On the positive zeros of the second derivative of Bessel functions

摘要

Let Jv(z) be the Bessel function of the first kind and of order v, Jv′(z) the derivative of Jv(z) and jv,1 its first positive zero. This paper examines the existence of zeros of Mv(z)=zJv′(z)+(βz2+α)Jv(z) in (0, jv,1) emphasis on the particular case where β=1 and α=−v2. In this case the zeros of Mv(z) are the zeros of the second derivative Jv″(z). Conditions are found under which the function jv″(z) has a unique zero in some subintervals of the interval (0, jv,1). The ordering relations that follow immediately and well-known bounds of the functions Jv+1(x)⧸Jv(x) lead to several upper and lower bounds for the first positive zero of Jv″(z), which are found to be much sharper than the well-known bounds in the literature.