This manuscript puts forward and verifies an analytical approach for the design of phononic crystals that feature a bandgap at the lowest possible frequencies, in the sense of finding the optimal layer thicknesses for a given set of materials in a prescribed layering order. The mathematical formulation rests upon the exact form of the half-trace function (half of the trace of the global transfer matrix) of the layered medium, which is directly connected to its bandgap structure. After showing that there is a tight relation between the frequency at which the first bandgap opens up and the curvature of the half-trace function at zero frequency, a new optimization strategy is proposed, based on the minimization of this curvature. Notably, the optimal solution is expressed as a closed-form equation and remains valid for any number of layers within the unit cell. We validate this analytical result by comparison with a numerically optimized design, finding remarkably good agreement between both solutions.