New sharp integral inequality: for f monic of degree n, U a component of E(f) with m zeros, s above the largest critical value modulus in U, one has ∫_{V_s} |f'/f| dA <= 2*pi*sqrt(mn)*s^{1/n} where V_s = {|f|<s} ∩ U, with equality for f = z^n. Proof: Cauchy–Schwarz splitting |f'/f| = (|f'| |f|^{(1/n-2)/2}) * (|f|^{-1/(2n)}), the degree-m pushforward area formula, and Pólya's area inequality via layer cake; the exponent 1/n is the unique one making the two factors proportional at z^n.