η α,(cid:17)α
τ1 τ1 τ2 τ2
from (2.9)-(2.12), which completes the proof. (cid:16) (cid:17) (cid:16) (cid:3)(cid:17)
Corollary 7. Under the assumptions of Theorem 4, and µ = µ = µ > 0, τ =
1 2
τ =τ >0 with µ+τ =1, then we have
1 2
f(a)+f(b) 1 b (b−2a) f′ mb m (2µ+41µ)(3µ+1) +2τ , η =1
− f(x)dx ≤
(cid:12)(cid:12) 2 b−aZa (cid:12)(cid:12)  (b−2a) f′ m(cid:12)(cid:12)b (cid:0)m (cid:1)((cid:12)(cid:12)2µ+h41µ)(3µ+1) +2τηln(ηατ(,αατi,)α−)1 , η <1
(cid:12) (cid:12) (cid:20) τ τ (cid:21)
(cid:12) (cid:12) (cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)Reference(cid:12)s (cid:12)
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(cid:3)Department of Mathematics, Faculty of Science and Arts, Kilis 7 Aralık Univer-
sity,Kilis, 79000,Turkey.
E-mail address: (cid:3)[email protected]
▽The Institute for Graduate Studies in Sciences and Engineering, Kilis 7 Aralık
University,Kilis, 79000,Turkey.
E-mail address: ▽[email protected]