A sharp convexity disk for a Dirichlet-polynomial zero displacement map

Fix alpha=log(3)/log(2), beta=alpha-1, and R=beta^beta/alpha^alpha. We prove that the germ z(0)=0 defined by 1-exp(-z)+u exp(-alpha z)=0 is a convex conformal map throughout |u|<R. Every smaller centered circle maps to a real-analytic Jordan curve of positive curvature. The map extends continuously and injectively to the closed critical disk; its image has exactly one nonsmooth boundary point, with interior angle pi/2. The proof excludes the complete nonpositive-curvature locus of a half-plane inverse by an explicit scalar inequality. The radius is sharp, and a two-term Puiseux expansion describes the corner. For 0<epsilon<=1/128, we identify all zeros of 1-2^(-s)+epsilon 3^(-s) in |Re(s log(2))|<=1/8 and obtain the convex oval forming the closure of their labeled displacements. The Fuss-Catalan identification and analytic radius are classical background for this geometric conclusion.