Determine the exact behavior of the degree-restricted constants

Determine whether the degree-restricted extremal constants for admissible univalent Blaschke pairs are strictly increasing, whether the degree-one constant equals the unrestricted constant, and whether the unrestricted constant equals 4.

Background

The paper reformulates Smale’s mean value problem as an extremal problem over admissible pairs consisting of a normalized univalent map W of the unit disk and a finite Blaschke product B such that WB is locally univalent. The unrestricted constant is the supremum of 1/B(0) over all such pairs, while the degree-restricted constant for degree d is the corresponding supremum over pairs with deg B ≤ d.

The paper proves that the unrestricted constant is at most 4, that the degree-restricted constants satisfy quantitative upper bounds below 4, and that the degree-one constant lies in a nontrivial numerical interval. It does not determine whether the degree-restricted sequence is strictly increasing, whether the degree-one problem already yields the unrestricted supremum, or whether the limiting unrestricted value reaches the upper bound 4.