Yes, this is so. And much stronger statements are available: See the beautiful survey paper
MR2567024 Rashkovskii, Alexander Classical and new loglog-theorems. Expo. Math. 27 (2009), no. 4, 271–287.
also available on the arxiv
All results in this paper are actually for subharmonic functions, so this settles the questrion for both analytic and harmonic functions.