Fekete–Szego and Second Hankel Determinant Problems for a Bounded-Second-Derivative Class of Univalent Functions, and Its Fractional-Derivative Extension

Let A denote the class of functions g(z)=z+∑k=2∞akzk analytic and normalized in U={z:|z|<1}, and let D={g∈A:|g′′(z)|≤1,z∈U}. We give a short, self-contained proof that D consists entirely of univalent functions by a direct line-integral estimate combined with the classical Noshiro–Warschawski theorem. We then solve, in closed form and with explicit extremal functions, two classical extremal coefficient problems for D: the Fekete–Szego problem and the second Hankel determinant, using the exact Schur parametrization of bounded analytic functions applied to g′′. We also establish a sharp bound on a single coefficient |an| for every n≥2, obtained by an elementary Parseval argument that does not require the full joint Schur parametrization needed for the Fekete–Szego and Hankel problems. We then compose D with the Owa–Srivastava fractional derivative operator Ωzδ (δ∈[0,1)), used to build fractional bi-univalent subclasses, to define the one-parameter family D(δ)={g∈A:|(Ωzδg)′′(z)|≤1}, and extend all three results to D(δ), again with explicit extremal functions. Because the underlying weights grow at different rates in the fractional parameter δ, the Fekete–Szego problem exhibits a genuine, monotone bifurcation as δ ranges over [0,1), while the Hankel and general-coefficient bounds decrease monotonically—effects that are structural consequences of the fractional operator rather than a simple rescaling of the classical case. All results are verified in two independent numerical ways: the Schur parametrization lemma itself is checked against an explicit, independently constructed rational Schur function whose Taylor coefficients are extracted numerically via the Cauchy integral formula, and the sharp bounds are checked by Monte Carlo maximization over the admissible Schur parameters. Both checks are explained in detail and agree with the closed forms to at least four decimal places. Unlike the Carathéodory class of functions with positive real part, whose coefficient body is fixed at the origin (value 1) and is classically parametrized without an extra recursive step, the Schur class B used here does not pin down φ(0), so its exact coefficient body genuinely requires the two-step recursive Schur parametrization applied below to g′′; identifying and exploiting this distinction is part of the technical contribution of the paper.