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Bessel函数及其导数的相位函数的显式一致双侧界。作为推论，我们得到了基于若干初等函数反函数值的Bessel函数及其导数的零点的新包围。除少数例外，这些界对所有非负指标的零点及所有Bessel函数均成立。我们提供的数值证据表明，我们的界要么优于先前已知的最佳结果，要么与之紧密匹配。