Hermitian pluriharmonic maps between almost Hermitian manifolds

In the case where both the domain and target manifolds are almost Hermitian, we introduce the concept of Hermitian pluriharmonic maps. We prove that any holomorphic or anti-holomorphic map between almost Hermitian manifolds is Hermitian pluriharmonic. We also establish some monotonicity formulae for the partial energies of Hermitian pluriharmonic maps into Kähler manifolds. As an application, under appropriate assumptions on the growth of the partial energies, some holomorphicity results are proven.