Upper bounds for the moments of zeta prime rho

Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $\zeta(s)$ for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments. The proof is based upon a recent method of K. Soundararajan that provides analogous bounds for continuous moments of the Riemann zeta-function as well as for moments L-functions at the central point, averaged over families.