Convexity of KL divergence between symmetric alpha-stable scales

Establish whether the function v ↦ D(g_v^(alpha) || g_s^(alpha)) is convex on (0, ∞) for every stability index alpha in (0, 2), where D denotes Kullback–Leibler divergence between two symmetric alpha-stable densities with the same alpha and scales v and s (fixed s > 0).

Background

Within the positivity analysis for Mixed Fractional Information (MFI), the paper considers the behavior of the relative entropy D(g_v^alpha || g_s^alpha) as a function of the scale parameter v for fixed s and alpha. The authors note that convexity is plausible and verified for special cases alpha = 1 (Cauchy) and alpha = 2 (Gaussian), but the general case remains unresolved.

Convexity would have useful implications for monotonicity and optimization properties of D(v) and could strengthen analytical results connected to MFI, but the main positivity proof does not depend on convexity. Hence, determining convexity for general alpha is posed as an open question.