Estimating the derivative of the Legendre polynomial (Q1759557)

Let \(P_{n}(x)\) be the classical Legendre polynomials, \(-1\leq x\leq 1\). The authors determine the best constant \(A\) in the inequality \[ (1-x^2)^{3/4}\,\Big|\frac{d\,P_{n}(x)}{dx}\Big|<A\,\sqrt{n+\frac{2}{3}},\quad n\geq 2. \] This is \(A=\max_{0\leq t<\infty}\sqrt{t}\,J_{1}(t)=0.825031\dots\), where \(J_{1}(t)\) is the usual Bessel function of the first kind.