  The main purpose of this paper is to develop some methods to investigate
equivalent norms and Hardy-Littlewood type Theorems on Lipschitz type spaces of
analytic functions and complex-valued harmonic functions. Initially, some
characterizations of equivalent norms on Lipschitz type spaces of analytic
functions and complex-valued harmonic functions will be given. In particular,
we give an answer to an open problem posed by Dyakonov in (Math. Z.249(2005),
597--611). Furthermore, some Hardy-Littlewood type Theorems of complex-valued
harmonic functions are established. The obtained results improve and extend the
main results in (Acta Math.178(1997),143--167).
  Additionally, we apply the equivalent norms and Hardy-Littlewood type
Theorems to study composition operators between Lipschitz type spaces.
