**Appendix A**

**Proof of Theorem 1.** Consider the function

$$\varphi(z) = \mu \sum\_{k=1}^{M} \sum\_{n=1}^{N} c\_{kn} q\_{kn} (z[sh])^{c\_{kn}} \,. \tag{A1}$$

which appears on the left-hand side of Equation (19). Taking advantage of the obvious inequalities,

$$
\varphi\_{-}(z) = MN\mu c\_{\min} q\_{\min}(z)^{\varepsilon\_{\min}} \\
<\varphi(z) \\
$$

The variables are 0 < *cmin* < 1, 0 < *cmax* < 1, *cmin* < *cmax*, and *cmin* < *ckn* < *cmax*. Consider the equations

$$
\varrho\_{-}(z) = T[sh], \quad \varrho(z) = T[sh], \quad \varrho\_{+}(z) = T[sh]. \tag{A3}
$$

The functions *ϕ*−(*z*), *ϕ*(*z*), and *ϕ*+(*z*) are strictly convex. Therefore, the solutions of these equations has the relationship

$$z\_- < z^\* < z\_{+"} \tag{A4}$$

which concludes the proof of Theorem 1.
