$$\begin{aligned}
W=V+\frac{dS}{dr^*}-S^2\,.\end{aligned}$$ We therefore see that if we can find a function $S$ such as $S(r^*=+\infty)\le 0$, $S(r^*=-\infty)\ge 0$ and $W\ge0$, we will have the left-hand side of eq. (\[eq:AB2\]) positive and hence stability of the solution.

Considering $$\begin{aligned}
S=\frac{1}{2}\frac{d \ln (C\cal F)}{dr^*}=\frac{\sqrt{AB}}{2}\Bigl(\frac{C'}{C}+\frac{\cal F'}{\cal F}\Bigr)\,,\end{aligned}$$ we find that $$\begin{aligned}
W=(l+2)(l-1)\frac{A}{C}\frac{\cal F}{\cal H}\,.\end{aligned}$$ If ${\cal F}/{\cal H}<0$, the solution is unstable. In fact, eq. (\[eq:fin\]) is necessarily negative for sufficiently large $l$ (because of $W$), while if ${\cal F}/{\cal H}>0$ and $S(r^*=+\infty)\le 0$, $S(r^*=-\infty)\ge 0$, all the terms of eq. (\[eq:fin\]) are positive for any $l$ which ensures the stability.

In conclusion, Horndeski black holes are linearly stable under odd perturbations if and only if ${\cal F}>0$, ${\cal G}>0$ and ${\cal H}>0$. Also we need the conditions $S(r^*=+\infty)\le 0$, $S(r^*=-\infty)\ge 0$ where $$\begin{aligned}
\label{eq:S-def}
S=\frac{\sqrt{AB}}{2}\Bigl(\frac{C'}{C}+\frac{\cal F'}{\cal F}\Bigr)\,.\end{aligned}$$

Application to particular models
================================

General Relativity
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We first apply the results to the most standard example to check the consistency of our calculations. In general relativity, the action can be written in the form (\[eq:action\]) with

$$K=0,~G_3=0,~G_4=\frac{1}{2},~G_5=0\,.$$

In this case, we have $${\cal F}={\cal G}={\cal H}=1\,,$$ which are all positive. The stability is therefore proved iff the S-deformation technique used is licit, 