Inequalities and monotonicity properties for zeros of Hermite functions

We study the variation of the zeros of the Hermite function H-lambda(t) wit h respect to the positive real variable lambda. We show that, for each non- negative integer n, H-lambda(t) has exactly n + 1 real zeros when n < lambd a less than or equal to n + 1, and that each zero increases from -infinity to infinity as lambda increases. We establish a formula for the derivative of a zero with respect to the parameter lambda; this derivative is a comple tely monotonic function of lambda. By-products include some results on the regular sign behaviour of differences of zeros of Hermite polynomials as we ll as a proof of some inequalities, related to work of W. K. Hayman and E. L. Ortiz for the largest zero of H-lambda(t). Similar results on zeros of c ertain confluent hypergeometric functions are given too. These specialize t o results on the first, second, etc., positive zeros of Hermite polynomials .