  We study mixed weak estimates of Sawyer type for maximal operators associated
to the family of Young functions $\Phi(t)=t^r(1+\log^+t)^{\delta}$, where
$r\geq 1$ and $\delta\geq 0$. More precisely, if $u$ and $v^r$ are $A_1$
weights, and $w$ is defined as $w=1/\Phi(v^{-1})$ then the following estimate
\[uw\left(\left\{x\in \mathbb{R}^n: \frac{M_\Phi(fv)(x)}{v(x)} >
t\right\}\right) \leq C\int_{\mathbb{R}^n}
\Phi\left(\frac{|f(x)|v(x)}{t}\right)u(x) \,dx\] holds for every positive $t$.
This extends mixed estimates to a wider class of maximal operators, since when
we put $r=1$ and $\delta=0$ we recover a previous result for the
Hardy-Littlewood maximal operator. This inequality generalizes some previous
results proved by Cruz Uribe, Martell and P\'erez in (Int. Math. Res. Not.
(30): 1849-1871, 2005). Moreover, it includes estimates for some maximal
operators related with commutators of Calder\'on-Zygmund operators.
