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A Harmonic Measure Interpretation of the Arithmetic–Geometric Mean

The American Mathematical Monthly, 2007
(2007). A Harmonic Measure Interpretation of the Arithmetic–Geometric Mean. The American Mathematical Monthly: Vol. 114, No. 7, pp. 610-622.
Byron L. Walden, Lesley A. Ward
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An approximation to the arithmetic-geometric mean

The Mathematical Gazette, 2014
Summary: Given positive numbers \(a>b\), consider the `agm iteration' given by \(a_0=a\), \(b_0=b\) and \[ a_{n+1}=\frac{1}{2}(a_n+b_n), \quad b_{n+1}=(a_nb_n)^{1/2}. \] At each stage, the two new numbers are the arithmetic and geometric means of the previous two.
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More Matrix Forms of the Arithmetic-Geometric Mean Inequality

SIAM Journal on Matrix Analysis and Applications, 1993
The authors prove the following arithmetic-geometric mean inequality: \(2||| A^* XB||| \leq ||| AA^* X+XBB^*|||\) for arbitrary \(n\times n\) matrices \(A\), \(B\), \(X\). They also show that the real function \(f(p):=||| A^{1+p} XB^{1-p}+A^{1-p} XB^{1+p}|||\), \(A,B\geq 0\) is convex on \([-1,1]\) and takes its minimum at \(p=0\), where \(|||\cdot|||\)
Rajendra Bhatia, Chandler Davis
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An Arithmetic-Geometric Mean of a Third Kind!

Lecture Notes in Computer Science, 2019
The concept of the generalized arithmetic-geometric mean (GAGM) embraces both the arithmetic-geometric mean (AGM) and the modified arithmetic-geometric mean (MAGM) as two special concepts. The GAGM is applied for attaining a unifying formula for calculating complete elliptic integrals (CEI), including those of the third kind, thereby providing a ...
Semjon Adlaj
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The Arithmetic-Geometric Mean of Gauss (1984)

2016
Paper 3: David A. Cox, “The arithmetic-geometric mean of Guass,” L’Enseignement Mathematique, vol. 30 (1984), p. 275–330. Reprinted by permission.
David A Cox
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The arithmetic-geometric mean and the elliptic mean error

Acta Geodaetica et Geophysica Hungarica, 2003
Without any special term, the mathematical definition of a measuring index for reliability of geodetic point was given by Lajos Homorodi. For this index, the term of the elliptic mean error was proposed by the author of the present paper and it was shown that the elliptic mean error is beneficial to the being for characterizing the reliability of ...
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Arithmetic-geometric means of positive matrices

Mathematical Proceedings of the Cambridge Philosophical Society, 1987
AbstractWe prove the existence of unique limits and establish inequalities for matrix generalizations of the arithmetic–geometric mean of Lagrange and Gauss. For example, for a matrix A = (aij) with positive elements aij, define (contrary to custom) A½ elementwise by [A½]ij = (aij)½.
Cohen, Joel E., Nussbaum, Roger D.
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Higher genus arithmetic-geometric means

The Ramanujan Journal, 2007
For any two complex numbers, one can define as usual their arithmetic-geometric mean. Due to the ambiguity of the square root, this is a multi-valued function. Given one value, Gauss determined all its values and moreover showed that they are closely related to certain elliptic integrals.
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Remarks on the matrix arithmetic–geometric mean inequality

Acta Scientiarum Mathematicarum
The author proves among others: (1) Let \(A\) and \(B\) be positive definite matrices. Then for every unitarily invariant norm \[ \left\vert {\left\Vert A^{\frac{1}{2}}B^{\frac{1}{2}} \right\Vert} \right\vert \leqslant \left\vert {\left\Vert {(AB)}^{\frac{1}{2}}+{(BA)}^{\frac{1}{2}} \right\Vert} \right\vert \leqslant (1/2) \left\vert {\left\Vert {A + B}
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Arithmetic-geometric means for electrical energies

Proceedings of the IEEE, 1980
The paper indicates an important arithmetic-geometric relation for the behavior of linear passive reciprocal n-ports. The active energies dissipated in the unit time interval in any linear passive time invariant reciprocal resistive system satisfy the inequalities P 1 + P 2 ≥ 2P 0 √P 1 P 2 ≥ P 0 where P 1 = (V 1 , I 1 ), P 2 = (V 2 , I 2 ), P 0 = (V 1 ,
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