On estimation of the derivative of the Legendre polynomial
Estimates of the derivative of the Legendre polynomial on the orthogonality segment −1 x  1 of the type 1 − x2 α dPn(x)/dx (n + a)δ are well known. In particular, if α = 0 then δ = 2, A = 1/2; if α = 1 then δ = 1/2, A = *2/π = 0.797885. In both cases the constants δ, A are unimprovable. An estimate δ = 1/2, A = *4/π = 1.128379 in case α = 3/4 occurs in literature. Here we have found exact values of main constants if α = 3/4: δ = 1 2 , A= max 0 t1(t) = 0.825031, J1 being the Bessel function. It is admissible to adopt a = 2/3 for all n  0. The set of a-values for which the inequality under consideration is true for sufficiently large n has the lower bound equal to 1/2, though the limiting value a = 1/2 is not acceptable.