Inequalities and Numerical Bounds for Zeros of Ultraspherical Polynomials

Abstract

We improve previous results concerning the monotonicity in $\lambda $ of $f(\lambda )x_{nk}^{(\lambda )} $ where $x_{nk}^{(\lambda )} $ is a positive zero of the ultraspherical polynomial $P_n^{(\lambda )} (x)$, and $f(\lambda )$ is a suitably chosen positive increasing function. The range of validity is extended to $ - \frac{1}{2} \leqq \lambda \leqq \frac{3}{2}$ rather than $0 \leqq \lambda \leqq 1$. In a certain sense the results are the best obtainable by the methods used. Some new elementary bounds for the zeros are obtained and compared with known results. An inequality for $\partial {{(\log x_{nk}^{(\lambda )} )} / {\partial \lambda }}$ is also derived.