Therefore we obtain via the arithmetic-geometric mean inequality that
Therefore by applying the arithmetic-geometric mean inequality and (48), we obtain the desired inequality.
Johann Carl Friedrich Gauss observed the connection between the arithmetic-geometric mean iteration and an elliptic integral.
The convergence of the arithmetic-geometric mean iteration follows directly from Theorem 4 by letting
They cover how inequalities behave, squares are never negative, the arithmetic-geometric mean
inequality, the harmonic mean, symmetry in algebra, the rearrangement inequality, and the Cauchy-Schwarz inequality.
To solve these problems, he used the well-known arithmetic-geometric mean
The arithmetic-geometric mean agm(x, y) is computed recursively (with very fast convergence) as
In this approximation of [H.sub.3/2] (from now on, simply [H.sub.3/2]), the arithmetic-geometric mean, agm, will be the mean of [H.sub.3/2],
Furuichi, A refinement of the arithmetic-geometric mean
inequality, arXiv: 0912.
As n [right arrow] [infinity], [a.sub.n] and [g.sub.n] converge quadratically to Gauss's arithmetic-geometric mean
, M([a.sub.0], [g.sub.0]), and
Introductory chapters cover the arithmetic-geometric mean
inequality, the Cachy-Schwartz inequality, H|lder's inequality for sums, Nesbitt's inequality, and the Rearrangement and Chebyshev inequalities.
It is known that the scalar arithmetic-geometric mean
agm(a, b) of two (nonnegative) numbers a and b is defined by starting with [a.sub.0] = a and [b.sub.0] = b and then iterating