Smale's Mean Value Conjecture and its Dual Conjecture for Complex Polynomials

Abstract

For the purposes of proving Smale's mean value conjecture, one may restrict consideration to an explicitly described family of so-called Schlicht normalized polynomials. For Schlicht normalized polynomials whose leading coefficient decays sufficiently slowly, we show that the conjectured upper bound $1$ becomes asymptotically valid as the degree $d$ tends to infinity. The key and novel input is a refined Koebe $\frac{1}{4}$-type theorem of Cunningham, tailored to Schlicht functions whose images have bounded logarithmic capacity. For the dual mean value conjecture, we obtain an improvement of Eremenko's Markov-type inequality for regions bounded by polynomial lemniscates on which the polynomial is univalent. As a consequence, we improve the best known unconditional lower bound $\frac{1}{d^2}$ of Dubinin to $\frac{(d-\frac{1}{2})^{1/d}}{d^2}$ for all $d \ge 2$. The strengthened Markov-type inequality constitutes one of the main technical contributions of this paper. Its proof combines the solution to a related extremal problem for the logarithmic capacity of polynomial lemniscates, with the quasi-conformal deformation method introduced by Eremenko and Hayman to establish the connectedness of extremal polynomial lemniscates arising in the Erdos, Herzog and Piranian's problem on maximal lemniscate length.