A Proof of Smale’s Mean Value Conjecture

≤ n− 1 n . Equality only occurs for p(z) = a1z + anz n with arbitrary a1, an ∈ C \ {0}. Here a proof will be given by a variational method which recently has been used in a similar way to prove Sendov’s conjecture (s. [2]). Former results and the backgrounds of both conjectures can be found in the survey article of Schmeisser [4]. Let n > 1 be fixed and define Fn as the class of nth degree monic complex polynomials p with p(0) = 0, p(0) 6= 0 and p(ζ) 6= 0 for all derivative zeros ζ of p. Obviously it suffices to consider polynomials p ∈ Fn in order to give a proof of Smale’s conjecture. For such p we define ρ(p) := min {