**Proposition 3.**

*(i) Suppose that* −1 < *ρ* < 0*. For μ*1 < *μ*2 *we have*

$$\mathcal{g}(\mathbf{t}\_0) = \mathcal{g}\_A(\mathbf{t}\_{A\prime}\mathbf{s}\_A) = 4(\mu\_2 + (1 - 2\rho)\mu\_1)\mu\_2$$

*where,* (*tA*,*sA*)=(*tA*(*ρ*),*sA*(*ρ*)) := <sup>1</sup>−2*<sup>ρ</sup> μ*1 , 1 *<sup>μ</sup>*2−2*μ*1*<sup>ρ</sup> is the unique minimizer of g*(*<sup>t</sup>*,*<sup>s</sup>*),(*<sup>t</sup>*,*<sup>s</sup>*) ∈ (0, <sup>∞</sup>)<sup>2</sup>*. For μ*1 = *μ*2 =: *μ we have*

$$\mathcal{g}(t\_0) = \mathcal{g}\_A(t\_{A'}s\_A) = \mathcal{g}\_B(t\_{B'}s\_B) = 8(1-\rho)\mu\_r$$

*where* (*tA*,*sA*) = <sup>1</sup>−2*<sup>ρ</sup> μ* , 1 (<sup>1</sup>−2*ρ*)*<sup>μ</sup>* ∈ *<sup>A</sup>*,(*tB*,*sB*) := 1 (<sup>1</sup>−2*ρ*)*μ* , 1−2*ρ μ* ∈ *B are the only two minimizers of g*(*<sup>t</sup>*,*<sup>s</sup>*),(*<sup>t</sup>*,*<sup>s</sup>*) ∈ (0, <sup>∞</sup>)<sup>2</sup>*.*

*(ii) Suppose that* 0 ≤ *ρ* < *ρ*ˆ1*. We have*

$$\mathcal{g}(\mathbf{t}\_0) = \mathcal{g}\_A(t\_{A\prime}\mathbf{s}\_A) = 4(\mu\_2 + (1 - 2\rho)\mu\_1)\mu\_2$$

*where* (*tA*,*sA*) ∈ *A is the unique minimizer of g*(*<sup>t</sup>*,*<sup>s</sup>*),(*<sup>t</sup>*,*<sup>s</sup>*) ∈ (0, <sup>∞</sup>)<sup>2</sup>*. (iii)Supposethatρ*=*ρ*ˆ1*.Wehave*

$$\lg(\mathbf{t}\_0) = \lg\_A(\mathbf{t}\_{A\prime}\mathbf{s}\_A) = 4(\mu\_2 + (1 - 2\rho)\mu\_1)\_{\rho}$$

*where* (*tA*,*sA*)=(*tA*(*ρ*ˆ1),*sA*(*ρ*ˆ1)) = (*t*<sup>∗</sup>(*ρ*ˆ1),*s*<sup>∗</sup>(*ρ*ˆ1)) ∈ *L, is the unique minimizer of g*(*<sup>t</sup>*,*<sup>s</sup>*),(*<sup>t</sup>*,*<sup>s</sup>*) ∈ (0, <sup>∞</sup>)<sup>2</sup>*, with* (*t*<sup>∗</sup>,*s*<sup>∗</sup>) *defined in* (6)*.*

*(iv) Suppose that ρ*ˆ1 < *ρ* < *ρ*ˆ2*. We have*

$$\lg(\mathbf{t}\_0) = \lg\_A(\mathbf{t}^\*, \mathbf{s}^\*) = \lg\_L(\mathbf{t}^\*) = \frac{2}{1+\rho}(\mu\_1 + \mu\_2 + 2/\mathbf{t}^\*),$$

*where* (*t*<sup>∗</sup>,*s*<sup>∗</sup>) ∈ *L is the unique minimizer of g*(*<sup>t</sup>*,*<sup>s</sup>*),(*<sup>t</sup>*,*<sup>s</sup>*) ∈ (0, <sup>∞</sup>)<sup>2</sup>*. (v) Suppose that ρ* = *ρ*ˆ2*. We have <sup>t</sup>*<sup>∗</sup>(*ρ*ˆ2) = *s*<sup>∗</sup>(*ρ*ˆ2) = 1/*μ*2 *and*

$$\lg(\mathbf{t}\_0) = \lg\_A(1/\mu\_2, 1/\mu\_2) = \lg\_L(1/\mu\_2) = \lg\_2(1/\mu\_2) = 4\mu\_2.$$

*where the minimum of g*(*<sup>t</sup>*,*<sup>s</sup>*),(*<sup>t</sup>*,*<sup>s</sup>*) ∈ (0, ∞)<sup>2</sup> *is attained at* (1/*μ*2, 1/*μ*2)*, with g*3(1/*μ*2, 1/*μ*2) = *g*2(1/*μ*2) *and* 1/*μ*2 *is the unique minimizer of g*2(*s*),*<sup>s</sup>* ∈ (0, <sup>∞</sup>)*.*

*(vi) Suppose that ρ*ˆ2 < *ρ* < 1*. We have*

$$\lg(t\_0) = \lg(1/\mu\_2) = 4\mu\_{2'} $$

*where the minimum of g*(*<sup>t</sup>*,*<sup>s</sup>*),(*<sup>t</sup>*,*<sup>s</sup>*) ∈ (0, ∞)<sup>2</sup> *is attained when g*(*<sup>t</sup>*,*<sup>s</sup>*) = *g*2(*s*)*.*

**Remark 4.** *In case that μ*1 = *μ*<sup>2</sup>*, we have ρ*ˆ1 = 0, *ρ*ˆ2 = 1 *and thus scenarios (ii) and (vi) do not apply.*
