In particular, the bounds for the arithmetic-geometric mean A G M have attracted the attention of many mathematicians.
In this article, we establish some new inequality chains for the ratio of certain bivariate means, and we present several sharp bounds for the arithmetic-geometric mean.
Recently, the bounds for the arithmetic-geometric mean (operatorname{AG}(a,b)) and Toader mean (T a,b)) have attracted the attention of many mathematicians.
Finally, we prove that (L_{-1}(a,b)) and (L_{-1/2}(a,b)) are the best possible lower and upper generalized logarithmic mean bounds for the arithmetic-geometric mean (operatorname{AG}(a,b)).
In this section, we present several sharp bounds for the arithmetic-geometric mean A G M. Theorem 1 can be derived from Propositions 1-4 and Lemma 7. Theorem 1 Let a, b > 0 with a ≠ b.
The aim of this paper is to establish the new inequality chains for the ratio of certain bivariate means, and we present the sharp bounds for the arithmetic-geometric mean A G M. In order to establish our main results we need several lemmas, which we present in this section.
Inequality (L_{-1}(a,b 0) with (aneq b), where (L_{-1}(a,b)) and (L_{-1/2}(a,b)) are tholdsst possible lower and upper generalized logarithmic mean bounds for the allthmetic-geometric mean (operatorname{AG}(a,b>0, respectively.
In [9] Vamanmurthy and Vuorinen presented the upper bounds for the arithmetic-geometric mean A G M in terms of the arithmetic mean A, geometric mean G and logarithmic mean L as follows: A G M ( a, b ) < L 2 ( a, b ) = ( A ( a, b ) L ( a, b ) ) 1 / 2, A G M ( a, b ) < I ( a, b ) < A ( a, b ), A G M ( a, b ) < A 1 / 2 ( a, b ). for all a, b > 0 with a ≠ b.
If reliable concentration slope estimates are not available, but bounds for the slope values can be obtained, then one can use interval arithmetic to derive upper and lower limits for the dependent fluxes and parameters using Equation (3) (or Equation (7) [ 28].
However, the association was slightly stronger for both men and women for the arithmetic test.
