  Let $f(z) = \sum_{k=0}^\infty a_k z^k$ be an analytic function in a disk
$D_R$ of radius $R>0$, and assume that $f$ is $p$-valent in $D_R$, i.e. it
takes each value $c\in{\mathbb C}$ at most $p$ times in $D_R$. We consider its
Borel transform $$ B(f)(z) = \sum_{k=0}^\infty \frac{a_k}{k!} z^k , $$ which is
an entire function, and show that, for any $R>1$, the valency of the Borel
transform $B(f)$ in $D_R$ is bounded in terms of $p,R$. We give examples,
showing that our bounds, provide a reasonable envelope for the expected
behavior of the valency of $B(f)$. These examples also suggest some natural
questions, whose expected answer will strongly sharper our estimates.
  We present a short overview of some basic results on multi-valent functions,
in connection with "Taylor domination", which, for $f(z) = \sum_{k=0}^\infty
a_k z^k$, is a bound of all its Taylor coefficients $a_k$ through the first few
of them. Taylor domination is our main technical tool, so we also discuss
shortly some recent results in this direction.
