New inequalities from classical Sturm theorems (Q707205)

The authors develop a systematic study of transformations of second order hypergeometric differential equations to normal form by means of Liouville transformations. Those transformations are chosen in such a way that the problem of computing the extrema or studying the monotonicity properties of the resulting coefficient reduces to solving a quadratic equation. Then, using Sturm comparison theorems, they obtain inequalities satisfied by the zeros of the solutions of those differential equations. They improve previously known inequalities, extend their range of validity and obtain new inequalities. In particular: (i) they complete Szegö's bounds on the zeros of Jacobi polynomials relaxing the restrictions for the parameter values, (ii) they show that Grosjean's inequality on the zeros of Legendre polynomials is also valid for Jacobi polynomials, (iii) they obtain bounds on ratios of consecutive zeros of Gauss and confluent hypergeometric functions and (iv) derive an inequality involving the geometric mean of zeros of Bessel functions.