关于Neuman-Sándor平均的一些特殊组合不等式

徐仁旭¹
机构：浙江建设职业技术学院人文与信息系，浙江杭州 311231
邮箱：55444765@qq.com.
作者简介：徐仁旭（1980—），https://orcid.org/0000-0003-0383-7917，男，硕士，讲师，主要从事解析不等式研究，E-mail：55444765@qq.com.
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徐会作²
机构：温州广播电视大学教师教学发展中心，浙江;温州 325000
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钱伟茂³
机构：湖州广播电视大学远程教育学院，浙江湖州 313000
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1. 浙江建设职业技术学院人文与信息系，浙江杭州 311231； 2. 温州广播电视大学教师教学发展中心，浙江;温州 325000； 3. 湖州广播电视大学远程教育学院，浙江湖州 313000

中图分类号: O174.6

DOI:10.3785/j.issn.1008⁃9497.2019.03.005

On some special combination inequalities for Neuman-Sándor mean

Renxu EN_AUTHOR:XU¹
Affiliation：Department of Humanity and Information,Zhejiang College of Construction,Hangzhou 311231,China
Email：55444765@qq.com.
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Huizuo XU²
Affiliation：Teachers’ Teaching Development Center, Wenzhou Broadcast and TV University, Wenzhou 325000, Zhejiang Province,China
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Weimao QIAN³
Affiliation：School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, Zhejiang Province,China
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1. Department of Humanity and Information,Zhejiang College of Construction,Hangzhou 311231,China； 2. Teachers’ Teaching Development Center, Wenzhou Broadcast and TV University, Wenzhou 325000, Zhejiang Province,China； 3. School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, Zhejiang Province,China

CLC: O174.6

DOI:10.3785/j.issn.1008⁃9497.2019.03.005

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摘要

研究了Neuman-Sándor平均NS(a,b)关于调和平均H(a,b)、算术平均A(a,b)、二次平均Q(a,b)若干特殊组合的序关系，给出最佳参数α1,α2,α3,α4,β1,β2,β3,β4∈0,1，使得下列双向不等式：α1Q2(a,b)+1-α1A2(a,b)<NS(a,b)<β1Q2(a,b)+1-β1A2(a,b),α2Q(a,b)+1-α2A(a,b)A(a,b)<NS(a,b)<β2Q(a,b)+1-β2A(a,b)A(a,b),α3Q2(a,b)+1-α3H2(a,b)<NS(a,b)<β3Q2(a,b)+1-β3H2(a,b),α4Q(a,b)+1-α4H(a,b)A(a,b)<NS(a,b)<β4Q(a,b)+1-β4H(a,b)A(a,b)对所有不同的正实数a和b均成立。

Abstract

In the article, we deal with the order relation on the Neuman-Sándor mean NS(a,b) with respect to the special combinations of the harmonic mean H(a,b),arithmetic mean A(a,b) and quadratic mean Q(a,b) ，and present the best possible parameters α1,α2,α3,α4,β1,β2,β3,β4∈0,1 such that the double inequalitiesα1Q2(a,b)+1-α1A2(a,b)<NS(a,b)<β1Q2(a,b)+1-β1A2(a,b),α2Q(a,b)+1-α2A(a,b)A(a,b)<NS(a,b)<β2Q(a,b)+1-β2A(a,b)A(a,b),α3Q2(a,b)+1-α3H2(a,b)<NS(a,b)<β3Q2(a,b)+1-β3H2(a,b),α4Q(a,b)+1-α4H(a,b)A(a,b)<NS(a,b)<β4Q(a,b)+1-β4H(a,b)A(a,b)hold for all a,b>0 with a≠b, where H(a,b),A(a,b),Q(a,b) and NS(a,b) are the harmonic, arithmetic, quadratic and Neuman-Sándor means of a and b, respectively.

关键词

Neuman-Sándor平均；调和平均；算术平均；二次平均

Keywords

Neuman-Sàndor mean; harmonic mean; arithmetic mean; quadratic mean

0　引言
设p∈R,a,b>0,则Neuman-Sándor平均NS(a,b)^[1],调和平均H(a,b),几何平均G(a,b)，算术平均A(a,b), 二次平均Q(a,b)和p阶幂平均Mp(a,b)可分别定义为
NS(a,b)=a-b2sinh-1a-b/a+b ,
(1)
H(a,b)=2aba+b ,G(a,b)=ab,
A(a,b)=a+b2 ,Q(a,b)=a2+b22,
(2)
以及
Mp(a,b)=ap+bp21/p,　p≠0,ab,p=0
其中sinh-1x=logx+x2+1为反双曲正弦函数。显然Mp(a,b)关于a,b是对称和一阶齐次的。众所周知，不等式:
H(a,b)=M-1(a,b)<G(a,b)=M0(a,b)<A(a,b)=M1(a,b)<NS(a,b)<Q(a,b)=M2(a,b)
对所有不同的正实数a和b成立。
近年来，Neuman-Sándor平均NS(a,b)和其他二元平均的比较得到了深入研究，涉及的重要不等式可参见文献[2⁃28,29,30]。
NEUMAN等 ^[1] 证明了不等式
A(a,b)<NS(a,b)<A(a,b)log1+2,
NS(a,b)<2A(a,b)+Q(a,b)3
对所有不同的正实数a和b均成立。
2012年，YANG^[31] 建立了不等式
Mlog2/loglog3+22(a,b)<NS(a,b)<M4/3(a,b)
对所有不同的正实数a和b成立，其中
log2/loglog3+22和4/3是最佳参数。
文献[32,33]证明了双向不等式
α1Q(a,b)+1-α1A(a,b)<NS(a,b)<β1Q(a,b)+1-β1A(a,b)
α2H(a,b)+1-α2Q(a,b)<NS(a,b)<β2H(a,b)+1-β2Q(a,b)
对所有不同的正实数a和b成立当且仅当
α1≤1-log1+2/2-1log1+2=0.3249⋅⋅⋅,　β1≥1/3,　α2≥2/9,β2≤1-2log1+2=0.1977⋅⋅⋅
本文的主要结果是给出以下几个最佳参数α1,α2,α3,α4,β1,β2,β3,β4∈0,1使得下列双向不等式：
α1Q2(a,b)+1-α1A2(a,b)<NS(a,b)<β1Q2(a,b)+1-β1A2(a,b)
α2Q(a,b)+1-α2A(a,b)A(a,b)<NS(a,b)<β2Q(a,b)+1-β2A(a,b)A(a,b),
α3Q2(a,b)+1-α3H2(a,b)<NS(a,b)<β3Q2(a,b)+1-β3H2(a,b),
α4Q(a,b)+1-α4H(a,b)A(a,b)<NS(a,b)<β4Q(a,b)+1-β4H(a,b)A(a,b)
对所有不同的正实数a和b均成立。

1　引理

为了证明主要结果，需引入以下相关引理。

引理1^[34,35]　设f,g:a,b→R，函数在a,b连续，在a,b可导，且g'x≠0。若f'x/g'x在a,b单调递增（递减），则函数fx-fa/gx-ga和fx-fb/gx-gb也在a,b单调递增（递减）。若f'x/g'x是严格单调的，则上述结论的单调性也是严格的。

引理2^[36,37]　设实幂级数Ax=∑n=0∞anxn和Bx=∑n=0∞bnxn在-r,rr>0上收敛，且an,bn>0对所有的n成立。若序列an/bn严格单调递增（递减），则函数x↦Ax/Bx在0,r严格单调递增（递减）。

引理3　函数fx=sinh2(x)-x2x2sinh2(x)在区间0,log1+2严格单调递减，且值域为1/log21+2-1,1/3。

证明　设f1x=sinh2(x)-x2,

f2x=x2sinh2(x)

经简单计算可得

fx=f1xf2x=f1x-f10+f2x-f20 ,

(3)

f'1xf'2x=sinh2x-2xx2sinh2x+xcosh2x-x=∑n=0∞22n+12n+1!x2n+1-2xx2∑n=0∞22n+12n+1!x2n+1+x∑n=0∞22n2n!x2n-x=∑n=0∞22n+32n+3!x2n∑n=0∞n+222n+22n+2!x2n=∑n=0∞anx2n∑n=0∞bnx2n

(4)

其中，

an=22n+3/2n+3!>0

bn=n+222n+2/2n+2!>0

(5)

且

an+1bn+1-anbn=-24n+9n+2n+32n+32n+5<0

(6)

对所有n≥0成立。

由引理2和等式(4)及不等式(6)得f'1x/f'2x在0,log1+2严格单调递减。

注意到

f0+=a0b0=13 ,

flog1+2=1log21+2-1 。

(7)

由引理1和等式(3)、(7)及f'1x/f'2x的单调性，引理3得证。

引理4　函数gx=sinh2x-x2x2coshx-x2

在区间0,log1+2严格单调递增，值域为2/3,1+21/log21+2-1。

证明　设g1x=sinh2(x)-x2,

g2x=x2coshx-x2

经简单计算可得

gx=g1xg2x=g1x-g10+g2x-g20,g'1xg'2x=sinh2x-2xx2sinhx+2xcoshx-2x。

(8)

利用幂级数展开可得

g'1xg'2x=∑n=0∞22n+12n+1!x2n+1-2xx2∑n=0∞x2n+12n+1!+2x∑n=0∞x2n2n!-2x=∑n=1∞22n+12n+1!x2n+1x2∑n=0∞x2n+12n+1!+2x∑n=1∞x2n2n!=∑n=0∞22n+32n+3!x2n+3∑n=0∞2n+22n+2!x2n+3=∑n=0∞22n+32n+3!x2n∑n=0∞2n+22n+2!x2n=∑n=0∞cnx2n∑n=0∞dnx2n,

(9)

其中，

cn=22n+32n+3!>0dn=2n+22n+2!>0

(10)

且

cn+1dn+1-cndn=6n2+17n+9n+2n+32n+32n+5>0

(11)

对所有n≥0成立。

由引理2和等式(9)及不等式(10)、(11)得g'1x/g'2x在0,log1+2严格单调递增。

注意到

g0+=c0d0=23,glog1+2-=1+21/log21+2-1

(12)

由引理1和等式(8)、(12)及g'1x/g'2x的单调性，引理4得证。

引理5　设p∈R，

hx=1-p2x8+24p2-7p+3x6+3p2+5p-8x4+22p-3x2+7-9p

(13)

则以下结论成立:

(i) 若p=7/9, 则当x∈0,1时hx<0。

(ii) 若p=1/2log21+2=0.6436⋅⋅⋅, 则存在λ0∈0,1使得当x∈0,λ0时hx>0；当x∈λ0,1时hx<0。

证明　(i) 当p=7/9时,等式(13)可化为

hx=281x22x6-2x4-93x-117<-281x293x+117<0 ,

(14)

对所有x∈0,1成立。即结论(i)得证。

(ii) 当p=1/2log21+2=0.6436⋯时,经数值计算得

4p2-7p+3=0.1515⋯>0 ,

(15)

3p2+5p-8=-3.5388⋯<0,

(16)

2p-3=-1.7126⋯<0 ,

(17)

17p-18=-7.0579⋯<0 ,

(18)

h0=7-9p=1.2071⋯>0 ,

(19)

h1=4p3p-4=-5.3269⋯<0 。

(20)

h'x=4x21-p2x6+34p2-7p+3x4+

4x3p2+5p-8x2+2p-3 。

(21)

由不等式(15) ~(18)和等式(21)可得

h'x<4x21-p2x2+34p2-7p+3x2+4x3p2+5p-8x2+2p-3x2=4p17p-18x3<0

(22)

对所有x∈0,1成立。

因此,由不等式(19)、(20)和(22)，结论(ii)得证。

引理 6　设p∈R,

kx=2p1-px3+9p2-16p+8x2+12p1-px+5p2-16p+8

(23)

则以下结论成立:

(i) 若p=8/9, 则当x∈1,2时,kx>0;

(ii)若p=2/2log21+2=0.9102⋯时,

则存在μ0∈1,2，当x∈1,μ0时，kx<0；而当x∈μ0,2时，kx>0。

证明　(i) 当p=8/9时，有

kx=881x-12x2+11x+23>0 ,

(24)

对所有x∈1,2成立。即结论(i)得证。

(ii) 当p=2/2log21+2=0.9102⋯时,经数值计算得

9p2-16p+8=0.8929⋯>0 。

(25)

k1=28-9p=-0.3846⋯<0 ,

(26)

k2=162p1-p+23p2-48p+24=1.2130⋯>0

(27)

由等式(23)和不等式(25)得

k'x=6p1-px2+29p2-16p+8x+12p1-p>0

(28)

对所有x∈1,2成立。

因此, 由不等式(26)~(28)，结论(ii)得证。

2　主要结果及证明
定理1　双向不等式
α1Q2(a,b)+1-α1A2(a,b)<NS(a,b)<
β1Q2(a,b)+1-β1A2(a,b)
(29)
对所有不同的正实数a和b成立当且仅当
α1≤1/log21+2-1=0.2873⋯,β1≥1/3
证明　不等式(29)可改写成
α1<NS2(a,b)-A2(a,b)Q2(a,b)-A2(a,b)<β1
(30)
由A(a,b),Q(a,b)和NS(a,b)是对称和一阶齐次的，不失一般性，假设a>b>0，设
v=a-b/a+b∈0,1 ，
x=sinh-1v∈0,log1+2 。
则不等式(30)可化为
α1<sinh2(x)-x2x2sinh2(x)<β1
(31)
由不等式(31)及引理3得，不等式(29)对所有不同的正实数a和b成立当且仅当
α1≤1/log21+2-1=0.2873⋯,β1≥1/3
定理2　双向不等式
α2Q(a,b)+1-α2A(a,b)A(a,b)<NS(a,b)<β2Q(a,b)+1-β2A(a,b)A(a,b),
(32)
对所有不同的正实数a和b成立当且仅当α2≤2/3，β2≥1+21/log21+2-1=0.6936⋯。
证明　不等式(32)可改写成
α2<NS2(a,b)-A2(a,b)A(a,b)Q(a,b)-A(a,b)<β2
(33)
不失一般性，假设a>b>0，v=a-b/a+b∈0,1，x=sinh-1v∈0,log1+2。则不等式(33)可化为
α2<sinh2(x)-x2x2cosh(x)-1<β2
(34)
由不等式(34)及引理4，可得不等式(32)对所有不同的正实数a和b成立当且仅当α2≤2/3，β2≥1+21/log21+2-1=0.6936⋯。
定理 3　双向不等式
α3Q2(a,b)+1-α3H2(a,b)<NS(a,b)<β3Q2(a,b)+1-β3H2(a,b),
对所有不同的正实数a和b成立当且仅当
α3≤1/2log21+2=0.6436⋯
且β3≥7/9。
证明　因H(a,b),Q(a,b)和NS(a,b)是对称和一阶齐次的，不失一般性，假设a>b>0，x=a-b/a+b∈0,1且p∈0,1。则根据式(1)和(2)得
NS2(a,b)-H2(a,b)Q2(a,b)-H2(a,b)=x/sinh-1x2-1-x221+x2-1-x22,
(35)
logpQ2(a,b)+1-pH2(a,b)NS(a,b)=12logp1+x2+1-p1-x22-logx+logsinh-1x=∶Hx
(36)
经简单计算得
H0+=0 ,
(37)
H1-=12log2p+loglog1+2 ,
(38)
H'x=p-1x4+1x1-px4+3p-2x2+1sinh-1xH1x,
(39)
其中，
H1x=x1-px4+3p-2x2+11+x2p-1x4+1-sinh-1x,H10=0H11-=2-log1+2>0
(40)
H'1x=-x21+x23/2p-1x4+12hx
(41)
其中hx由等式(13)定义。
以下分2种情形证明。
情形1　p=1/2log21+2。由引理5(ii)和等式(41)可知，存在λ0∈0,1，使得H1x在0,λ0严格单调递减，在λ0,1严格单调递增。
由等式(39)、 (40)及H1x的分段单调性可得，存在λ∈0,1使得Hx在0,λ严格单调递减，在λ,1严格单调递增。
注意到此时等式(38) 变成
H1-=0 。
(42)
由等式(36)、(37)和(42)及Hx的分段单调性可得
NS(a,b)>12log21+2Q2(a,b)+1-12log21+2H2(a,b)12
(43)
对所有不同的正实数a和b成立。
情形2　p=7/9。由引理5(i)和等式(41)可得，H1x在0,1严格单调递增。
由等式(36)、(37)、(39)、(40)及H1x的单调性可得
NS(a,b)<79Q2(a,b)+29H2(a,b)
(44)
对所有不同的正实数a和b成立。
注意到
limx→0+x/sinh-1x2-1-x221+x2-1-x22=79
(45)
limx→1-x/sinh-1x2-1-x221+x2-1-x22=12log21+2
(46)
由等式(35)、(45)、(46)和不等式(43)、(44)，定理3得证。
定理4　双向不等式
α4Q(a,b)+1-α4H(a,b)A(a,b)<NS(a,b)<β4Q(a,b)+1-β4H(a,b)A(a,b)
对所有不同的正实数a和b成立当且仅当α4≤8/9，β4≥2/2log21+2=0.9102⋯。
证明　因H(a,b),A(a,b),Q(a,b)和NS(a,b)是对称和一阶齐次的，不失一般性，假设a>b>0，v=a-b/a+b∈0,1，x=1+v2∈1,2，且p∈0,1，则由等式(1)和(2)得
NS2(a,b)-A(a,b)H(a,b)A(a,b)Q(a,b)-H(a,b)=x2-1/sinh-1x2-12-2-x2x2+x-2,
(47)
logpQ(a,b)+1-pH(a,b)A(a,b)NS(a,b)=12logp-1x2+px+21-p-12logx2-1+logsinh-1x2-1=∶Kx
(48)
经简单计算可得
K1+=0 ,
(49)
K2-=14log2p2+loglog1+2 ,
(50)
K'x=[px2+21-px+p]/2x2-1×p-1x2+px+21-psinh-1x2-1K1x,
(51)
其中，
K1x=2x2-1p-1x2+px+21-ppx2+21-px+p-sinh-1x2-1,　K11=0,
(52)
K12=22p221-p+3p-log1+2
(53)
K'1x=-x2-1px2+21-px+p2kx
(54)
kx由等式(23)定义。
以下分2种情形证明。
情形1　p=8/9。由等式(54)和引理6(i)可知，K1x在1,2严格单调递减。
由等式(48)、(49)、(51)、(52)及K1x的单调性得到
NS(a,b)>89Q(a,b)+19H(a,b)A(a,b)
(55)
对所有不同的正实数a和b成立。
情形 2　p=2/2log21+2。由等式(54)和引理6(ii)得,存在μ0∈1,2使得K1x在1,μ0严格递增，在μ0,2严格递减。
由等式(53) 得到
K12=-0.0187⋯<0
(56)
由等式(51)、(52)和不等式(56)及K1x的分段单调性可知，存在μ∈1,2使得Kx在1,μ单调递增，在μ,2单调递减。
注意到等式(50)变为
K2-=0 。
(57)
因此,由式(48)、(49)和(57)及Kx的分段单调性可得
NS(a,b)<22log21+2Q(a,b)+1-22log21+2H(a,b)A(a,b)12
(58)
对所有不同的正实数a和b成立。注意到
limx→1+x2-1/sinh-1x2-12-2-x2x2+x-2=89
(59)
limx→2-x2-1/sinh-1x2-12-2-x2x2+x-2=22log21+2
(60)
由等式(47)、(59)、(60)和不等式(55)、(58)，定理4得证。
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基本信息

DOI:10.3785/j.issn.1008⁃9497.2019.03.005

中图分类号: O174.6

文献标识码: A

基金信息

国家自然科学基金资助项目（61673169, 61374086, 11371125, 11401191）；浙江广播电视大学科学研究课题（XKT-17Z04，XKT-17G26）；浙江省教育厅2017年度高校访问学者“教师专业发展项目”（FX2017108，FX2017084）.

稿件历史

纸刊出版 : 2019-05-25
收稿日期 : 2018-08-27

徐仁旭

机构：浙江建设职业技术学院人文与信息系，浙江杭州 311231

Affiliation：Department of Humanity and Information,Zhejiang College of Construction,Hangzhou 311231,China

邮箱：55444765@qq.com.

作者简介：徐仁旭（1980—），https://orcid.org/0000-0003-0383-7917，男，硕士，讲师，主要从事解析不等式研究，E-mail：55444765@qq.com.

徐会作

机构：温州广播电视大学教师教学发展中心，浙江;温州 325000

Affiliation：Teachers’ Teaching Development Center, Wenzhou Broadcast and TV University, Wenzhou 325000, Zhejiang Province,China

钱伟茂

机构：湖州广播电视大学远程教育学院，浙江湖州 313000

Affiliation：School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, Zhejiang Province,China



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参考文献（References）

1
NEUMAN E, SÁNDOR J. On the Schwab-Borchardt mean[J]. Mathematica Pannonica, 2003, 14(2):253-266.
2
NEUMAN E, SÁNDOR J. On the Schwab-Borchardt meanⅡ[J]. Mathematica Pannonica, 2006, 17(1): 49-59.
3
LI Y M, LONG B Y, CHU Y M. Sharp bounds for the Neuman-Sándor mean in terms of generalized logarithmic mean[J]. Journal of Mathematical Inequalities, 2012, 6(4): 567-577.doi:10.7153/jmi-06-54
4
CHU Y M, LONG B Y, GONG W M, et al. Sharp bounds for Seiffert and Neuman- Sándor means in terms of generalized logarithmic means[J]. Journal of Inequalities and Applications, 2013, Article 10.doi:10.1186/1029-242x-2013-10
5
CHU Y M , LONG B Y. Bounds of the Neuman-Sándor mean using power and identic means[J]. Abstract and Applied Analysis, 2013, Article ID 832591.doi:10.1155/2013/832591
6
CHU Y M, WANG M K,LIU B Y. Sharp inequalities for the Neuman-Sándor mean in terms of arithmetic and contra-harmonic means[J]. Revue D’Analyse Numérique Et De Théorie De L’ Approximation, 2013, 42(2):115-120.
7
QIAN W M, CHU Y M. On certain inequalities for Neuman-Sándor mean[J].Abstract and Applied Analysis, 2013, Article ID 790783.doi:10.1155/2013/790783
8
ZHANG F, CHU Y M ,QIAN W M. Bounds for the arithmetic mean in terms of the Neuman-Sándor and other bivariate means[J]. Journal of Applied Mathematics, 2013, Article ID 582504.doi:10.1155/2013/582504
9
HE Z Y, QIAN W M, JIANG Y L, et al. Bounds for the combinations of Neuman-Sándor, arithmetic, and second Seiffert means in terms of Contraharmonic mean[J]. Abstract and Applied Analysis, 2013, Article ID 903982.doi:10.1155/2013/903982
10
XIA W F, CHU Y M. Optimal inequalities between Neuman-Sándor, centroidal and harmonic means[J]. Journal of Mathematical Inequalities, 2013, 7(4): 593-600.doi:10.7153/jmi-07-56
11
YANG Z H. Estimates for Neuman-Sándor mean by power means and their relative errors[J]. Journal of Mathematical Inequalities, 2013, 7(4): 711-726.doi:10.7153/jmi-07-65
12
JIANG W D, QI F. Sharp bounds for the Neuman-Sándor mean in terms of the power and Contraharmonic means[J]. Cogent Mathematics, 2015, 2 (1): 995951.doi:10.1080/23311835.2014.995951
13
HUANG H Y, WANG N , LONG B Y. Optimal bounds for Neuman-Sándor mean in terms of the geometric convex combination of two Seiffert means[J]. Journal of Inequalities and Applications, 2016:14.doi:10.1186/s13660-015-0955-2
14
CHEN J J , LEI J J ,LONG B Y. Optimal bounds for Neuman-Sándor mean in terms of the convex combination of the logarithmic and the second Seiffert means[J]. Journal of Inequalities and Applications, 2017(1):251.doi:10.1186/s13660-017-1516-7
15
LEI J J ,CHEN J J , LONG B Y. Optimal bounds for the first Seiffert mean in terms of the convex combination of the logarithmic and Neuman-Sándor mean[J]. Journal of Mathematical Inequalities, 2018, 12(2): 365-377.doi:10.7153/jmi-2018-12-27
16
WANG M K, CHU Y M, QIU Y F, et al. An optimal power mean inequality for the complete elliptic integrals[J]. Applied Mathematics Letters, 2011, 24(6):887-890.doi:10.1016/j.aml.2010.12.044
17
WANG M K, QIU S L, CHU Y M, et al. Generalized Hersch-Pfluger distortion Function and complete elliptic integrals[J]. Journal Mathematical Analysis Applysis and Applications, 2012, 385(1): 221-229.doi:10.1016/j.jmaa.2011.06.039
18
ZHAO T H, WANG M K, ZHANG W, et al. Quadratic transformation inequalities for Gaussian hypergeometric function[J]. Journal of Inequalities Applications, 2018: Article 251.doi:10.1186/s13660-018-1848-y
19
WANG M K, QIU S L, CHU Y M. Infinite series formula for Hubner upper bound function with applications to Hersch-Pfluger distortion function[J]. Mathematical Inequalities and Applications, 2018, 21(3): 629-648.doi:10.7153/mia-2018-21-46
20
XU H Z, CHU Y M, QIAN W M. Sharp bounds for Sandor-Yang means in terms of arithmetic and contra-harmonic means[J]. Journal of Inequalities and Applications, 2018, Article 127.doi:10.1186/s13660-018-1719-6
21
WANG M K, LI Y M, CHU Y M. Inequalities and infinite product formula for Ramanujan generalized modular equation function[J]. Ramanujan Journal, 2018, 46(1):189-200.doi:10.1007/s11139-017-9888-3
22
WANG M K, CHU Y M. Landen inequalities for a class of hypergeometric functions with applications[J].Mathematical Inequalities and Applications, 2018, 21(2):521-537.doi:10.7153/mia-2018-21-38
23
YANG Z H, QIAN W M, CHU Y M, et al. On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind[J]. Journal Mathematical Analysis Applications, 2018, 462(2): 1714-1726.doi:10.1016/j.jmaa.2018.03.005
24
QIAN W M, CHU Y M. Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters[J]. Journal Inequalities and Applications, 2017(1): Article 274.doi:10.1186/s13660-017-1550-5
25
YANG Z H,QIAN W M, CHU Y M , et al. On rational bounds for the gamma function[J].Journal of Inequalities and Applications,2017(1): 210.doi:10.1186/s13660-017-1484-y
26
WANG M K, CHU Y M. Refinements of transformation inequalities for zero-balanced hypergeometric functions[J]. Acta Mathematica Scientia, 2017, 37B(3):607-622.doi:10.1016/s0252-9602(17)30026-7
27
CHU Y M , QIU Y F ,WANG M K. Holder mean inequalities for the complete elliptic integrals[J]. Integral Transform Specical Functions, 2012, 23(7):521-527.doi:10.1080/10652469.2011.609482
28
CHU Y M, WANG M K, JIANG Y P , et al. Concavity of the complete elliptic integrals of the second kind with respect to Holder mean[J]. Journal of Mathematical Analysis and Applications, 2012, 395(2): 637-642.doi:10.1016/j.jmaa.2012.05.083
29
CHU Y M ,WANG M K. Optimal Lehmer mean bounds for the Toader mean[J]. Results in Mathematics, 2012, 61(3/4): 223-229.doi:10.1007/s00025-010-0090-9
30
CHU Y M, WANG M K, QIU S L. Optimal combinations bounds of root-square and arithmetic means for Toader mean[J]. Proceedings-Mathematical Sciences, 2012, 122(1): 41-51.doi:10.1007/s12044-012-0062-y
31
YANG Z H. Sharp power means bounds for Neuman-Sándor mean[J]. Mathematics Subject Classification, 2012, arXiv:1208.0895v1.doi:10.1155/2015/172867
32
NEUMAN E. A note on a certain bivariate mean[J]. Journal of Mathematical Inequalities, 2012, 6(4):637-643.doi:10.7153/jmi-06-62
33
ZHAO T H, CHU Y M, LIU B Y. Optimal bounds for Neuman-Sándor mean in terms of the convex Combinations of harmonic, geometric, quadratic, and contraharmonic means[J]. Abstract and Applied Analysis, 2012, Article ID 302635.doi:10.1155/2012/302635
34
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关于Neuman-Sándor平均的一些特殊组合不等式

On some special combination inequalities for Neuman-Sándor mean

摘要

Abstract

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2 主要结果及证明

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参考文献（References）

基本信息

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稿件历史

参考文献（References）

0　引言

1　引理

2　主要结果及证明