scholarly journals Optimal Lower Power Mean Bound for the Convex Combination of Harmonic and Logarithmic Means

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Yu-Ming Chu ◽  
Shan-Shan Wang ◽  
Cheng Zong
Keyword(s):  
Convex Combination ◽  
Power Mean ◽  
Lower Power ◽  
Power Means ◽  
Logarithmic Means

We find the least valueλ∈(0,1)and the greatest valuep=p(α)such thatαH(a,b)+(1−α)L(a,b)>Mp(a,b)forα∈[λ,1)and alla,b>0witha≠b, whereH(a,b),L(a,b), andMp(a,b)are the harmonic, logarithmic, andp-th power means of two positive numbersaandb, respectively.

scholarly journals The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Wei-Feng Xia ◽  
Yu-Ming Chu ◽  
Gen-Di Wang
Keyword(s):  
Convex Combination ◽  
Arithmetic Mean ◽  
Power Mean ◽  
Logarithmic Mean ◽  
Positive Real ◽  
Lower Power ◽  
Double Inequality ◽  
Logarithmic Means

Forp∈ℝ, the power meanMp(a,b)of orderp, logarithmic meanL(a,b), and arithmetic meanA(a,b)of two positive real valuesaandbare defined byMp(a,b)=((ap+bp)/2)1/p, forp≠0andMp(a,b)=ab, forp=0,L(a,b)=(b-a)/(log⁡b-log⁡a), fora≠bandL(a,b)=a, fora=bandA(a,b)=(a+b)/2, respectively. In this paper, we answer the question: forα∈(0,1), what are the greatest valuepand the least valueq, such that the double inequalityMp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b)holds for alla,b>0?

scholarly journals Optimal Inequalities for Power Means

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Yong-Min Li ◽  
Bo-Yong Long ◽  
Yu-Ming Chu ◽  
Wei-Ming Gong
Keyword(s):  
Power Mean ◽  
Power Means ◽  
Optimal Inequalities

We present the best possible power mean bounds for the productMpα(a,b)M-p1-α(a,b)for anyp>0,α∈(0,1), and alla,b>0witha≠b. Here,Mp(a,b)is thepth power mean of two positive numbersaandb.

scholarly journals VARIANTS OF ANDO–HIAI TYPE INEQUALITIES FOR DEFORMED MEANS AND APPLICATIONS

2020 ◽  
pp. 1-18 ◽  
Author(s):  
MOHSEN KIAN ◽  
MOHAMMAD SAL MOSLEHIAN ◽  
YUKI SEO
Keyword(s):  
Hilbert Space ◽  
Norm Inequality ◽  
Power Mean ◽  
Variable Operator ◽  
Linear Map ◽  
Power Means ◽  
Operator Mean ◽  
Positive Linear Map ◽  
Euclidean Mean

Abstract For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.

scholarly journals Sharp Bounds for Power Mean in Terms of Generalized Heronian Mean

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Hongya Gao ◽  
Jianling Guo ◽  
Wanguo Yu
Keyword(s):  
Power Mean ◽  
Sharp Bounds ◽  
Power Means ◽  
Heronian Mean

For1<r<+∞, we find the least valueαand the greatest valueβsuch that the inequalityHα(a,b)<Ar(a,b)<Hβ(a,b)holds for alla,b>0witha≠b. Here,Hω(a,b)andAr(a,b)are the generalized Heronian and the power means of two positive numbersaandb, respectively.

scholarly journals A lower bound for ratio of power means

2004 ◽  
Vol 2004 (1) ◽  
pp. 49-53
Author(s):  
Feng Qi ◽  
Bai-Ni Guo ◽  
Lokenath Debnath
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Positive Real Number ◽  
Positive Sequence ◽  
Power Mean ◽  
Natural Numbers ◽  
Positive Real ◽  
Power Means

Letnandmbe natural numbers. Suppose that{ai}i=1n+mis an increasing, logarithmically convex, and positive sequence. Denote the power meanPn(r)for any given positive real numberrbyPn(r)=((1/n)∑i=1nair)1/r. ThenPn(r)/Pn+m(r)≥an/an+m. The lower bound is the best possible.

scholarly journals Bounds of the Neuman-Sándor Mean Using Power and Identric Means

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yu-Ming Chu ◽  
Bo-Yong Long
Keyword(s):  
Power Mean ◽  
Sharp Bounds ◽  
Lower Power

In this paper we find the best possible lower power mean bounds for the Neuman-Sándor mean and present the sharp bounds for the ratio of the Neuman-Sándor and identric means.

scholarly journals Sharp Power Mean Bounds for Sándor Mean

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Yu-Ming Chu ◽  
Zhen-Hang Yang ◽  
Li-Min Wu
Keyword(s):  
Power Mean ◽  
Double Inequality ◽  
Power Means

We prove that the double inequalityMp(a,b)<X(a,b)<Mq(a,b)holds for alla,b>0witha≠bif and only ifp≤1/3andq≥log 2/(1+log 2)=0.4093…, whereX(a,b)andMr(a,b)are the Sándor andrth power means ofaandb, respectively.

scholarly journals Inequalities between Power Means and Convex Combinations of the Harmonic and Logarithmic Means

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Wei-Mao Qian ◽  
Zhong-Hua Shen
Keyword(s):  
Power Means ◽  
Logarithmic Means

We prove thatαH(a,b)+(1−α)L(a,b)>M(1−4α)/3(a,b)forα∈(0,1)and alla,b>0witha≠bif and only ifα∈[1/4,1)andαH(a,b)+(1−α)L(a,b)<M(1−4α)/3(a,b)if and only ifα∈(0,3345/80−11/16), and the parameter(1−4α)/3is the best possible in either case. Here,H(a,b)=2ab/(a+b),L(a,b)=(a−b)/(log a−log b), andMp(a,b)=((ap+bp)/2)1/p(p≠0)andM0(a,b)=abare the harmonic, logarithmic, andpth power means ofaandb, respectively.

scholarly journals One-Kind Hybrid Power Means of the Two-Term Exponential Sums and Quartic Gauss Sums

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Xiaoxue Li ◽  
Li Chen
Keyword(s):  
Exponential Sums ◽  
Gauss Sums ◽  
Asymptotic Formulas ◽  
Linear Recurrence ◽  
Power Mean ◽  
Hybrid Power ◽  
Recurrence Formulas ◽  
Analytic Methods ◽  
Power Means ◽  
Hybrid Power Mean

The main purpose of this article is using the analytic methods and the properties of the classical Gauss sums to study the calculating problem of the hybrid power mean of the two-term exponential sums and quartic Gauss sums and then prove two interesting linear recurrence formulas. As applications, some asymptotic formulas are obtained.

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