On Lemniscates and Critical Points of Finite Blaschke Products

Let $$B$$ be a finite Blaschke product of degree $$n\geq 2$$ . We discuss the behavior of the area of the set $$E(B,t)=\{z\colon |B(z)|\leq t\}$$ as the parameter $$t$$ changes, $$0<t<1$$ , and also a condition on $$t$$ under which the connected component of $$E(B,t)$$ , $$B(0)=0$$ , containing the origin contains at least one critical point of $$B$$ . For real products $$B$$ , $$B(0)=0$$ , $$B'(0)\not= 0$$ , we establish sharp lower bounds for the moduli of some critical points in terms of the moduli of the corresponding critical values and the quantity $$|B'(0)|$$ . Cases are considered separately in which the critical points are located on the same radius of the disk $$|z|<1$$ and in which they lie on the same diameter of this disk on different sides of the origin. These estimates do not depend on the degree of the product $$B$$ .