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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-Littlewoodmaximal 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 Youngfunction as above, then the inequality  \[uv^r\left(\left\{x\in \mathbb{R}^n:\frac{M_\Phi(fv)(x)}{v(x)}>t\right\}\right)\leqC\int_{\mathbb{R}^n}\Phi\left(\frac{|f(x)|}{t}\right)u(x)v^r(x)\,dx\]  holds for every positive $t$.  A motivation for studying these type of estimates is to find an alternativeway to prove the boundedness properties of $M_\Phi$. Moreover, it is well-knownthat 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 HarmonicAnalysis.

Improvements on Sawyer type estimates for generalized maximal functions