The Cubic Erdős–Herzog–Piranian Lemniscate Inequality

Let p be a monic polynomial of degree three and putΓp = {z ∈ C : |p(z)| = 1}, L(p) = H1(Γp).We prove thatL(p) ≤ L(z3 − 1) = 21/3B16,12.The proof uses the extremal reduction of Eremenko and Hayman, which permits one topass to a normalized cubic whose critical points lie on the lemniscate. The remainingone-parameter family is reduced by the Chebyshev identity for T3 and the Joukowskimap to an elementary one-dimensional estimate.