We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel funct
ion $J_ u (x)$ and the Airy function $Ai(x)$ and find a sharp approximation for their zeros. We also answer the question raised by Olenko by showing that $$c_1 | u^2-1/4,| < sup_{x ge 0} x^{3/2}|J_ u(x)-sqrt{frac{2}{pi x}} , cos (x-frac{pi u}{2}-frac{pi}{4},)| <c_2 | u^2-1/4,|, $$ $ u ge -1/2 , ,$ for some explicit numerical constants $c_1$ and $c_2.$
We establish some new Turans type inequalities for orthogonal polynomials defined by a three-term recurrence with monotonic coefficients. As a corollary we deduce asymptotic bounds on the extreme zeros of orthogonal polynomials with polynomially growing coefficients of the three-term recurrence.
