A point \(\zeta\in \mathbb C\) is said to be a critical point of a complex polynomial \(p\) if \(p^\prime(\zeta)=0\). By S. Smale’s inequality [Bull. Am. Math. Soc., New Ser. 4, 1–36 (1981; Zbl 0456.12012)], if \(\deg p\geq 2\) and \(x\in\mathbb C\) is not a critical point of \(p\), then there exists a critical point \(\zeta\) of \(p\) such that \[ \left|\frac{p(\zeta)-p(x)}{\zeta-x}\right|\leq 4|p^\prime(x)|. \] Let \[ S(p,x):=\min\left(\left|\frac{p(\zeta)-p(x)}{(\zeta-x)p^\prime(x)}\right|:\;p^\prime(\zeta)=0\right) \] and for each \(d\geq2\), let \[ K(d):=\sup\{S(p,x):\;\deg p=d,\;p^\prime(x)\neq0\}. \] Smale conjectured that \(K(d)=1\) or possibly \(K(d)\leq 1-1/d\) and himself proved that always \(K(d)\leq 4\).
There are many partial results in connection with this conjecture. Some of them have been listed in the paper under review. In particular, it is known that Smale’s conjecture is true for \(d=2,3,4\) [see Q. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs. New Series 26. Oxford: Oxford University Press (2002; Zbl 1072.30006)]. The main result of the present paper is that in case \(d\geq 8\) one has \[ K(d)<4-\frac{2.263}{\sqrt d}, \] which improves previously known results. If \(d\leq 8\), there are known better bounds than the one given above, e.g. G. Schmeisser [in Approximation theory, DARBA, Sofia, 353–369 (2002; Zbl 1033.30005)] gave an elementary proof that \[ K(d)\leq \frac{2^d-(d+1)}{d(d-1)}, \] and this bound is smaller than \(4-2.263/\sqrt d\) for \(2\leq d\leq 7\).