Smale’s Problem for Critical Points on Certain Two Rays

Abstract

Let f be a polynomial of degree n ≥ 2 with f (0) = 0 and f (0) = 1. We prove that there is a critical point ζ of f with | f (ζ )/ζ | ≤ 1/2 provided that the critical points of f lie in the sector {r e iθ : r > 0, |θ| ≤ π/6}, and | f (ζ )/ζ | < 2/3 if they lie in the union of the two rays {1 + r e ±iθ : r ≥ 0}, where 0 < θ ≤ π/2.2000 Mathematics subject classification: primary 30C10; secondary 30C15.

Polynomials Versus Finite Blaschke Products

“…The estimate of the constant N has been considered by many people and we refer the reader to [10,11,20,23,29] and [6] for more details.…”

Smale’s mean value conjecture for finite Blaschke products

“…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]).…”