scholarly journals Best Possible Inequalities between Generalized Logarithmic Mean and Classical Means

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Yu-Ming Chu ◽  
Bo-Yong Long
Keyword(s):  
Logarithmic Mean ◽  
Double Inequality ◽  
Classical Means ◽  
Harmonic Means

We answer the question: forα,β,γ∈(0,1)withα+β+γ=1, what are the greatest valuepand the least valueq, such that the double inequalityLp(a,b)<Aα(a,b)Gβ(a,b)Hγ(a,b)<Lq(a,b)holds for alla,b>0witha≠b? HereLp(a,b),A(a,b),G(a,b), andH(a,b)denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbersaandb, respectively.

scholarly journals Sharp Bounds by the Generalized Logarithmic Mean for the Geometric Weighted Mean of the Geometric and Harmonic Means

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Wei-Mao Qian ◽  
Bo-Yong Long
Keyword(s):  
Logarithmic Mean ◽  
Weighted Mean ◽  
Sharp Bounds ◽  
Harmonic Means

We present sharp upper and lower generalized logarithmic mean bounds for the geometric weighted mean of the geometric and harmonic means.

scholarly journals Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Wei-Mao Qian ◽  
Yu-Ming Chu
Keyword(s):  
Unique Solution ◽  
Trigonometric Functions ◽  
Logarithmic Mean ◽  
Double Inequality ◽  
Logarithmic Means ◽  
Inverse Trigonometric Functions

We prove that the double inequalityLp(a,b)<U(a,b)<Lq(a,b)holds for alla,b>0witha≠bif and only ifp≤p0andq≥2and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, wherep0=0.5451⋯is the unique solution of the equation(p+1)1/p=2π/2on the interval(0,∞),U(a,b)=(a-b)/[2arctan⁡((a-b)/2ab)], andLp(a,b)=[(ap+1-bp+1)/((p+1)(a-b))]1/p  (p≠-1,0),L-1(a,b)=(a-b)/(log⁡a-log⁡b)andL0(a,b)=(aa/bb)1/(a-b)/eare the Yang, andpth generalized logarithmic means ofaandb, respectively.

scholarly journals Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Yu-Ming Chu ◽  
Li-Min Wu ◽  
Ying-Qing Song
Keyword(s):  
Harmonic Mean ◽  
Power Mean ◽  
Double Inequality ◽  
The One ◽  
Harmonic Means

We present the best possible parametersα=α(r)andβ=β(r)such that the double inequalityMα(a,b)<Hr(a,b)<Mβ(a,b)holds for allr∈(0, 1/2)anda, b>0witha≠b, whereMp(a, b)=[(ap+bp)/2]1/p  (p≠0)andM0(a, b)=abandHr(a, b)=2[ra+(1-r)b][rb+(1-r)a]/(a+b)are the power and one-parameter harmonic means ofaandb, respectively.

scholarly journals Note on weighted Carleman-type inequality

2005 ◽  
Vol 2005 (3) ◽  
pp. 475-481 ◽  
Author(s):  
Chao-Ping Chen ◽  
Wing-Sum Cheung ◽  
Feng Qi
Keyword(s):  
Type Inequality ◽  
Arithmetic Mean ◽  
Logarithmic Mean ◽  
Double Inequality ◽  
Carleman Type

A double inequality involving the constanteis proved by using an inequality between the logarithmic mean and arithmetic mean. As an application, we generalize the weighted Carleman-type inequality.

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?

Determination of the Probability Distribution of the Mean Sound Level

2010 ◽  
Vol 35 (4) ◽  
pp. 543-550 ◽  
Author(s):  
Wojciech Batko ◽  
Bartosz Przysucha
Keyword(s):  
Probability Distribution ◽  
Probability Distribution Function ◽  
Mean Value ◽  
Sound Level ◽  
Function Form ◽  
Noise Levels ◽  
Logarithmic Mean ◽  
The Mean ◽  
Estimation Of Uncertainty

AbstractAssessment of several noise indicators are determined by the logarithmic mean <img src="/fulltext-image.asp?format=htmlnonpaginated&src=P42524002G141TV8_html\05_paper.gif" alt=""/>, from the sum of independent random resultsL1;L2; : : : ;Lnof the sound level, being under testing. The estimation of uncertainty of such averaging requires knowledge of probability distribution of the function form of their calculations. The developed solution, leading to the recurrent determination of the probability distribution function for the estimation of the mean value of noise levels and its variance, is shown in this paper.

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