I'm looking for a reference for an inequality related to the "fundamental inequality" about entropy and rate of escape of random walks (on the Cayley graph of a group). Namely,

$\textbf{Question}$: Under which conditions (on the measure $P$), do the following inequalities hold: $$ l_P^2 \leq h_P \leq v l_P. $$

The second inequality (called "fundamental", due to Vershik and/or Guivarc'h, I have unfortunately no access to any of these paper) is valid for a large class of measures, but my concern is about the first one.

The above quantities are defined by: $$ H_{n,P} = -\sum_{g \in G} P^{(n)}(g) \ln P^{(n)}(g), \qquad \text{and } \qquad L_{n,P} = \sum_{g \in G} |g| P^{(n)}(g) $$ where $P^{(n)}$ is the convolution of $P$ with itself $n$ times [EDIT:and $|g|$ is the word length of $g$ for the generating set of the Cayley graph]. Then define the [asympotic] entropy, the rate of escape (or drift, or Green speed) and the exponential volume growth to be $$ h_P = \lim_{n \to \infty} \tfrac{1}{n} H_{n,P}, \qquad l_P = \lim_{n \to \infty} \tfrac{1}{n} L_{n,P}, \qquad \text{and } v = \lim_{n \to \infty} \tfrac{1}{n} \log |B_n| $$ where $|B_n|$ is the cardinality of the ball of radius $n$ around the identity.

PS: If I understood correctly, a paper of Blachère, Haïssinsky and Mathieu shows that (if $\sum_{g \in G} P^{(1)}(g) \ln P^{(1)}(g)$ is finite) $l_P=0$ is equivalent to $h_P=0$. This would be a consequence of the above inequalities, so I assume they are not too trivial in this generality.