COUNTEREXAMPLES TO TISCHLER'S STRONG FORM OF SMALE'S MEAN VALUE CONJECTURE

Abstract

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 θ of P. A stronger conjecture due to Tischler asserts that min θ | 1 2 − P ( θ ) θ ċ P ′ ( 0 ) | ⩽ K 1 with K 1 = 1 2 − 1 / d . Tischler's conjecture is known to be true: (i) for local perturbations of the extremum P0(z)=zd − 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 L2-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. 2000 Mathematics Subject Classification 30C15.