Turánians and Wronskians for the Zeros of Bessel Functions

Paul Turán [On the zeros of the polynomials of Legendre, Časopis pro Peěstovani Mat. a Fys., 75 (1950), pp. 113–122] proved that the Legendre polynomials satisfy the inequality $P_n (x)P_{n + 2} (x) - [P_{n + 1} (x)]^2 < 0, - 1 < x < 1$. Here it is shown that the positive zeros of arbitrary real Bessel functions satisfy sjmilar inequalities, even in a more general form. An analogous result is established for the corresponding Wronskian. In § 8, Remark 3, the monotonicity results established in the course of the proofs here are used to complement those derived by Sturm methods in [LEE LORCH, Elementary comparison techniques for certain classes of Sturm–Liouville equations, Proc. Uppsala 1977 Inter. Conf. Dill. Equations, Symposia Univ. Upsaliensis Annum Quingentesimum Celebrantis 7, Acta Univ. Upsaliensis, Uppsala 1977, pp. 125–133].