An upper bound on Jacobi polynomials

Let ${\bf P}_k^{(\alpha, \beta)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality \begin{equation*} \max_{x \in [\delta_{-1},\delta_1]}\sqrt{(x- \delta_{-1})(\delta_1-x)} (1-x)^{\alpha}(1+x)^{\beta} ({\bf P}_{k}^{(\alpha, \beta)} (x))^2 < \frac{3 \sqrt{5}}{5}, \end{equation*} where $\delta_{-1}<\delta_1$ are appropriate approximations to the extreme zeros of ${\bf P}_k^{(\alpha, \beta)} (x) .$ As a corollary we confirm, even in a stronger form, T. Erd\'{e}lyi, A.P. Magnus and P. Nevai conjecture [Erd\'{e}lyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614], by proving that \begin{equation*} \max_{x \in [-1,1]}(1-x)^{\alpha+{1/2}}(1+x)^{\beta+{1/2}}({\bf P}_k^{(\alpha, \beta)} (x))^2 < 3 \alpha^{1/3} (1+ \frac{\alpha}{k})^{1/6}, \end{equation*} in the region $k \ge 6, \alpha, \beta \ge \frac{1+ \sqrt{2}}{4}.

An upper bound for nonnegative rank

We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices, which allows us to prove that [6(n+1)/7] linear inequalities suffice to describe a convex n-gon up to a linear projection

An upper bound on Reidemeister moves

We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. This leads to a conceptually simple solution to the equivalence problem for links.Comment: 40 pages, 14 figures; v2: very minor change