Logarithmic convexity for a coupled system with two subdiffusion orders

We prove a logarithmic convexity estimate for a pair of coupled Caputo subdiffusion equations with different time orders. For every bounded smooth domain, every $0<\alpha_1<\alpha_2<1$, and every final time $T>0$, the product $L^2$ norm at time $t$ is bounded by a constant times the geometric interpolation of the initial and final norms, with exponent $t/T$. The estimate holds for arbitrary $L^2$ initial pairs and includes domains with growing low-frequency modes. This gives an affirmative answer to the Dirichlet norm-interpolation question for the system posed by Chorfi. The proof uses a positive matrix integral representation to establish finite-time invertibility and uniform bounds for all large spatial frequencies.