We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.

翻译：我们证明了贝塞尔函数及其导数相位函数的显式统一双侧界。由此，我们获得了贝塞尔函数及其导数零点的新界域，这些界域通过某些初等函数逆值进行表征。除少数例外情况外，这些界适用于所有零点和所有非负阶数的贝塞尔函数。我们提供的数值证据表明，所得界要么优于先前已知的最佳结果，要么与之高度吻合。