Exponential taylor domination

Let (Equation presented) 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∈C at most p times in D_R. We consider its Borel transform (Equation presented) 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 (Equation presented), 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.