Exponential Taylor domination

Abstract

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.