Logarithmic Curvature as an Asymptotic Diagnostic Tool

We introduce the scale-normalized logarithmic curvature index C(n) = n Φ''(n) / Φ'(n), where Φ(n) = log f(n) for a positive function f(n) with sufficiently regular growth. This index serves as a diagnostic for the asymptotic behavior of f(n). Under suitable conditions (including membership in a Hardy field and rapid enough convergence of C(n) to a constant c > -1), we rigorously prove that Φ(n) ∼ (1/(1+c)) n^{1+c}. We classify standard growth regimes, discuss boundary cases requiring logarithmic corrections, and extend the framework to convex duality in large deviation theory and discrete sequences. The curvature index C(n) measures the elasticity of the logarithmic slope and bridges regular variation, exponential growth, and structures in large deviations. Its main contribution is enabling exponent extraction from sequences whose asymptotics remain theoretically open. Keywords: asymptotic analysis, Hardy fields, logarithmic curvature, regular variation, large deviations, growth classification.