  In this paper, we extend Gang Liu's three circle theorem for K\"ahler
manifolds to almost Hermitian manifolds. As applications of the three circle
theorem, we obtain sharp dimension estimates for spaces of holomorphic
functions of polynomial growth and rigidity for the estimates, Liouville
theorems for pluri-subharmonic functions of sub-logarithmic growth and
Liouville theorems for holomorphic functions of Cheng-type, and fundamental
theorem of algebra on almost Hermitian manifolds with slightly negative
holomorphic sectional curvature. The results are generalization of Liu's
results and some of them are new even for the K\"ahler case. We also discuss
the converse of the three circle theorem on Hermitian manifolds which turns out
to be rather different with the K\"ahler case. In order to obtain the three
circle theorem on almost Hermitian manifolds, we also establish a general
maximum principle in the spirit of Calabi's trick so that a general three
circle theorem is a straight forward consequence of the general maximum
principle. The general maximum principle and three circle theorem established
may be of independent interests.
