Bounds for the Ratios of Differences of Power Means in Two Arguments

Using methods from classical analysis, sharp bounds for the ratio of differences of Power Means are obtained. Our results generalize and extend previous ones due to S. Wu(2005), and to S. Wu and L. Debnath.Comment: 1 figur

Partial Fraction Expansions for Newton's and Halley's Iterations for Square Roots

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).Comment: 10 page

An Inequality for Bounded Functions

In this note we prove optimal inequalities for bounded functions in terms of their deviation from their mean. These results extend and generalize some known inequalities due to Thong (2011) and Perfetti (2011)Comment: 7 page