Some inequalities for polynomials and rational functions associated with lemniscates

On the Harmonic Measure and Capacity of Rational Lemniscates

“…A map between two topological spaces F : X → Y is called proper if the inverse image F −1 (K) of every compact subset K of Y is a compact subset of X. Dubinin [8], among other results, generalized a result of Pólya for the area of a polynomial lemniscate by proving the following inequality for a proper holomorphic map F from a domain D onto a circular ring…”

“…A map between two topological spaces F : X → Y is called proper if the inverse image F −1 (K) of every compact subset K of Y is a compact subset of X. Dubinin [8], among other results, generalized a result of Pólya for the area of a polynomial lemniscate by proving the following inequality for a proper holomorphic map F from a domain D onto a circular ring {z ∈ C : t 1 < |z| < t 2 } (0 < t 1 < t 2 < +∞): if E is the union of all those connected components of Ĉ \D whose boundaries contain points corresponding, under the holomorphic function F, to points on the circle {z ∈ C : |z| = t 1 } and ∞ ∈ E, then…”