ON THE POSITIVE ZEROS OF THE 2ND DERIVATIVE OF BESSEL-FUNCTIONS

Let J-nu(z) be the Bessel function of the first kind and of order nu, J-nu'(z) the deriavative of J-nu(z) and j-nu,1 its first positive zero. This paper examines the existence of zeros of M-nu(z) = z(J)nu'(z) + (beta-z2 + alpha)J-nu(z) in (0, j-nu,1) with emphasis on the particular case where beta = 1 and alpha = - nu2. In this case the zeros of M-nu(z) are the zeros of the second derivative J-nu"(z) of J-nu(z). Conditions are found under which the function J-nu"(z) has a unique zero in some subintervals of the interval (0, j-nu,1). The ordering relations that follow immediately and well-known bounds of the functions J-nu + 1(x)/J-nu(x) lead to several upper and lower bounds for the first positive zero of J-nu"(z), which are found to be much sharper than the well-known bounds in the literature.