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边界。作为推论，我们以若干初等函数的反函数形式获得了Bessel函数及其导数零点的新的包围。除少数例外，这些界对所有零点和所有非负指数的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、要么紧密匹配先前已知的最佳结果。