Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions

In this paper an upper estimate of the number of limit cycles of the Abel equation = v(x,t), x ∊ , t ∊ S¹ is given. Here v is a polynomial in x with the higher coefficient one and periodic in t with period one. The bound depends on the degree n of the polynomial and the magnitude of its coefficients. In the second part we give an explicit upper estimate of the number of zeros of a holomorphic function in a compact subset of its domain through the growth rate of the function and some geometric constant that is expressed here by means of the Poincaré metric. This improves the estimate given in Ilyashenko and Yakovenko (1996 J. Differ. Equ. 126 87-105).