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Let $S$ be a compact orientable surface, and $\Mod(S)$ its mapping classgroup. Then there exists a constant $M(S)$, which depends on $S$, with thefollowing property.  Suppose $a,b \in \Mod(S)$ are independent (i.e., $[a^n,b^m]\not=1$ for any$n,m \not=0$) pseudo-Anosov elements. Then for any $n,m \ge M$, the subgroup$<a^n,b^m>$ is free of rank two, and convex-cocompact in the sense ofFarb-Mosher. In particular all non-trivial elements in $<a^n,b^m>$ arepseudo-Anosov.  We also show that there exists a constant $N$, which depends on $a,b$, suchthat $<a^n,b^m>$ is free of rank two and convex-cocompact if $|n|+|m| \ge N$and $nm \not=0$.

Subgroups generated by two pseudo-Anosov elements in a mapping class group. II. Uniform bound on exponents