Exact Thresholds for Global Univalence, Convexity, and Positive Real Part of a Modified Second Einstein Function

We consider the parameterized modified second Einstein mapping in the unit disk and ask where its main geometric properties actually change as the parameter varies. The answer gives three different sharp thresholds. One determines global univalence, a second one convexity, and the third positive real part. Each threshold is necessary and sufficient. After normalization, the function is of bounded turning on the whole univalence range and is therefore close-to-convex there. The endpoint behavior is also exact: the derivative and convexity criteria reach equality at the positive real boundary point, while the positive-real-part condition reaches equality at the negative real boundary point. Earlier work used decimal cutoffs and plots to describe the admissible range. We replace these decimal cutoffs by exact analytic thresholds, including necessity and boundary equalities; the numerical refinement is small, especially at the upper endpoint. For the proof, we combine Bernoulli expansions with explicit tail estimates and harmonic minimum arguments. Real-axis points give necessity, while monotonicity in the parameter and a Sturm calculation settle the convexity endpoint. In this way, the three transitions and their endpoint behavior are determined exactly for this family.