Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture

Smale's mean value conjecture asserts thatmin 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 θ of P . A stronger conjecture due to Tischler asserts thatmin θ |1 2 − P ( θ )θ ċ P ′ ( 0 )| ⩽ K 1withK 1 = 1 2 − 1 / d . Tischler's conjecture is known to be true: (i) for local perturbations of the extremum P 0 ( 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 P 1 ( z )=( z − 1) d − (−1) d , but counterexamples to the conjecture are given in each degree d ⩾ 5. In addition, estimates for certain weighted L 1 ‐ and L 2 ‐averages of the quantities1 2 − P ( θ ) / θ . P ′ ( 0 ) are established, which lead to the best currently known value for K 1 in the case d =5. 2000 Mathematics Subject Classification 30C15.