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Publication - Dr Edward Crane

    Mean value conjectures for rational maps


    Crane, E, 2006, ‘Mean value conjectures for rational maps’. Complex Variables and Elliptic Equations, vol 51 (1)., pp. 41 - 50


    Let p be a polynomial in one complex variable. Smale's mean value conjecture estimates |p'(z)| in terms of the gradient of a chord from (z,p(z)) to some stationary point on the graph of p. The conjecture does not immediately generalize to rational maps since its formulation is invariant under the group of affine maps, not the full Mobius group. Here we give two possible generalizations to rational maps, both of which are Mobius invariant. In both cases we prove a version with a weaker constant, in parallel to the situation for Smale's mean value conjecture. Finally, we discuss some candidate extremal rational maps, namely rational maps all of whose critical points are fixed points.

    Full details in the University publications repository