We prove this bound by first using the unitary Ichino-Ikeda formula of N. Harris to relate the central L-value to an automorphic period integral.  There is a `trivial' bound for this integral, which turns out to correspond to the convexity bound for the L-value if the test vector is chosen correctly.  We are able to improve the bound for the period integral using a technique called arithmetic amplification, which uses the action of the Hecke operators, and this yields a subconvex bound.