On the positive zeros of the second derivative of Bessel functions

Abstract

Let J v ( z) be the Bessel function of the first kind and of order v, J v ′( z) the derivative of J v ( z) and j v,1 its first positive zero. This paper examines the existence of zeros of M v ( z)= zJ v ′( z)+( βz 2+ α) J v ( z) in (0, j v,1 ) emphasis on the particular case where β=1 and α=− v 2. In this case the zeros of M v ( z) are the zeros of the second derivative J v ″( z). Conditions are found under which the function j v ″( z) has a unique zero in some subintervals of the interval (0, j v,1 ). The ordering relations that follow immediately and well-known bounds of the functions J v+1 ( x)⧸ J v ( x) lead to several upper and lower bounds for the first positive zero of J v ″( z), which are found to be much sharper than the well-known bounds in the literature.