An upper bound for the first zero of Bessel functions

It is shown, using the Rayleigh-Ritz method of the calculus of variations, that an upper bound for the first zero ${j_v}$, of ${z^{ - v}}{J_v}(z)$, $v > - 1$, is given by

$${(v + 1)^{1/2}}\{ {(v + 2)^{1/2}} + 1\} ,$$

and that for large v, $v,{j_v} = v + O({v^{1/2}})$.