  In this paper we prove mixed inequalities for the maximal operator $M_\Phi$,
for general Young functions $\Phi$ with certain additional properties,
improving and generalizing some previous estimates for the Hardy-Littlewood
maximal operator proved by E. Sawyer. We show that given $r\geq 1$, if $u,v^r$
are weights belonging to the $A_1$-Muckenhoupt class and $\Phi$ is a Young
function as above, then the inequality
  \[uv^r\left(\left\{x\in \mathbb{R}^n:
\frac{M_\Phi(fv)(x)}{v(x)}>t\right\}\right)\leq
C\int_{\mathbb{R}^n}\Phi\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\]
  holds for every positive $t$.
  A motivation for studying these type of estimates is to find an alternative
way to prove the boundedness properties of $M_\Phi$. Moreover, it is well-known
that for the particular case $\Phi(t)=t(1+\log^+t)^m$ with $m\in\mathbb{N}$
these maximal functions control, in some sense, certain operatos in Harmonic
Analysis.
