  The main aim of this paper is to investigate the Hardy-Littlewood type
Theorem and the Heinz type inequality on functions induced by a differential
operator. We first prove a more general Hardy-Littlewood type theorem for the
Dirichlet solution of a differential operator which depends on $\alpha >0$ over
the unit ball $\mathbb{B}^n$ of $\mathbb{R}^n$ with $n\geq 2$, related to the
Lipschitz type space defined by a fast majorant. We find that the case
$\alpha>0$ is completely different from the case $\alpha=0$. Then a more
general Heinz type inequality for the Dirichlet solution of a differential
operator will also be established in the case $\alpha>n-2$.
