On the zeros of derivatives of Bessel functions

The positive zeros of Jν‴(x) and Jν(n+1)(x) are studied by using classical analysis and the properties of Jν(x). It is proved that Jν‴(x) has a unique zero in specific intervals. Regarding Jν(n+1)(x), it is proved that its positive zero jν,m(n+1) is an increasing function with respect to ν, for ν>n. Moreover, the first two Rayleigh sums for jν,m(n) are calculated. The obtained results extend and complement previously known results and also answer an open problem regarding the monotonicity of jν,m(n). As a consequence of these results, a lower bound for jν,1(n+1) is deduced, as well as an inequality between jν,1(n+1) and jν,1(n).