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 Global Bounds for the Generalized Jensen Functional with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2105
Author(s):  
Slavko Simić ◽  
Bandar Bin-Mohsin
Keyword(s):  
Power Mean ◽  
Jensen Functional ◽  
Exact Bounds ◽  
Harmonic Means ◽  
Stolarsky Means ◽  
Generalized Arithmetic

In this article we give sharp global bounds for the generalized Jensen functional Jn(g,h;p,x). In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means.

scholarly journals Considerations for performance metrics of metagenomic next generation sequencing analyses

2020 ◽  
Author(s):  
Jason G. Kralj ◽  
Stephanie L. Servetas ◽  
Samuel P. Forry ◽  
Scott A. Jackson
Keyword(s):  
Performance Metrics ◽  
Limit Of Detection ◽  
Clinical Performance ◽  
Ground Truth ◽  
Negative Control ◽  
Fitness For Purpose ◽  
Performance Metric ◽  
The One ◽  
Sensitivity Specificity ◽  
Harmonic Means

AbstractEvaluating the performance of metagenomics analyses has proven a challenge, due in part to limited ground-truth standards, broad application space, and numerous evaluation methods and metrics. Application of traditional clinical performance metrics (i.e. sensitivity, specificity, etc.) using taxonomic classifiers do not fit the “one-bug-one-test” paradigm. Ultimately, users need methods that evaluate fitness-for-purpose and identify their analyses’ strengths and weaknesses. Within a defined cohort, reporting performance metrics by taxon, rather than by sample, will clarify this evaluation. An estimated limit of detection, positive and negative control samples, and true positive and negative true results are necessary criteria for all investigated taxa. Use of summary metrics should be restricted to comparing results of similar cohorts and data, and should employ harmonic means and continuous products for each performance metric rather than arithmetic mean. Such consideration will ensure meaningful comparisons and evaluation of fitness-for-purpose.

Remarks on the Neglected Mean

1973 ◽  
Vol 66 (3) ◽  
pp. 253-255
Author(s):  
Joseph L. Ercolano
Keyword(s):  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Real Numbers ◽  
Positive Real ◽  
Golden Proportion ◽  
The One

The harmonic mean of two positive, real numbers was known to early Greek mathematicians. In fact, it is alleged that “Pythagoras learned in Mesopotamia of three means—the arithmetic, the geometric, and the subcontrary (later called the harmonic)—and of the ‘golden proportion’ relating two of these: the first of two numbers is to their arithmetic mean as their harmonic mean is to the second of the numbers” (Boyer 1968). Archytas, a disciple of Pythagoras (whose most important contribution to mathematics may very well have been his intervention with Dionysius to save the life of his friend, Plato), wrote on the application of these three means to music, and is possibly the one who is responsible for renaming the suhcontrary mean the harmonic mean (Boyer 1968).

scholarly journals Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Yu-Ming Chu ◽  
Ye-Fang Qiu ◽  
Miao-Kun Wang
Keyword(s):  
Power Mean ◽  
Geometric Means ◽  
Double Inequality

We answer the question: forα∈(0,1), what are the greatest valuepand the least valueqsuch that the double inequalityMp(a,b)<Pα(a,b)G1−α(a,b)<Mq(a,b)holds for alla,b>0witha≠b. Here,Mp(a,b),P(a,b), andG(a,b)denote the power of orderp, Seiffert, and geometric means of two positive numbersaandb, respectively.

scholarly journals A Best Possible Double Inequality for Power Mean

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yong-Min Li ◽  
Bo-Yong Long ◽  
Yu-Ming Chu
Keyword(s):  
Power Mean ◽  
Double Inequality

We answer the question: for anyp,q∈ℝwithp≠qandp≠-q, what are the greatest valueλ=λ(p,q)and the least valueμ=μ(p,q), such that the double inequalityMλ(a,b)<Mp(a,b)Mq(a,b)<Mμ(a,b)holds for alla,b>0witha≠b? WhereMp(a,b)is thepth power mean of two positive numbersaandb.

scholarly journals Considerations for Performance Metrics of Metagenomic Next Generation Sequencing Analyses

2021 ◽  
Author(s):  
Jason Kralj ◽  
Stephanie L. Servetas ◽  
Samuel P. Forry ◽  
Scott A. Jackson
Keyword(s):  
Performance Metrics ◽  
Clinical Performance ◽  
Ground Truth ◽  
Negative Control ◽  
Fitness For Purpose ◽  
Performance Metric ◽  
The One ◽  
Sensitivity Specificity ◽  
Harmonic Means ◽  
Generation Sequencing

Abstract Background: Evaluating the performance of metagenomics analyses has proven a challenge, due in part to limited ground-truth standards, broad application space, and numerous evaluation methods and metrics. Application of traditional clinical performance metrics (i.e. sensitivity, specificity, etc.) using taxonomic classifiers do not fit the “one-bug-one-test” paradigm. Ultimately, users need methods that evaluate fitness-for-purpose and identify their analyses’ strengths and weaknesses. Within a defined cohort, reporting performance metrics by taxon, rather than by sample, will clarify this evaluation.Results: For a complete assessment, estimated limits of detection, positive and negative control samples, and true positive and negative true results are necessary criteria for all investigated taxa. Use of summary metrics should be restricted to comparing results of similar, or ideally the same, cohorts and data, and should employ harmonic means and continuous products for each performance metric rather than arithmetic mean. Conclusions: Organism-centric analysis and reporting will enable clear performance assessment and meaningful comparisons between methods in evaluating fitness for purpose of metagenomic analyses with their intended applications.

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 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 two-parameter geometric and arithmetic mean bounds for the Sándor–Yang mean

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wei-Mao Qian ◽  
Yue-Ying Yang ◽  
Hong-Wei Zhang ◽  
Yu-Ming Chu
Keyword(s):  
Arithmetic Mean ◽  
Sharp Bounds ◽  
Tangent Function ◽  
The One ◽  
Harmonic Means ◽  
Two Parameter

Abstract In the article, we provide the sharp bounds for the Sándor–Yang mean in terms of certain families of the two-parameter geometric and arithmetic mean and the one-parameter geometric and harmonic means. As applications, we present new bounds for a certain Yang mean and the inverse tangent function.

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