Dennis C. Russell

A note on Mathieu's inequality

SummaryThe function defined by the real infinite series

Tauberian-Type Results for Convolution of Sequences

S(t): = \sum\limits_{n = 1}^\infty {2n(n^2 + t^2 )^{ - 2} }

A Problem Concerning the Gamma Function

has been the subject of a number of estimates, starting with an 1890 conjecture by Mathieu (in connection with work on elasticity of solid bodies) thatS(t) < t−2. More recently it appeared in work on Mercerian theorems for Cesàro summability. A combination of the best of the various published inequalities so far (for larget) is

On generalized Cesàro means of integral order

\frac{1}{{t^2 }} - \frac{5}{{16t^4 }}< S(t) = \frac{1}{{t^2 }} - \frac{1}{{6t^2 }} + O\left( {\frac{1}{{t^6 }}} \right).

Inclusion Theorems for Section-Bounded Matrix Transformations

However, one has only to observe that