Extremal Polynomials in Smale’s Mean Value Conjecture

On Critical Values of Polynomials with Real Critical Points

“…Beardon, Minda and Ng [1] reduced the constant 4 to 4 1−1/n , which was slightly improved in the papers [2], [7]. At this time the best upper estimate belongs to Crane (see [4], [5]). In the case when the critical points of f have equal modulus or the values of f at the critical points have equal modulus, Sheil-Small ( [10], pp.…”

“…Cases of equality. If min{h 1 (u, A), h 2 (u, A)} = 2/3, then by (4) and the monotonicity properties of h 1 and h 2 , this can only be when A = A 0 (u); on the other hand, the analysis above shows that this can occur only when A = 0, so we must have A = A 0 (u) = 0 and thus u = 1. Furthermore, to have the equality min{|f (1)|, |f (ue iθ )|/u} = 2/3, we must have, for all t ∈ (0, 1), equality in at least one of the estimates which are equal and are < 2/3 unless e iθ = −1, so that the critical points of f are then ±1.…”

Smale’s mean value conjecture and the coefficients of univalent functions

“…The conjecture is true for d = 2, 3, 4; see [11]. Furthermore, the case where d = 5 was recently proved in [3] by using a method based on the results in [2]. The best known result for an arbitrary d is found in [1], where the factor 4 in (1.1) is replaced by 4…”

Smale’s Problem for Critical Points on Certain Two Rays

“…The best known upper bound is due to Crane (see [4,5]). If the critical points of f all have the same modulus, or if the values of f at the critical points have equal modulus, then a theorem of Sheil-Small [11, pp.…”