Smale’s mean value conjecture for finite Blaschke products

Abstract

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale's mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by V. Dubinin and T. Sugawa for the dual mean value conjecture for polynomials.≤ c|P ′ (a)| for at least one critical point b of P (zero of P ′ ). Let M be the least possible value of the factor c for all non-linear polynomials and M n be the corresponding value for polynomials of degree n. It was proven by Smale [30] that 1 ≤ M ≤ 4 and he conjectured that M = 1 or even M n = n−1 n and pointed out that the number n−1 n would, if true, be the best possible bound here as it is attained (for any nonzero λ) when P (z) = z n − λz and a = 0. The conjecture was repeated in [31,28] and it is also listed as one of the three minor problems in Smale's famous problem list [32]. The conjecture is now known as Smale's mean value conjecture which has remained open since 1981 even though it was proven to be true for many classes of polynomials (see [5],[26],[14],[15],[17],[25],[27],[33],[34] and [37]).

Distortion and critical values of the finite Blaschke product

“…In [16], a lower bound for the quantity max f (ζ) ζf ′ (0) : f ′ (ζ) = 0 was found for all finite Blaschke products of degree n 2 such that f (0) = 0, f ′ (0) = 0. The following sharp lower bound complements the research [16] on the critical values of Blaschke products.…”

“….., n. These products and their applications have been studied by many authors (see, for example, [12,[14][15][16] and the references therein). At the same time, problems associated with critical values have not been fully studied [16; 17, p. 365].…”