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PureMathematicsnØêÆ,2021,11(1),94-102
PublishedOnlineJanuary2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.111014
k'(α,m)-éêà¼êHermite-Hadamard
.È©Øª
‘‘‘§§§¸¸¸§§§ààà···§§§•••§§§©©©
∗
§§§ŸŸŸjjjÜÜÜ
Ü•“‰ÆêƆÚOÆ,S Ü•
ÂvFϵ2020c1214F¶¹^Fϵ2021c115F¶uÙFϵ2021c125F
Á‡
©Ì‡ïá˜#9(α,m)éêà¼êHermite-Hadamard.È©Øª"
'…c
Hermite-Hadamard.Øª§(α,m)-éêà¼ê§Øª
IntegralInequalitiesofHermite-Hadamard
Stylefor(α,m)-Logarithmically
ConvexFunctions
YingHuang,PanpanHan,JingQi,FangWang,WenWang
∗
,MaosenWei
SchoolofMathematicsandStatistics,HefeiNormalUniversity,HefeiAnhui
Received:Dec.14
th
,2020;accepted:Jan.15
th
,2021;published:Jan.25
th
,2021
∗ÏÕŠö"
©ÙÚ^:‘,¸,à·,•,©,ŸjÜ.k'(α,m)-éêà¼êHermite-Hadamard.È©Øª[J].
nØêÆ,2021,11(1):94-102.DOI:10.12677/pm.2021.111014
‘
Abstract
Theaimofthispap eristoestablishseveralnewHermite-Hadamardstyleinequalities
for(α,m)-logarithmicallyconvexfunction.
Keywords
Hermite-Hadamard’sInequality,(α,m)-LogarithmicallyConvexFunction,Inequality
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
©¥R= (−∞,∞),R
+
= [0,∞),R
++
= (0,∞).
Äk,·‚£Á˜-‡Vg:
½Â1.1[1][2]e¼êf: I⊂R→R÷v
f(tx+(1−t)y) ≤tf(x)+(1−t)f(y),∀x,y∈I,t∈[0,1],(1.1)
K¡¼êf´Iþà¼ê.d,XJ−f´à¼ê,Ò¡f´]¼ê.
½Â1.2[3]e¢¼êf: I⊂R
+
→R
++
÷v
f(tx+(1−t)y) ≤[f(x)]
t
m
[f(y)]
m(1−t)
m
,∀x,y∈I,m∈(0,1],t∈[0,1],(1.2)
Ò¡¼êf´m-éêà¼ê.
CÏ,©z[4]òþ¡½Â?˜í2(α,m)-éêà¼ê,Xe:
½Â1.3[4]e¢¼êf: I⊂R
+
→R
++
f(tx+m(1−t)y) ≤[f(x)]
t
α
[f(y)]
m(1−t
α
)
,(1.3)
holdsforallx,y∈I,(α,m) ∈(0,1]×(0,1],t∈[0,1].Ò¡f´(α,m)-éêà¼ê.
DOI:10.12677/pm.2021.11101495nØêÆ
‘
¢¼êf: I⊂R→R´à¼ê,KͶHermite-HadamardÈ©Øª•
f
a+b
2
≤
1
b−a
b
a
f(x)dx≤
f(a)+f(b)
2
,a,b∈I,a<b.(1.4)
d,¯õÆöí2à¼êVg,k'AÛà¼ê!NÚà¼ê!m-à¼ê±
9(α,m)-à¼êHermite-Hadamard.È©Øª,•[ŒÖƒ'©z[5–10].
Bai-Qi-Xi3©z[4]¥¼(α,m)-éê༼êHermite-Hadamard.Øª,Xe:
½nA[4]I⊂R
+
´m«m,…¼êf:I→R
+
´IþŒ‡¼ê,…f
0
∈L([a,b])(
0≤a<b<R
+
).q∈[1,+∞),(α,m)∈(0,1] ×(0,1],e|f
0
(x)|
q
´[0,
b
m
]þ(α,m)-éêà¼
ê,Kk
f(a)+f(b)
2
−
b
a
f(x)dx≤
a+b
2
f
0
b
m
1
2
1−1/q
[E
1
(α,mq)]
1/q
,
E(α,m,q) =
1
2
,µ= 1,
F(µ,αq),0 <µ<1,
µ
(1−α)q
F(µ,
q
α
),µ>1,
µ=
|f
0
(a)|
|f
0
(
b
m
)|
m
,
F(u,v) =
1
v
2
(lnu)
2
v(u
v
−1)lnu−2(u
v/2
−1)
2
,u,v>0,u6= 1)
©Ì‡ïá˜#9(α,m)-éêà¼êHermite-Hadamard.È©Øª.
2.̇(J
©Ì‡(JXeµ
½n2.1f: I⊂R
+
→R
++
´(α,m)-éêà¼ê,…
J
1
=
af(a)
(b−a)
2
b
a
(b−x)f(x)dx,
J
2
=
bf(b)
(b−a)
2
b
a
(x−a)f(x)dx.
e(α,m) ∈(0,1],Kk
J
1
+J
2
≤
[f(
a
m
)]
m
[af(a)h(δ
α
)+bf(b)g(δ
α
)],δ≤1;
[f(
a
m
)]
m
af(a)h δ
1
α
+bf(b)gδ
1
α
,δ≥1.
(2.1)
DOI:10.12677/pm.2021.11101496nØêÆ
‘
Ù¥a,b∈I…a<b,δ=
f(b)
[f(
a
m
)]
m
,
h(β) =
1
0
(1−t)β
t
dt=
1
2
,β= 1;
β−1−lnβ
(lnβ)
2
,β6= 1.
(2.2)
g(β) =
1
0
tβ
t
dt=
1
2
,β= 1;
βlnβ−lnβ+1
(lnβ)
2
,β6= 1.
(2.3)
.
y²:x= (1−t)a+tb= m(1−t)
a
m
+tb(t∈[0,1],m∈(0,1]),ØJ
J
1
=
af(a)
(b−a)
2
b
a
(b−x)f(x)dx= af(a)
1
0
(1−t)f(m(1−t)
a
m
+tb)dt,(2.4)
J
2
=
bf(b)
(b−a)
2
b
a
(x−a)f(x)dx= bf(b)
1
0
tf(m(1−t)
a
m
+tb)dt.(2.5)
Ï•f(x)´(α,m)-éêà¼ê,d(2.2)Œ
J
1
= af(a)
1
0
(1−t)f(m(1−t)
a
m
+tb)dt
≤af(a)
1
0
(1−t)[f(
a
m
)]
m(1−t
α
)
[f(b)]
t
α
dt
= af(a)[f(
a
m
)]
m
1
0
(1−t)
[f(b)]
[f(
a
m
)]
m
t
α
dt.(2.6)
±9
J
2
= bf(b)
1
0
tf(m(1−t)
a
m
+tb)dt
≤bf(b)
1
0
t[f(
a
m
)]
m(1−t
α
)
[f(b)]
t
α
dt
= bf(b)[f(
a
m
)]
m
1
0
t
[f(b)]
[f(
a
m
)]
m
t
α
dt.(2.7)
e0 <u≤1 ≤v,0 <α,s≤1,K
u
α
s
≤u
αs
,v
α
s
≤v
α
s
.(2.8)
DOI:10.12677/pm.2021.11101497nØêÆ
‘
(1)e
[f(b)]
[f(
a
m
)]
m
≤1,d(2.6-2.8)Œ
J
1
≤af(a)[f(
a
m
)]
m
1
0
(1−t)
[f(b)]
[f(
a
m
)]
m
tα
dt= af(a)[f(
a
m
)]
m
h
f(b)
[f(
a
m
)]
m
α
(2.9)
J
2
≤bf(b)[f(
a
m
)]
m
1
0
t
[f(b)]
[f(
a
m
)]
m
tα
dt= bf(b)[f(
a
m
)]
m
g
f(b)
[f(
a
m
)]
m
α
.(2.10)
(2)e
[f(b)]
[f(a)]
m
≥1,d(2.6-2.8)Œ
J
1
≤af(a)[f(
a
m
)]
m
1
0
(1−t)
[f(b)]
[f(
a
m
)]
m
t
α
dt= af(a)[f(
a
m
)]
m
h
f(b)
[f(
a
m
)]
m
1
α
(2.11)
J
2
≤bf(b)[f(
a
m
)]
m
1
0
t
[f(b)]
[f(
a
m
)]
m
t
α
dt= bf(b)[f(
a
m
)]
m
g
f(b)
[f(
a
m
)]
m
1
α
.(2.12)
d(2.9)Ú(2.12)Œ(2.1).
½n2.2¢¼êf: I⊂R
+
→R
++
.e[f(x)]
q
(q≥1)´Iþ(α;m)-éêà¼ê,Kk
J
1
+J
2
≤
1
p+1
1
p
[f(
a
m
)]
m
af(a)[h(δ
qα
)]
1
q
+bf(b)[g(δ
qα
)]
1
q
,δ≤1;
1
p+1
1
p
[f(
a
m
)]
m
af(a)h δ
q
α
1
q
+bf(b)gδ
q
α
1
q
,δ≥1.
(2.13)
Ù¥a,b∈I…a<b,(α,m) ∈(0,1],p≥1…
1
p
+
1
q
= 1.
y²:Ï•[f(x)]
q
´(α;m)éêà¼ê,¿(Ü(2.2),(2.3),±9H¨olderØª§Œ
J
1
= af(a)
1
0
(1−t)f(m(1−t)
a
m
+tb)dt
≤af(a)
1
0
(1−t)
p
dt
1
p
1
0
[f(m(1−t)
a
m
+tb)]
q
dt
1
q
≤af(a)
1
0
(1−t)
p
dt
1
p
1
0
[f(
a
m
)]
qm(1−t
α
)
[f(b)]
qt
α
dt
1
q
=
1
p+1
1
p
af(a)[f(
a
m
)]
m
1
0
[f(b)]
[f(
a
m
)]
m
qt
α
dt
1
q
.(2.14)
DOI:10.12677/pm.2021.11101498nØêÆ
‘
Ú
J
2
= bf(b)
1
0
tf(m(1−t)
a
m
+tb)dt
≤bf(b)
1
0
t
p
dt
1
p
1
0
[f(m(1−t)
a
m
+tb)]
q
dt
1
q
≤bf(b)
1
0
t
p
dt
1
p
1
0
[f(
a
m
)]
qm(1−t
α
)
[f(b)]
qt
α
dt
1
q
=
1
p+1
1
p
bf(b)[f(
a
m
)]
m
1
0
[f(b)]
[f(
a
m
)]
m
qt
α
dt
1
q
.(2.15)
(1)e
[f(b)]
[f(
a
m
)]
m
≤1,|^(2.8),(2.14)Ú(2.15)Œ
J
1
≤
1
p+1
1
p
af(a)[f(
a
m
)]
m
1
0
[f(b)]
[f(
a
m
)]
m
qtα
dt
1
q
=
1
p+1
1
p
af(a)[f(
a
m
)]
m
h
f(b)
[f(a)]
m
qα
1
q
(2.16)
J
2
≤
1
p+1
1
p
bf(b)[f(
a
m
)]
m
1
0
[f(b)]
[f(
a
m
)]
m
qtα
dt
1
q
=
1
p+1
1
p
bf(b)[f(
a
m
)]
m
g
f(b)
[f(
a
m
)]
m
qα
1
q
.(2.17)
(2)e
[f(b)]
[f(
a
m
)]
m
≥1,2d(2.8),(2.14)Ú(2.15)Œ
J
1
≤
1
p+1
1
p
af(a)[f(
a
m
)]
m
1
0
[f(b)]
[f(
a
m
)]
m
qt
α
dt
1
q
=
1
p+1
1
p
af(a)[f(
a
m
)]
m
h
f(b)
[f(a)]
m
q
α
1
q
(2.18)
J
2
≤
1
p+1
1
p
bf(b)[f(
a
m
)]
m
1
0
[f(b)]
[f(
a
m
)]
m
qt
α
dt
1
q
=
1
p+1
1
p
bf(b)[f(
a
m
)]
m
g
f(b)
[f(
a
m
)]
m
q
α
1
q
.(2.19)
d(2.16)Ú(2.19)Œ(2.13).
DOI:10.12677/pm.2021.11101499nØêÆ
‘
½n2.3¼êf: I⊂R
+
→R
++
´(α,m)éêà¼ê.ea,b∈I…a<b,Kk
1
b−a
b
a
f(x)f(a+b−x)dx
≤
1
2
[f(
a
m
)]
2m
p(u
2α
)+[f(
b
m
)]
2m
p(v
2
α
),u≤1,v≥1;
1
2
[f(
a
m
)]
2m
p(u
2α
)+[f(
b
m
)]
2m
p(v
2α
),u≤1,v≤1;
1
2
[f(
a
m
)]
2m
p(u
2
α
)+[f(
b
m
)]
2m
p(v
2α
),u≥1,v≤1;
1
2
[f(
a
m
)]
2m
p(u
2
α
)+[f(
b
m
)]
2m
p(v
2
α
),u≥1,v≥1.
,(2.20)
Ù¥u=
f(b)
f(
a
m
)
,v=
f(a)
f(
b
m
)
,
p(β) =
1
0
β
t
dt=
1,β= 1;
β−1
lnβ
,β6= 1.
y²:Ï•f´(α,m)éêà¼ê,Ké?¿a,b∈I,t∈[0,1],(α,m) ∈(0,1]×(0,1],k
f(m(1−t)a+tb) ≤[f(a)]
m(1−t
α
)
[f(b)]
t
α
,(2.21)
f((ta+m(1−t)b) ≤[f(a)]
t
α
[f(b)]
m(1−t
α
)
,(2.22)
x=(1 −t)a+ tb=m(1 −t)
a
m
+ tb(t∈[0,1],m∈(0,1]),¿A^ÄØª
√
uv≤
u
2
+v
2
2
(u,v>0),Œ
1
b−a
b
a
f(x)f(a+b−x)dx
=
1
b−a
1
0
f((1−t)a+tb)f((ta+(1−t)b)(b−a)dt
≤
1
2
1
0
{[f((1−t)a+tb)]
2
+[f((ta+(1−t)b)]
2
}dt.(2.23)
ò(2.21)Ú(2.22)“\(2.23),Œ
1
b−a
b
a
f(x)f(a+b−x)dx
≤
1
2
1
0
{[f(
a
m
)]
m(1−t
α
)
[f(b)]
t
α
}
2
dt+
1
0
{[f(a)]
t
α
[f(
b
m
)]
m(1−t
α
)
}
2
dt
=
1
2
[f(
a
m
)]
2m
1
0
f(b)
f(
a
m
)
2t
α
dt+[f(
b
m
)]
2m
1
0
f(a)
f(
b
m
)
2t
α
dt.(2.24)
DOI:10.12677/pm.2021.111014100nØêÆ
‘
(1)e
f(b)
f(
a
m
)
≤1,
f(a)
f(
b
m
)
≥1,d(2.8)Ú(2.24)Œ
1
b−a
b
a
f(x)f(a+b−x)dx
≤
1
2
[f(
a
m
)]
2m
1
0
f(b)
f(
a
m
)
2tα
dt+[f(
b
m
)]
2m
1
0
f(a)
f(
b
m
)
2t
α
dt
=
1
2
[f(
a
m
)]
2m
p
f(b)
f(
a
m
)
2α
+[f(
b
m
)]
2m
p
f(a)
f(
b
m
)
2
α
.(2.25)
(2)e
f(b)
f(
a
m
)
≤1,
f(a)
f(
b
m
)
≤1,K
1
b−a
b
a
f(x)f(a+b−x)dx
≤
1
2
[f(
a
m
)]
2m
1
0
f(b)
f(
a
m
)
2tα
dt+[f(
b
m
)]
2m
1
0
f(a)
f(
b
m
)
2tα
dt
=
1
2
[f(
a
m
)]
2m
p
f(b)
f(
a
m
)
2α
+[f(
b
m
)]
2m
p
f(a)
f(
b
m
)
2α
.(2.26)
(3)e
f(b)
f(
a
m
)
≥1,
f(a)
f(
b
m
)
≤1,K
1
b−a
b
a
f(x)f(a+b−x)dx
≤
1
2
[f(
a
m
)]
2m
1
0
f(b)
f(
a
m
)
2t
α
dt+[f(
b
m
)]
2m
1
0
f(a)
f(
b
m
)
2tα
dt
=
1
2
[f(
a
m
)]
2m
p
f(b)
f(
a
m
)
2
α
+[f(
b
m
)]
2m
p
f(a)
f(
b
m
)
2α
.(2.27)
(4)e
f(b)
f(
a
m
)
≥1,
f(a)
f(
b
m
)
≥1,K
1
b−a
b
a
f(x)f(a+b−x)dx
≤
1
2
[f(
a
m
)]
2m
1
0
f(b)
f(
a
m
)
2t
α
dt+[f(
b
m
)]
2m
1
0
f(a)
f(
b
m
)
2t
α
dt
=
1
2
[f(
a
m
)]
2m
p
f(b)
f(
a
m
)
2
α
+[f(
b
m
)]
2m
p
f(a)
f(
b
m
)
2
α
.(2.28)
d(2.25)Ú(2.28)=Œ(2.20)¤á.
DOI:10.12677/pm.2021.111014101nØêÆ
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