Let Mf(x) be the Hardy-Littlewood maximal function on $mathbb{R}^n$. Let $Phi$ and $Psi$ be functions satisfying $Phi$(t) = ${int^t}_0$a(s)ds and $Psi(t)$ = ${int^t}_0$b(s)ds, where a(s) and b(s) are positive continuous such that ${int^infty}_0frac{a(s)}{s}ds$ = $infty$ and b(s) is quasi-increasing. We show that if there exists a constant $c_1$ so that ${int^s}_0frac{a(t)}{t}dt;c_1b(c_1s)$ for all $sgeq0$, then there exists a constant $c_1$ such that(0.1) $int_{mathbb{R^{n}}$ $Phi(Mf(x))dx;leq;c_2$ $int_mathbb{R^{n}}$$Psi(c_2midf(x)mid)dx$ for all $fepsilonL^1(R^n_$. Conversely, if there exists a constant $c_2$ satisfying the condition (0.1), then there exists a constant $c_1$ so that ${int^s}_deltafrac{a(t)}{t}dt=;leq;c_1b(c_1s$ for all $delta$ > 0 and $sgeqdelta$.