scholarly journals Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhi-Jun Guo ◽  
Yu-Ming Chu ◽  
Ying-Qing Song ◽  
Xiao-Jing Tao
Keyword(s):  
Convex Combination ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Sharp Bounds ◽  
Contraharmonic Mean

We give several sharp bounds for the Neuman meansNAHandNHA(NCAandNAC) in terms of harmonic meanH(contraharmonic meanC) or the geometric convex combination of arithmetic meanAand harmonic meanH(contraharmonic meanCand arithmetic meanA) and present a new chain of inequalities for certain bivariate means.

Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means

2020 ◽  
Vol 70 (5) ◽  
pp. 1097-1112
Author(s):  
Hong-Hu Chu ◽  
Tie-Hong Zhao ◽  
Yu-Ming Chu
Keyword(s):  
Elliptic Integral ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Complete Elliptic Integral ◽  
Quadratic Mean ◽  
Sharp Bounds ◽  
Toader Mean ◽  
The One

AbstractIn the article, we present the best possible parameters α1, β1, α2, β2 ∈ ℝ and α3, β3 ∈ [1/2, 1] such that the double inequalities$$\begin{array}{} \begin{split} \displaystyle \alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b), \\ \alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b), \\ C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b) \end{split} \end{array}$$hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean, $\begin{array}{} \displaystyle Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2} \end{array}$ is the quadratic mean, C(a, b) = (a2 + b2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and $\begin{array}{} T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3} \end{array}$ is the Toader mean of order 3.

A Neutral Model With Fluctuating Population Size and Its Effective Size

Genetics ◽  
2002 ◽  
Vol 161 (1) ◽  
pp. 381-388
Author(s):  
Masaru Iizuka ◽  
Hidenori Tachida ◽  
Hirotsugu Matsuda
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Diffusion Model ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Effective Size ◽  
Neutral Model ◽  
Average Heterozygosity ◽  
Special Case ◽  
Concrete Model

Abstract We consider a diffusion model with neutral alleles whose population size is fluctuating randomly. For this model, the effects of fluctuation of population size on the effective size are investigated. The effective size defined by the equilibrium average heterozygosity is larger than the harmonic mean of population size but smaller than the arithmetic mean of population size. To see explicitly the effects of fluctuation of population size on the effective size, we investigate a special case where population size fluctuates between two distinct states. In some cases, the effective size is very different from the harmonic mean. For this concrete model, we also obtain the stationary distribution of the average heterozygosity. Asymptotic behavior of the effective size is obtained when the population size is large and/or autocorrelation of the fluctuation is weak or strong.

scholarly journals Impact of different centroid means on the accuracy of orthometric height modelling by geometric geoid method

2020 ◽  
Vol 6 (4) ◽  
pp. 124 ◽  
Author(s):  
Oluyori P. Dare ◽  
Eteje S. Okiemute
Keyword(s):  
Root Mean Square ◽  
Arithmetic Mean ◽  
Orthometric Height ◽  
Harmonic Mean ◽  
Geometric Method ◽  
Mean Square ◽  
Geoid Model ◽  
Microsoft Excel ◽  
Orthometric Heights ◽  
The Impact

<p class="abstract"><strong>Background:</strong> Orthometric height, as well as geoid modelling using the geometric method, requires centroid computation. And this can be obtained using various models, as well as methods. These methods of centroid mean computation have impacts on the accuracy of the geoid model since the basis of the development of the theory of each centroid mean type is different. This paper presents the impact of different centroid means on the accuracy of orthometric height modelling by geometric geoid method.</p><p class="abstract"><strong>Methods:</strong> DGPS observation was carried out to obtain the coordinates and ellipsoidal heights of selected points. The centroid means were computed with the coordinates using three different centroid means models (arithmetic mean, root mean square and harmonic mean). The computed centroid means were entered accordingly into a Microsoft Excel program developed using the Multiquadratic surface to obtain the model orthometric heights at various centroid means. The root means square error (RMSE) index was applied to obtain the accuracy of the model using the known and the model orthometric heights obtained at various centroid means.  </p><p class="abstract"><strong>Results:</strong> The computed accuracy shows that the arithmetic mean method is the best among the three centroid means types.</p><p class="abstract"><strong>Conclusions:</strong> It is concluded that the arithmetic mean method should be adopted for centroid computation, as well as orthometric height modelling using the geometric method.</p>

When Averaging Multiples, the Arithmetic Mean Is Inferior to the Harmonic Mean

2021 ◽  
Vol 40 (2) ◽  
pp. 61-67
Author(s):  
Gilbert E. Matthews
Keyword(s):  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Central Tendency

This article posits that using the arithmetic mean to average multiples is mathematically inferior. A multiple is an inverted ratio with price in the numerator. The harmonic mean is a statistically sound method for averaging inverted ratios. It should be used as a measure of central tendency for multiples, along with the median. Empirically, the harmonic mean and the median of a set of multiples are usually similar. Because the harmonic mean can be overly affected by abnormally low multiples, the valuator must use judgment to exclude outliers.

scholarly journals Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Yu-Ming Chu ◽  
Shou-Wei Hou
Keyword(s):  
Sharp Bounds ◽  
Double Inequality ◽  
Seiffert Mean ◽  
Contraharmonic Mean

We find the greatest valueαand the least valueβin(1/2,1)such that the double inequalityC(αa+(1-α)b,αb+(1-α)a)<T(a,b)<Cβa+1-βb,βb+(1-βa)holds for alla,b>0witha≠b. Here,T(a,b)=(a-b)/[2 arctan((a-b)/(a+b))]andCa,b=(a2+b2)/(a+b)are the Seiffert and contraharmonic means ofaandb, respectively.

Comparison of Harmonic Mean versus Arithmetic Mean Clearance Values

1986 ◽  
Vol 75 (4) ◽  
pp. 427-429 ◽  
Author(s):  
L.J. Schaaf ◽  
F.C. Lam ◽  
D.G. Perrier
Keyword(s):  
Arithmetic Mean ◽  
Harmonic Mean

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).

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