Smale's mean value constant exceeds 1.1559, and other miscellanea

We prove K* > 1.1559837 for Smale's mean value constant K*, improving the bound 1.1018 proved by Jatar and Ng. They showed that K* equals the supremum of 1/B(0) over pairs (W, B) of a normalised univalent map W of the unit disk and a finite Blaschke product B with B(0) > 0 such that (WB)′ has no zero in the disk; we write ℬ_d for this supremum over pairs with deg B ≤ d. Our bound comes from an explicit computer-assisted pair built from a polynomial of degree 18. The Dubinin (2024) conjecture fails in every large degree; this already follows from the earlier bound K* > 1 by a remark of Dubinin, and we give a direct short argument. Our other results are of three kinds. Bounds: 1.1363375 < ℬ₁ < 1.177801, ℬ₂ < 100/61, and ℬ_d ≤ 4 − 0.00597/(d log d) for every d ≥ 2. Extremal pairs: if (W, B) attains ℬ_d with d ≥ 1, then W does not extend holomorphically past the circle and is not rational; if 1/B_n(0) → 4, then after rotation W_n tends to the Koebe function; and a locally best pair built from a polynomial of degree n has at least n weighted boundary defects. Structure: for W meromorphic past the circle, an exact count from the circle bounds deg B by a number read off W, a pocket formula expresses the winding number of (WB)′/(W′B) along the circle through the points where W/(zW′) < 0, and we describe Blaschke models and symmetric pairs.