Counterexamples to Tischler's strong form of smale's mean value conjecture

Smale's mean value conjecture asserts that min_θ |P(θ)/θ| ≤ K|P′(0)| for every polynomial P of degree d satisfying P(0) = 0, where K = (d-1)/d and the minimum is taken over all critical points 6 of P. A stronger conjecture due to Tischler asserts that min_θ|1/2 - P(θ)/θ·P′(0)| ≤ K ₁ with K₁ = 1/2 - 1/d. Tischler's conjecture is known to be true: (i) for local perturbations of the extremum P₀(z) = z ^d - dz, and (ii) for all polynomials of degree d ≤ 4. In this paper, Tischler's conjecture is verified for all local perturbations of the extremum P₁(z) = (z - 1)^d -(-1)^d, but counterexamples to the conjecture are given in each degree d ≥ 5. In addition, estimates for certain weighted L¹- and L ²-averages of the quantities 1/2 - P(θ)/θ · P′(0) are established, which lead to the best currently known value for K₁ in the case d = 5.