Some upper estimates of the number of limit cycles of planar vector fields with applications to Liénard equations

We estimate the number of limit cycles of planar vector fields through the size of the domain of the Poincaré map, the increment of this map, and the width of the complex domain to which the Poincaré map may be analytically extended. The estimate is based on the relationship between the growth and zeros of holomorphic functions [IYa], [I]. This estimate is then applied to getting the upper bound of the number of limit cycles of the Liénard equation $\dot x=y-F(x)$, $\dot y=-x$ through the (odd) power of the monic polynomial $F$ and magnitudes of its coefficients.