The Sharp Bound for the Fourth Hankel Determinant over Functions with Positive Real-Part Derivative

Abstract

Let D be the open unit disk and let R denote the class of normalized analytic functions whose derivatives have positive real part. We determine the sharp upper bound 8/125 for the absolute value of the fourth Hankel determinant H(4,1). The determinant is represented as an integer polynomial in the first six Carathéodory coefficients divided by 6,048,000; the numerator contains sixteen monomials, each of weighted degree twelve. A finite Schur parametrization eliminates the final parameter exactly and yields a finite Fourier-phase radial majorant. Rotation covariance reduces the problem to a five-dimensional radial domain. A conic neighborhood of the extremal stratum is certified by exact quantifier elimination over the reals, while the complementary region is covered by an exact rational Bernstein partition-and-bound certificate. Every terminal box has a strictly positive rational gap, and none remains unresolved. Equality is attained by an explicit rotational family, proving sharpness. The argument uses exact arithmetic throughout and does not rely on Monte Carlo sampling or floating-point optimization.