Counting periodic solutions of the forced pendulum equation

Let h be a holomorphic function with h(0) = 1. The number of zeros of h on a disk centered at the origin can be controlled by the maximum value of |h| on a larger disk. This is a classical result in complex analysis that is sometimes called Jensen’s inequality (see for instance [4]). In [2] Il’yashenko and Yakovenko applied this result together with the theory of conformal mappings to count the zeros of the solutions of linear di erential equations. The same techniques will be applied in this paper to the forced pendulum equation. The result will be an upper estimate on the number of periodic solutions. The forced pendulum equation can be written in the form