*3.3. Detailed Analyses of the Exact Recursive Values, i.e., for the Cases* (*β*A, *β*H, *α*A, *α*H) ∈ P*NI* ∪ P*SP,1*

In the no-immigration-case (*β*A, *β*H, *α*A, *α*H) ∈ PNI and in the equal-fraction-case (*β*A, *β*H, *α*A, *α*H) ∈ PSP,1, the Hellinger integral can be calculated exactly in terms of *Hλ*(*P*A,*n*||*P*H,*n*) = *Vλ*,*X*0,*<sup>n</sup>* (cf. (39)), as proposed in part (a) of Theorem 1. This quantity depends on the behaviour of the sequence *a* (*q E λ* ) *n n*∈N , with *q E λ* := *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>gt; 0, and of the sum *α*A *β*A ∑ *n k*=1 *a* (*q E λ* ) *k n*∈N . The last expression is equal to zero on PNI. On PSP,1, this sum is unequal to zero. Using Lemma A1 we conclude that *q E <sup>λ</sup>* < *β<sup>λ</sup>* (resp. *q E <sup>λ</sup>* > *βλ*) iff *λ* ∈]0, 1[ (resp. *λ* ∈ R\[0, 1]), since on PNI ∪ PSP,1 there holds *<sup>β</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>β</sup>*H. Thus, from Properties <sup>1</sup> (P1) we can see that the sequence *a* (*q E λ* ) *n n*∈N is strictly negative, strictly decreasing and it converges to the unique solution *x* (*q E λ* ) <sup>0</sup> ∈] − *βλ*, *q E <sup>λ</sup>* − *βλ*[ of the Equation (44) if *<sup>λ</sup>* <sup>∈</sup>]0, 1[. For *<sup>λ</sup>* <sup>∈</sup> <sup>R</sup>\[0, 1], (P3) implies that the sequence *a* (*q E λ* ) *n n*∈N is strictly positive, strictly increasing and converges to the smallest positive solution *x* (*q E λ* ) <sup>0</sup> ∈]0, − log(*q E λ* )] of the Equation (44) in case that (P3a) is satisfied, otherwise it diverges to ∞. Thus, we have shown the following detailed behaviour of Hellinger integrals:

**Proposition 2.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*NI*×]0, 1[ *and all initial population sizes X*<sup>0</sup> ∈ N *there holds*

$$H(a) = H\_{\lambda}(P\_{\mathcal{A},1}||P\_{\mathcal{H},1}) = \exp\left\{ \left(\beta\_{\mathcal{A}}^{\lambda}\beta\_{\mathcal{H}}^{1-\lambda} - \lambda\beta\_{\mathcal{A}} - (1-\lambda)\beta\_{\mathcal{H}}\right)X\_0 \right\} < 1,$$

(*b*) *the sequence* (*Hλ*(*P*A,*n*||*P*H,*n*)) *n*∈N *given by*

$$H\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) = \exp\left\{a\_n^{(q\_{\lambda}^{\mathbb{E}})} \mathbf{X}\_0\right\} =: V\_{\lambda, \mathbf{X}\_0, n}$$

*is strictly decreasing,*

$$\mu(\mathcal{C}) \qquad \lim\_{n \to \infty} H\_{\lambda}(P\_{\mathcal{A},n} || P\_{\mathcal{H},n}) \; = \; \exp \left\{ \mathbf{x}\_0^{(q\_{\lambda}^{\mathbb{E}})} X\_0 \right\} \in ]0, 1[\; \lambda]$$

$$(d) \qquad \lim\_{n \to \infty} \frac{1}{n} \log H\_{\lambda}(P\_{\mathcal{A},n} || P\_{\mathcal{H},n}) = 0$$

(*e*) *the map X*<sup>0</sup> 7→ *Vλ*,*X*0,*<sup>n</sup> is strictly decreasing*.

**Proposition 3.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*NI* × (R\[0, 1]) *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with q<sup>E</sup> λ* := *β λ* A *β* 1−*λ* H

$$I(\boldsymbol{a}) = \inf\_{\boldsymbol{\mathcal{H}}, \boldsymbol{\lambda}} (\mathcal{P}\_{\mathcal{A}, 1} || \mathcal{P}\_{\mathcal{H}, 1}) \\ = \exp\left\{ \left( \mathcal{P}\_{\mathcal{A}}^{\lambda} \mathcal{P}\_{\mathcal{H}}^{1-\lambda} - \mathcal{P}\_{\lambda} \right) \cdot \mathcal{X}\_0 \right\} \\ > 1, \boldsymbol{\lambda}$$

(*b*) *the sequence* (*Hλ*(*P*A,*n*||*P*H,*n*)) *n*∈N *given by*

$$H\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) = \exp\left\{a\_{n}^{(q\_{\lambda}^{\mathbb{E}})} \cdot \mathcal{X}\_{0}\right\} =: V\_{\lambda, \mathcal{X}\_{0}, n}$$

*is strictly increasing,*

$$(\mathbf{c}) \qquad \lim\_{\boldsymbol{\mathfrak{u}} \to \infty} H\_{\boldsymbol{\lambda}}(\mathbf{P}\_{\boldsymbol{\lambda}, \boldsymbol{\mathfrak{u}}} || \mathbf{P}\_{\boldsymbol{\mathfrak{u}}, \boldsymbol{\mathfrak{u}}}) = \begin{cases} \; \exp \left\{ \mathbf{x}\_{0}^{(q\_{\boldsymbol{\lambda}}^{\mathbb{E}})} \cdot \mathbf{X}\_{0} \right\} > 1, & \text{if} \quad \boldsymbol{\lambda} \in [\boldsymbol{\lambda}\_{-}, \boldsymbol{\lambda}\_{+}] \backslash [0, 1], \\\; \text{\textbf{\reflectbox{ $\mathfrak{u}$ }}} & \text{if} \quad \boldsymbol{\lambda} \in [-\infty, \boldsymbol{\lambda}\_{-}] \cup [\boldsymbol{\lambda}\_{+}, \infty[ , \boldsymbol{\lambda}\_{-}]] \end{cases}$$

$$(d) \qquad \lim\_{n \to \infty} \frac{1}{n} \log H\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) \quad = \begin{cases} 0, & \text{if } \quad \lambda \in [\lambda\_{-}, \lambda\_{+}] \backslash [0, 1],\\ \infty, & \text{if } \quad \lambda \in [-\infty, \lambda\_{-}] \cup [\lambda\_{+}, \infty], \end{cases}$$

(*e*) *the map X*<sup>0</sup> 7→ *Vλ*,*X*0,*<sup>n</sup> is strictly increasing*.

In the case (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP,1, the sequence *a* (*q E λ* ) *n n*∈N under consideration is formally the same, with the parameter *q E λ* := *β λ* A *β* 1−*λ* <sup>H</sup> <sup>&</sup>gt; 0. However, in contrast to the case <sup>P</sup>NI, on <sup>P</sup>SP,1 both the sequence *a* (*q E λ* ) *n n*∈N and the sum *α*A *β*A ∑ *n k*=1 *a* (*q E λ* ) *k n*∈N are strictly decreasing in case that *λ* ∈]0, 1[, and strictly increasing in case that *λ* ∈ R\[0, 1]. The respective convergence behaviours are given in Properties 1 (P1) and (P3). We thus obtain

**Proposition 4.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,1*×]0, 1[ *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with q<sup>E</sup> λ* := *β λ* A *β* 1−*λ* H

$$\begin{aligned} (a) \qquad &H\_{\lambda}(P\_{\mathcal{A},1}||P\_{\mathcal{H},1}) = \exp\left\{ \left(\mathcal{P}\_{\mathcal{A}}^{\lambda}\mathcal{J}\_{\mathcal{H}}^{1-\lambda} - \beta\_{\lambda}\right) \cdot \left(X\_0 + \frac{a\_{\mathcal{A}}}{\beta\_{\mathcal{A}}}\right) \right\} < 1, \\ \ldots & \quad \cdot \end{aligned}$$

$$\begin{aligned}(b) \quad &\quad \text{the sequence } \left(H\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n})\right)\_{n\in\mathbb{N}} \text{ given by} \\ &\quad H\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) = \exp\left\{a\_{n}^{(q\_{\lambda}^{\mathbb{E}})} \cdot \mathbf{X}\_{0} + \frac{\mathfrak{a}\_{\mathcal{A}}}{\mathfrak{P}\_{\mathcal{A}}} \sum\_{k=1}^{n} a\_{k}^{(q\_{\lambda}^{\mathbb{E}})}\right\} =: V\_{\lambda, \mathbf{X}\_{0}, n}, \\ &\quad \text{is strictly decreasing} \end{aligned}$$

*is strictly decreasing,*

$$(c) \qquad \lim\_{n \to \infty} H\_{\lambda}(P\_{\mathcal{A},n} || P\_{\mathcal{H},n}) = \begin{array}{c} 0 \end{array}$$

$$\mu(d) \quad \lim\_{\substack{n \to \infty \ \mathsf{T} \ \mathsf{T}}} \frac{1}{n} \log H\_{\mathsf{A}}(P\_{\mathsf{A},n} || P\_{\mathsf{H},n}) \; = \; \frac{\mathsf{a}\_{\mathsf{A}}}{\mathsf{\mathsf{P}}\_{\mathsf{A}}} \cdot \mathbf{x}\_{0}^{(q\_{\mathsf{A}}^{\mathsf{E}})} < \; \mathsf{0} \; \mathsf{A}$$

(*e*) *the map X*<sup>0</sup> 7→ *Vλ*,*X*0,*<sup>n</sup> is strictly decreasing*.

**Proposition 5.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ P*SP,1* × (R\[0, 1]) *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with q<sup>E</sup> λ* := *β λ* A *β* 1−*λ* H

$$\begin{aligned} (a) \qquad &H\_{\lambda}(P\_{\mathcal{A},1}||P\_{\mathcal{H},1}) = \exp\left\{ \left(\mathcal{P}\_{\mathcal{A}}^{\lambda}\mathcal{A}\_{\mathcal{H}}^{1-\lambda} - \beta\_{\lambda}\right) \cdot \left(X\_0 + \frac{a\_{\mathcal{A}}}{\beta\_{\mathcal{A}}}\right) \right\} > 1, \\ (b) \qquad &the \text{ sequence } (H\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}))\_{n \in \mathbb{N}} \text{ given by} \end{aligned}$$

$$\begin{aligned}(b) \qquad & \quad \text{the sequence } \left(H\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n})\right)\_{n\in\mathbb{N}} \text{ given by} \\ & \quad H\_{\lambda}(P\_{\mathcal{A},n}||P\_{\mathcal{H},n}) = \exp\left\{a\_{n}^{(q\_{\lambda}^{\mathbb{E}})} \cdot \mathbf{X}\_{0} + \frac{\mathbf{a}\_{\mathcal{A}}}{\beta\_{\mathcal{A}}} \sum\_{k=1}^{n} a\_{k}^{(q\_{\lambda}^{\mathbb{E}})}\right\} =: \quad V\_{\lambda, \mathbf{X}\_{0}, n} \end{aligned}$$

*is strictly increasing,*

$$(c) \qquad \lim\_{n \to \infty} H\_{\lambda}(P\_{\mathcal{A},n} || P\_{\mathcal{H},n}) \ = \infty,$$

$$(d) \qquad \lim\_{n \to \infty} \frac{1}{n} \log H\_{\lambda}(P\_{\mathcal{A}, \mathfrak{n}} || P\_{\mathcal{H}, \mathfrak{n}}) = \begin{cases} \frac{a\_{\mathcal{A}}}{\mathcal{P}\_{\mathcal{A}}} \cdot x\_{0}^{(q\_{\lambda}^{\mathbb{E}})} > 0, & \text{if} \quad \lambda \in [\lambda\_{-}, \lambda\_{+}] \backslash [0, 1],\\\infty, & \text{if} \, \lambda \in ]-\infty, \lambda\_{-}[\cup ]\lambda\_{+}, \infty[\lrcorner, \infty]. \end{cases}$$

(*e*) *the map X*<sup>0</sup> 7→ *Vλ*,*X*0,*<sup>n</sup> is strictly increasing*.

Due to the nature of the equal-fraction-case PSP,1, in the assertions (a), (b), (d) of the Propositions 4 and 5, the fraction *α*A/*β*<sup>A</sup> can be equivalently replaced by *α*H/*β*H.

**Remark 2.** *For the (to our context) incompatible setup of GWI with Poisson offspring but nonstochastic immigration of constant value 1, an "analogue" of part (d) of the Propositions 4 resp. 5 was established in Linkov & Lunyova [53].*

*3.4. Some Preparatory Basic Facts for the Remaining Cases* (*β*A, *β*H, *α*A, *α*H) ∈ P*SP*\P*SP,1*

The bounds *B L λ*,*X*0,*n* , *B U λ*,*X*0,*n* for the Hellinger integral introduced in formula (40) in Theorem 1 can be chosen arbitrarily from a (*p L λ* , *q L λ* , *p U λ* , *q U λ* )-indexed set of context-specific parameters satisfying (34), or equivalently (35).

In order to derive bounds which are optimal, with respect to goals that will be discussed later, the following monotonicity properties of the sequences *a* (*q*) *n n*∈N and *b* (*p*,*q*) *n n*∈N (cf. (36), (37)) for general, context-independent parameters *q* and *p*, will turn out to be very useful:

#### **Properties 2.**

*(P10) For* <sup>0</sup> <sup>≤</sup> *<sup>q</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>q</sup>*<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup> *there holds a*(*q*<sup>1</sup> ) *<sup>n</sup>* < *a* (*q*2) *<sup>n</sup> for all n* ∈ N*.*

$$\text{(P11)}\quad \text{For each fixed } q \ge 0 \text{ and } 0 \le p\_1 < p\_2 < \infty \text{ there holds } b\_n^{(p\_1, q)} < b\_n^{(p\_2, q)}, \text{ for all } n \in \mathbb{N}.$$



*(P14) For arbitrary* 0 < *p*1, *p*<sup>2</sup> *and* 0 ≤ *q*1, *q*<sup>2</sup> ≤ min{1,*e <sup>β</sup>λ*−1} *suppose that* log(*p*1) + *<sup>x</sup>* (*q*<sup>1</sup> ) <sup>0</sup> < log(*p*2) + *x* (*q*2) 0 *. Then there holds*

$$p\_1 \cdot e^{x\_0^{(q\_1)}} - a\_\lambda = \lim\_{n \to \infty} \frac{1}{n} \sum\_{k=1}^n b\_k^{(p\_1, q\_1)} < \lim\_{n \to \infty} \frac{1}{n} \sum\_{k=1}^n b\_k^{(p\_2, q\_2)} = p\_2 \cdot e^{x\_0^{(q\_2)}} - a\_\lambda \cdot e$$

From (P10) to (P12) one deduces that both sequences *a* (*q*) *n n*∈N and *b* (*p*,*q*) *n n*∈N are monotone in the general parameters *p*, *q* ≥ 0. Thus, for the upper bound of the Hellinger integral *B U λ*,*X*0,*n* we should use nonnegative context-specific parameters *p U <sup>λ</sup>* = *p <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) and *<sup>q</sup> U <sup>λ</sup>* = *q <sup>U</sup>* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) which are as small as possible, and for the lower bound *<sup>B</sup> L <sup>λ</sup>*,*X*0,*<sup>n</sup>* we should use nonnegative context-specific parameters *p L <sup>λ</sup>* = *p L* (*β*A, *β*H, *α*A, *α*H, *λ*) and *q L <sup>λ</sup>* = *q L* (*β*A, *β*H, *α*A, *α*H, *λ*) which are as large as possible, of course, subject to the (equivalent) restrictions (34) and (35).

To find "optimal" parameter pairs, we have to study the following properties of the function *φλ*(·) = *φ*(·, *β*A, *β*H, *α*A, *α*H, *λ*) defined on [0, ∞[ in (30) (which are also valid for the previous parameter context (*β*A, *β*H, *α*A, *α*H) ∈ (PNI ∪ PSP,1)):

#### **Properties 3.**

*(P15) One has*

$$\phi\_{\lambda}(\mathbf{x}) = (\mathbf{a}\_{\mathcal{A}} + \beta\_{\mathcal{A}}\mathbf{x})^{\lambda} \left( \mathbf{a}\_{\mathcal{H}} + \beta\_{\mathcal{H}}\mathbf{x} \right)^{1-\lambda} - \lambda (\mathbf{a}\_{\mathcal{A}} + \beta\_{\mathcal{A}}\mathbf{x}) + (1-\lambda)(\mathbf{a}\_{\mathcal{H}} + \beta\_{\mathcal{H}}\mathbf{x}) \left\{ \begin{array}{c} \le 0, \quad \text{if } \lambda \in [0, 1], \\\\ \ge 0, \quad \text{if } \lambda \in \mathbb{R}, [0, 1], \end{array} \right\}$$

*where equality holds iff f*A(*x*) = *f*H(*x*) *for some x* ∈ [0, ∞[ *iff x* = *x* ∗ := *α*A−*α*<sup>H</sup> *β*H−*β*<sup>A</sup> ∈ [0, ∞[ *.*

*(P16) There holds*

$$\phi\_{\lambda}(0) = \mathfrak{a}\_{\mathcal{A}}^{\lambda} \mathfrak{a}\_{\mathcal{H}}^{1-\lambda} - \mathfrak{a}\_{\lambda} \begin{cases} \leq 0, & \text{if } \lambda \in ]0, \mathbf{1}[\mathsf{L}], \\ \geq 0, & \text{if } \lambda \in \mathbb{R} \backslash [0, 1]. \end{cases}$$

*with equality iff α*<sup>A</sup> = *α*<sup>H</sup> *together with β*<sup>A</sup> 6= *β*<sup>H</sup> *(cf. Lemma A1).*

*(P17) For all λ* ∈ R\{0, 1} *one gets*

$$\phi'\_{\lambda}(\mathbf{x}) = \lambda \beta\_{\mathcal{A}} \left( f\_{\mathcal{A}}(\mathbf{x}) \right)^{\lambda - 1} \left( f\_{\mathcal{H}}(\mathbf{x}) \right)^{1 - \lambda} + (1 - \lambda) \beta\_{\mathcal{H}} \left( f\_{\mathcal{A}}(\mathbf{x}) \right)^{\lambda} \left( f\_{\mathcal{H}}(\mathbf{x}) \right)^{-\lambda} - \beta\_{\lambda} \dots$$

*(P18) There holds*

$$\lim\_{\lambda \to \infty} \phi'\_{\lambda}(\mathbf{x}) = \beta\_{\mathcal{A}}^{\lambda} \beta\_{\mathcal{H}}^{1-\lambda} - \beta\_{\lambda} \begin{cases} \le 0, & \text{if } \lambda \in ]0, 1[\\ \ge 0, & \text{if } \lambda \in \mathbb{R} \backslash [0, 1]\_{\star} \end{cases}$$

*with equality iff β*<sup>A</sup> = *β*<sup>H</sup> *together with α*<sup>A</sup> 6= *α*<sup>H</sup> *(cf. Lemma A1).*

*(P19) There holds*

$$\mathfrak{q}\_{\lambda}^{\prime\prime}(\mathbf{x}) = -\lambda(1-\lambda) \left( f\_{\mathcal{A}}(\mathbf{x}) \right)^{\lambda-2} \left( f\_{\mathcal{H}}(\mathbf{x}) \right)^{-\lambda-1} \left( \mathfrak{a}\_{\mathcal{A}} \mathfrak{f}\_{\mathcal{H}} - \mathfrak{a}\_{\mathcal{H}} \mathfrak{f}\_{\mathcal{A}} \right)^{2} \left\{ \begin{array}{l} \leq 0, \quad \text{if } \lambda \in [0, 1], \\\geq 0, \quad \text{if } \lambda \in \mathbb{R} \left\langle [0, 1], \frac{1}{2} \right\rangle \end{array} \right\}$$

*with equality iff* (*β*A, *β*H, *α*A, *α*H) ∈ (P*NI* ∪ P*SP,1*)*. Hence, for* (*β*A, *β*H, *α*A, *α*H) ∈ P*SP*\P*SP,1, the function φ<sup>λ</sup> is strictly concave (convex) for λ* ∈]0, 1[ *(λ* ∈ R\[0, 1]*). Notice that φ* 0 *λ* (0) = *λβ*<sup>A</sup> *α*A *α*H *λ*−<sup>1</sup> + (1 − *λ*)*β*<sup>H</sup> *α*A *α*H *λ* − *β<sup>λ</sup> can be either negative (e.g., for the setup* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (4, 2, 3, 1, 0.5) *,* (4, 2, 5, 1, 2) *, or zero (e.g., for* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (4, 2, 4, 1, 0.5),(4, 2, 3, 1, 2) *), or positive (e.g., for* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (4, 2, 5, 1, 0.5)*,*(4, 2, 2, 1, 2) )*, where the exemplary parameter*

*constellations have concrete interpretations in our running-example epidemiological context of Section 2.3. Accordingly, for λ* ∈]0, 1[*, due to concavity and (P17), the function φλ*(·) *can be either strictly decreasing, or can obtain its global maximum in* ]0, ∞[*, or–only in the case β*<sup>A</sup> = *β*H*—can be strictly increasing. Analogously, for λ* ∈ R\[0, 1]*, the function φλ*(·) *can be either strictly increasing, or can obtain its global minimum in* ]0, ∞[*, or–only in the case β*<sup>A</sup> = *β*H*—can be strictly decreasing. (P20) For all λ* ∈ R\{0, 1} *one has*

$$\begin{split} & \lim\_{\mathbf{x} \to \mathbf{x}} \left( \boldsymbol{\phi}\_{\lambda} (\mathbf{x}) - (\widetilde{r}\_{\lambda} + \widetilde{s}\_{\lambda} \, \mathbf{x}) \right) = \; \mathbf{0} \; \mathsf{A} \\ & \quad \text{for} \quad \widetilde{r}\_{\lambda} := \; \widetilde{p\_{\lambda}} - \mathfrak{a}\_{\lambda} := \; \lambda \mathsf{a}\_{\mathcal{A}} \left( \frac{\boldsymbol{\beta}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{H}}} \right)^{\lambda - 1} + (1 - \lambda) \mathsf{a}\_{\mathcal{H}} \left( \frac{\boldsymbol{\beta}\_{\mathcal{A}}}{\boldsymbol{\beta}\_{\mathcal{H}}} \right)^{\lambda} - \mathsf{a}\_{\lambda} \\ & \quad \text{and} \quad \widetilde{s}\_{\lambda} := \; \widetilde{q}\_{\lambda} - \boldsymbol{\beta}\_{\lambda} := \; \boldsymbol{\beta}\_{\mathcal{A}}^{\lambda} \boldsymbol{\beta}\_{\mathcal{H}}^{1 - \lambda} - \boldsymbol{\beta}\_{\lambda} \; . \end{split}$$

*The linear function <sup>φ</sup>*f*λ*(*x*) :<sup>=</sup> *<sup>r</sup>*e*<sup>λ</sup>* <sup>+</sup> *<sup>s</sup>*e*<sup>λ</sup>* · *<sup>x</sup> constitutes the* asymptote *of <sup>φ</sup>λ*(·)*. Notice that if <sup>β</sup>*<sup>A</sup> <sup>=</sup> *<sup>β</sup>*<sup>H</sup> *one has* <sup>e</sup>*s<sup>λ</sup>* <sup>=</sup> <sup>0</sup> <sup>=</sup> <sup>e</sup>*rλ; if <sup>β</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>β</sup>*<sup>H</sup> *we have* <sup>e</sup>*s<sup>λ</sup>* <sup>&</sup>lt; <sup>0</sup> *in the case <sup>λ</sup>* <sup>∈</sup>]0, 1[ *and* <sup>e</sup>*s<sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup> *if <sup>λ</sup>* <sup>∈</sup> <sup>R</sup>\[0, 1]*. Furthermore, <sup>φ</sup>λ*(0) <sup>&</sup>lt; *<sup>r</sup>*e*<sup>λ</sup> if <sup>λ</sup>* <sup>∈</sup>]0, 1[ *and <sup>φ</sup>λ*(0) <sup>&</sup>gt; *<sup>r</sup>*e*<sup>λ</sup> if <sup>λ</sup>* <sup>∈</sup> <sup>R</sup>\[0, 1]*, (cf. Lemma A1(c1) and (c2)). If <sup>α</sup>*<sup>A</sup> <sup>=</sup> *<sup>α</sup>*<sup>H</sup> *(and thus <sup>β</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>β</sup>*H*), then the intercept <sup>r</sup>*e*<sup>λ</sup> is strictly positive if λ* ∈]0, 1[ *resp. strictly negative if λ* ∈ R\[0, 1]*. In contrast, for the case α*<sup>A</sup> 6= *α*H*, the intercept <sup>r</sup>*e*<sup>λ</sup> can assume any sign, take e.g.,* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) ∈ {(3.7, 0.9, 2.0, 1.0, 0.5),(4, 2, 1.6, 1, 2)} *for <sup>r</sup>*e*<sup>λ</sup>* <sup>&</sup>gt; <sup>0</sup>*,* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) ∈ {(3.6, 0.9, 2.0, 1.0, 0.5),(4, 2, 1.5, 1, 2)} *for <sup>r</sup>*e*<sup>λ</sup>* <sup>=</sup> <sup>0</sup>*, and* (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H, *<sup>λ</sup>*) ∈ {(3.5, 0.9, 2.0, 1.0, 0.5),(4, 2, 1.4, 1, 2)} *for <sup>r</sup>*e*<sup>λ</sup>* <sup>&</sup>lt; <sup>0</sup>*; again, the exemplary parameter constellations have concrete interpretations in our running-example epidemiological context of Section 2.3.*

The properties (P15) to (P20) above describe in detail the characteristics of the function *φλ*(·) = *φ*(·, *β*A, *β*H, *α*A, *α*H, *λ*). In the previous parameter setup PNI ∪ PSP,1, this function is linear, which can be seen from (P19). In the current parameter setup PSP\PSP,1, this function can basically be classified into four different types. From (P16) to (P20) it is easy to see that for all current parameter constellations the particular choices

$$p^A\_\lambda := \mathfrak{a}^\lambda\_\mathcal{A} \mathfrak{a}^{1-\lambda}\_\mathcal{H} > 0, \qquad q^A\_\lambda := \mathfrak{f}^\lambda\_\mathcal{A} \mathfrak{f}^{1-\lambda}\_\mathcal{H} > 0,\tag{45}$$

which correspond to the following choices in (35)

$$r\_{\lambda}^{A} := a\_{\mathcal{A}}^{\lambda} a\_{\mathcal{H}}^{1-\lambda} - a\_{\lambda} \le 0 \quad (resp. \ge 0), \qquad s\_{\lambda}^{A} := \mathcal{S}\_{\mathcal{A}}^{\lambda} \mathcal{S}\_{\mathcal{H}}^{1-\lambda} - \mathcal{S}\_{\lambda} \le 0 \quad (resp. \ge 0),$$

– where *A* = *L* (resp. *A* = *U*)–lead to the tightest lower bound *B L λ*,*X*0,*n* (resp. upper bound *B U λ*,*X*0,*n* ) for *Hλ*(*P*A,*n*||*P*H,*n*) in (40) in the case *λ* ∈]0, 1[ (resp. *λ* ∈ R\[0, 1]). Notice that for the previous parameter setup (*β*A, *β*H, *α*A, *α*H) ∈ (PNI ∪ PSP,1) these choices led to the exact values of the Hellinger integral and to the simplification *p E λ* /*q E λ* · *β<sup>λ</sup>* − *α<sup>λ</sup>* = 0, which implies *b* (*p E λ* ,*q E λ* ) *<sup>n</sup>* = (*α*A/*β*A) · *a* (*q E λ* ) *<sup>n</sup>* . In contrast, in the current parameter setup (*β*A, *β*H, *α*A, *α*H) ∈ PSP\PSP,1 we only derive the *optimal* lower (resp. upper) bound for *λ* ∈]0, 1[ (resp. *λ* ∈ R\[0, 1]) by using the parameters *p A λ* , *q A λ* for *A* = *L* (resp. *A* = *U*) and *p A λ* /*q A λ* · *β<sup>λ</sup>* − *α<sup>λ</sup>* 6= 0. For a better distinguishability and easier reference we thus stick to the *L*−notation (resp. *U*−notation) here.

## *3.5. Lower Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[

The discussion above implies that the lower bound *B L λ*,*X*0,*n* for the Hellinger integral *Hλ*(*P*A,*n*||*P*H,*n*) in (40) is optimal for the choices *p L λ* , *q L <sup>λ</sup>* > 0 defined in (45). If *β*<sup>A</sup> 6= *β*H, due to Properties <sup>1</sup> (P1) and Lemma A1, the sequence *a* (*q L λ* ) *n n*∈N is strictly negative and strictly decreasing and converges to the unique negative solution of the Equation (44). Furthermore, due to (P5), the sequence *b* (*p L λ* ,*q L λ* ) *n n*∈N , as defined in (37), is strictly decreasing. Since *b* (*p L λ* ,*q L λ* ) <sup>1</sup> = *p L <sup>λ</sup>* − *α<sup>λ</sup>* ≤ 0 by Lemma A1, with equality iff *<sup>α</sup>*<sup>A</sup> <sup>=</sup> *<sup>α</sup>*H, the sequence *b* (*p L λ* ,*q L λ* ) *n n*∈N is also strictly negative (with the exception *b* (*p L λ* ,*q L λ* ) <sup>1</sup> = <sup>0</sup> for *α*<sup>A</sup> = *α*H) and strictly decreasing. If *β*<sup>A</sup> = *β*<sup>H</sup> and thus *α*<sup>A</sup> 6= *α*H, due to (P2), (P6) and Lemma A1, there holds *a* (*q L λ* ) *<sup>n</sup>* ≡ 0 and *b* (*q L λ* ) *<sup>n</sup>* ≡ *p L <sup>λ</sup>* − *α<sup>λ</sup>* < 0. Thus, analogously to the cases PNI ∪ PSP,1 we obtain

**Proposition 6.** *For all* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[ *and all initial population sizes X*<sup>0</sup> ∈ N *there holds with p<sup>L</sup> λ* := *α λ* A *α* 1−*λ* H , *q L λ* := *β λ* A *β* 1−*λ* H


$$\mathcal{B}^{L}\_{\lambda, \mathbf{X}\_{0:\mathcal{U}}} = \ \exp\left\{ a\_{\mathbf{n}}^{(q\_{\lambda}^{L})} \cdot \mathbf{X}\_{0} + \frac{p\_{\lambda}^{L}}{q\_{\lambda}^{L}} \sum\_{k=1}^{n} a\_{k}^{(q\_{\lambda}^{L})} + n \cdot \left(\frac{p\_{\lambda}^{L}}{q\_{\lambda}^{L}} \cdot \boldsymbol{\beta}\_{\lambda} - \boldsymbol{a}\_{\lambda}\right) \right\} \quad \text{is strictly decreasing.}$$

(*c*) lim*n*→<sup>∞</sup> *B L <sup>λ</sup>*,*X*0,*<sup>n</sup>* = 0 ,

$$\mu(d) = \lim\_{n \to \infty} \frac{1}{n} \log B^{L}\_{\lambda, X\_{0}, n} = \frac{p\_{\lambda}^{L}}{q\_{\lambda}^{L}} \cdot \left(\mathbf{x}\_{0}^{(q\_{\lambda}^{L})} + \boldsymbol{\beta}\_{\lambda}\right) - \boldsymbol{a}\_{\lambda} = \left. p\_{\lambda}^{L} \cdot e^{\mathbf{x}\_{0}^{(q\_{\lambda}^{L})}} - \boldsymbol{a}\_{\lambda} < \mathbf{0} \right.$$

$$\text{(\(e\))}\qquad \text{the map}\quad \mathcal{X}\_0 \mapsto \mathcal{B}^{L}\_{\lambda, \mathcal{X}\_0, \mathfrak{n}} \quad \text{is strictly decreasing.}$$

*3.6. Goals for Upper Bounds for the Cases* (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (P*SP*\P*SP,1*)×]0, 1[

For parameter constellations (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (PSP\PSP,1)×]0, 1[, in contrast to the treatment of the lower bounds (cf. the previous Section 3.5), the fine-tuning of the *upper bounds* of the Hellinger integrals *Hλ*(*P*A,*n*||*P*H,*n*) is much more involved. To begin with, let us mention that the monotonicity-concerning Properties 2 (P10) to (P12) imply that for a tight upper bound *B U λ*,*X*0,*n* (cf. (40)) one should choose parameters *p U <sup>λ</sup>* ≥ *p L <sup>λ</sup>* > 0, *q U <sup>λ</sup>* ≥ *q L <sup>λ</sup>* > 0 as small as possible. Due to the concavity (cf. Properties 3 (P19)) of the function *φλ*(·), the linear upper bound *φ U λ* (·) (on the ultimately relevant subdomain N0) thus must hit the function *φλ*(·) in at least one point *x* ∈ N0, which corresponds to some "discrete tangent line" of *φλ*(·) in *x*, or in at most two points *x*, *x* + 1 ∈ N0, which corresponds to the secant line of *φλ*(·) across its arguments *x* and *x* + 1. Accordingly, there is in general *no overall best* upper bound; of course, one way to obtain "good" upper bounds for *Hλ*(*P*A,*n*||*P*H,*n*) is to solve the optimization problem

$$\left(\overline{p\_{\boldsymbol{\lambda}}^{\rm II}}, \overline{q\_{\boldsymbol{\lambda}}^{\rm II}}\right) := \operatorname\*{\arg\min}\_{\left(p\_{\boldsymbol{\lambda}}^{\rm II}, q\_{\boldsymbol{\lambda}}^{\rm II}\right)} \left\{ \exp\left\{a\_{\boldsymbol{n}}^{\left(q\_{\boldsymbol{\lambda}}^{\rm II}\right)} \cdot \mathbf{X}\_{\boldsymbol{0}} + \sum\_{k=1}^{n} b\_{k}^{\left(p\_{\boldsymbol{\lambda}}^{\rm II}, q\_{\boldsymbol{\lambda}}^{\rm II}\right)}\right\},\tag{46}$$

subject to the constraint (35). However, the corresponding result generally depends on the particular choice of the initial population *X*<sup>0</sup> ∈ N and on the observation time horizon *n* ∈ N. Hence, there is in general no overall optimal choice of *p U λ* , *q U <sup>λ</sup>* without the incorporation of further goal-dependent constraints such as lim*n*→<sup>∞</sup> *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* = 0 in case of lim*n*→<sup>∞</sup> *Hλ*(*P*A,*n*||*P*H,*n*) = 0. By the way, mainly because of the non-explicitness of the sequence *a* (*q U λ* ) *n n*∈N (due to the generally not explicitly solvable recursion (36)) and the discreteness of the constraint (35), this optimization problem seems to be not straightforward to solve, anyway. The choice of parameters *p U λ* , *q U λ* for the upper bound *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* ≥ *Hλ*(*P*A,*n*||*P*H,*n*) can be made according to different, partially incompatible ("optimality-" resp. "goodness-") criteria and goals, such as:


Further goals–with which we do not deal here for the sake of brevity–are for instance (i) a very good tightness of the upper bound *B U λ*,*X*0,*n* for *n* ≥ *N* for some fixed large *N* ∈ N, or (ii) the criterion (G1) with *fixed* (rather than arbitrary) initial population size *X*<sup>0</sup> ∈ N.

Let us briefly discuss the three Goals (G1) to (G3) and their challenges: due to Theorem 1, Goal (G1) can only be achieved if the sequence *a* (*q U λ* ) *n n*∈N is non-increasing, since otherwise, for each fixed observation horizon *n* ∈ N there is a large enough initial population size *X*<sup>0</sup> such that the upper bound component *B*e (*p U λ* ,*q U λ* ) *λ*,*X*0,*n* becomes larger than 1, and thus *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* = <sup>1</sup> (cf. (40)). Hence, Properties <sup>1</sup> (P1) and (P2) imply that one should have *q U <sup>λ</sup>* <sup>≤</sup> *<sup>β</sup>λ*. Then, the sequence *b* (*p U λ* ,*q U λ* ) *n n*∈N is also non-increasing. However, since *b* (*p U λ* ,*q U λ* ) *<sup>n</sup>* might be positive for some (even all) *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, the sum ∑ *n k*=1 *b* (*p U λ* ,*q U λ* ) *k n*∈N is not necessarily decreasing. Nevertheless, the restriction

$$q\_{\lambda}^{\text{II}} - \beta\_{\lambda} \le 0 \quad \text{and} \quad p\_{\lambda}^{\text{II}} - a\_{\lambda} \le 0,\qquad \text{where at least one of the inequalities is strict},\tag{47}$$

ensures that both sequences *a* (*q U λ* ) *n n*∈N and *b* (*p L λ* ,*q U λ* ) *n n*∈N are nonpositive and decreasing, where at least one sequence is strictly negative, implying that the sum ∑ *n k*=1 *b* (*p U λ* ,*q U λ* ) *k n*∈N is strictly negative for *n* ≥ 2 and strictly decreasing. To see this, suppose that (47) is satisfied with two strict inequalities. Then, *a* (*q U λ* ) *n n*∈N as well as *b* (*p L λ* ,*q U λ* ) *n n*∈N are strictly negative and strictly decreasing. If *q U <sup>λ</sup>* = *β<sup>λ</sup>* and *p U <sup>λ</sup>* < *αλ*, we see from (P2) and (P6) that *a* (*q U λ* ) *<sup>n</sup>* ≡ 0 and that *b* (*p U λ* ,*q U λ* ) *<sup>n</sup>* ≡ *p U <sup>λ</sup>* − *α<sup>λ</sup>* < 0 (notice that *α<sup>λ</sup>* = 0 is not possible in the current setup PSP\PSP,1 and for *λ* ∈]0, 1[). In the last case *q U <sup>λ</sup>* < *β<sup>λ</sup>* and *p U <sup>λ</sup>* <sup>=</sup> *<sup>α</sup>λ*, from (P1) and (P5) it follows that *a* (*q U λ* ) *n n*∈N is strictly negative and strictly decreasing, as well as that *b* (*p U λ* ,*q U λ* ) <sup>1</sup> <sup>=</sup> <sup>0</sup> and *b* (*p L λ* ,*q U λ* ) *n n*∈N is strictly decreasing and strictly negative for *n* ≥ 2. Thus, whenever (47) is satisfied, the sum ∑ *n k*=1 *b* (*p U λ* ,*q U λ* ) *k n*∈N is strictly negative for *n* ≥ 2 and strictly decreasing.

To achieve Goal (G2), we have to require that the sequence *a* (*q U λ* ) *n n*∈N converges, which is the case if either *q U <sup>λ</sup>* ≤ *β<sup>λ</sup>* or *β<sup>λ</sup>* < *q U <sup>λ</sup>* ≤ min{1,*e <sup>β</sup>λ*−1} (cf. Properties <sup>1</sup> (P1) to (P3)). From the upper bound component *B*e (*p U λ* ,*q U λ* ) *λ*,*X*0,*n* (42) we conclude that Goal (G2) is met if the sequence *b* (*p U λ* ,*q U λ* ) *n n*∈N converges to a negative limit, i.e., lim*n*→<sup>∞</sup> *b* (*p U λ* ,*q U λ* ) *<sup>n</sup>* = *p U λ* · *e x* (*qU λ* ) <sup>0</sup> − *α<sup>λ</sup>* < 0. Notice that this condition holds true if (47) is satisfied: suppose that *q U <sup>λ</sup>* < *βλ*, then *x* (*q U λ* ) <sup>0</sup> < 0 and *p U λ* · *e x* (*qU λ* ) <sup>0</sup> − *α<sup>λ</sup>* < *p U <sup>λ</sup>* − *α<sup>λ</sup>* ≤ 0. On the other hand, if *p U <sup>λ</sup>* − *α<sup>λ</sup>* < 0, one obtains *x* (*q U λ* ) <sup>0</sup> ≤ 0 leading to *p U λ* · *e x* (*qU λ* ) <sup>0</sup> − *α<sup>λ</sup>* ≤ *p U <sup>λ</sup>* − *α<sup>λ</sup>* < 0.

The examination of Goal (G2) above enters into the discussion of Goal (G3): if the sequence *a* (*q U λ* ) *n n*∈N converges and lim*n*→<sup>∞</sup> *B U <sup>λ</sup>*,*X*0,*<sup>n</sup>* = 0, then there holds

$$\lim\_{n\to\infty} \frac{1}{n} \log \left( \mathcal{B}\_{\lambda, X\_0, n}^{\mathrm{II}} \right) \\ = \lim\_{n\to\infty} \frac{1}{n} \log \left( \hat{\mathcal{B}}\_{\lambda, X\_0, n}^{(p\_\lambda^{\mathrm{II}}, q\_\lambda^{\mathrm{II}})} \right) \\ = \left. p\_\lambda^{\mathrm{II}} \cdot e^{\mathbf{z}\_0^{(q\_\lambda^{\mathrm{II}})}} - \mathbf{z}\_\lambda \right|. \tag{48}$$

For the case (*β*A, *β*H, *α*A, *α*H, *λ*) ∈ (PSP\PSP,1)×]0, 1[, let us now start with our comprehensive investigations of the upper bounds, where we focus on fulfilling the condition (47) which tackles Goals (G1) and (G2) simultaneously; then, the Goal (G3) can be achieved by (48). As indicated above, various different parameter subcases can lead to different Hellinger-integral-upper-bound details, which we work out in the following. For better transparency, we employ the following notations (where the first four are just reminders of sets which were already introduced above)

PNI := n (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ [0, <sup>∞</sup>[ 4 : *<sup>α</sup>*<sup>A</sup> = *<sup>α</sup>*<sup>H</sup> = 0; *<sup>β</sup>*<sup>A</sup> > 0; *<sup>β</sup>*<sup>H</sup> > 0; *<sup>β</sup>*<sup>A</sup> 6= *<sup>β</sup>*<sup>H</sup> o , PSP := n (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ ]0, <sup>∞</sup>[ 4 : (*α*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>α</sup>*H) or (*β*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>β</sup>*H) or both <sup>o</sup> , P := PNI ∪ PSP, PSP,1 := (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> 6= *<sup>α</sup>*H, *<sup>β</sup>*<sup>A</sup> 6= *<sup>β</sup>*H, *α*A *β*A = *α*H *β*H , PSP,2 := { (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> = *<sup>α</sup>*H, *<sup>β</sup>*<sup>A</sup> 6= *<sup>β</sup>*<sup>H</sup> } , PSP,3 := (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> 6= *<sup>α</sup>*H, *<sup>β</sup>*<sup>A</sup> 6= *<sup>β</sup>*H, *α*A *β*A 6= *α*H *β*H = PSP,3a ∪ PSP,3b ∪ PSP,3c , PSP,3a := (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> 6= *<sup>α</sup>*H, *<sup>β</sup>*<sup>A</sup> 6= *<sup>β</sup>*H, *α*A *β*A 6= *α*H *β*H , *α*<sup>A</sup> − *α*<sup>H</sup> *β*<sup>H</sup> − *β*<sup>A</sup> ∈ ] − ∞, 0[ , PSP,3b := (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> 6= *<sup>α</sup>*H, *<sup>β</sup>*<sup>A</sup> 6= *<sup>β</sup>*H, *α*A *β*A 6= *α*H *β*H , *α*<sup>A</sup> − *α*<sup>H</sup> *β*<sup>H</sup> − *β*<sup>A</sup> ∈ ]0, ∞[\N , PSP,3c := (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> 6= *<sup>α</sup>*H, *<sup>β</sup>*<sup>A</sup> 6= *<sup>β</sup>*H, *α*A *β*A 6= *α*H *β*H , *α*<sup>A</sup> − *α*<sup>H</sup> *β*<sup>H</sup> − *β*<sup>A</sup> ∈ N , PSP,4 := { (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> 6= *<sup>α</sup>*<sup>H</sup> > 0, *<sup>β</sup>*<sup>A</sup> = *<sup>β</sup>*<sup>H</sup> } = PSP,4a ∪ PSP,4b , PSP,4a := { (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> 6= *<sup>α</sup>*<sup>H</sup> > 0, *<sup>β</sup>*<sup>A</sup> = *<sup>β</sup>*<sup>H</sup> ∈ ]0, 1[ } , <sup>P</sup>SP,4b :<sup>=</sup> { (*β*A, *<sup>β</sup>*H, *<sup>α</sup>*A, *<sup>α</sup>*H) ∈ PSP : *<sup>α</sup>*<sup>A</sup> <sup>6</sup><sup>=</sup> *<sup>α</sup>*<sup>H</sup> <sup>&</sup>gt; 0, *<sup>β</sup>*<sup>A</sup> <sup>=</sup> *<sup>β</sup>*<sup>H</sup> <sup>∈</sup> [1, <sup>∞</sup>[ } ; (49)

notice that because of Lemma A1 and of the Properties 3 (P15) one gets on the domain ]0, ∞[ the relation *φλ*(*x*) = 0 iff *f*A(*x*) = *f*H(*x*) iff *x* = *x* ∗ := *α*H−*α*<sup>A</sup> *β*A−*β*<sup>H</sup> ∈ ]0, ∞[.
