Precise values of growth indices for σ widehat⋆ τ in extreme cases

Determine the exact values of the growth indices γ(σ widehat⋆ τ) and \overline{γ}(σ widehat⋆ τ) in the extreme regimes where the bounds of Theorem 4.28 become formally trivial, in particular when γ(σ) ≤ \overline{γ}(τ) (e.g., \overline{γ}(τ) = +∞) and when \overline{γ}(σ) = +∞.

Background

The paper studies how the generalized conjugates modify the growth indices γ(⋅) and \overline{γ}(⋅). Theorem 4.28 provides inequalities: γ(σ) ≤ γ(σ widehat⋆ τ) + \overline{γ}(τ) and \overline{γ}(σ widehat⋆ τ) + γ(τ) ≤ \overline{γ}(σ).

In certain extreme scenarios (e.g., \overline{γ}(τ) = +∞ or \overline{γ}(σ) = +∞), these inequalities become formally trivial and do not determine the actual values of the indices for σ widehat⋆ τ. The authors explicitly note that the precise values in these cases are unclear.

Resolving these values would complete the understanding of index behavior under the generalized upper Legendre conjugate, especially in slowly varying or unbounded-index regimes.