Let $c_n$ be the number of self-avoiding walks in ${\mathbb Z}^2$ of length $n$. Because $c_n$ is a submultiplcative sequence ($c_{n+m} \leq c_nc_m$ for all $n, m \geq 1$), Fekete's lemma tells us that $\lim_{n \rightarrow \infty} c_n^{1/n}$ exists and equals $\inf c_n^{1/n}$. So we can define the connective constant $\mu = \lim_{n \rightarrow \infty} c_n^{1/n}$ that governs the growth rate of $c_n$, and if we happen to know a particular $c_n$ that gives a rigourous upper bound on $\mu$, namely $\mu \leq c_n^{1/n}$.
On page 10 of Madras and Slade's 1993 book "The self-avoiding walk", a better bound for $\mu$ is given: $ \mu \leq \left(\frac{c_n}{c_1}\right)^\frac{1}{n-1} ~~~~~(n \geq 2). $
This bound is attributed to Alm, but the only reference is to an unpublished manuscript of Ahlberg and Janson from 1980.
Does anyone know a good proof/reference for this bound? It would be implied by submultiplicativity of $a_n := c_{n+1}/c_1$, which in turn would be implied by the inequality $c_{n+m-1}c_1 \leq c_nc_m$ for all $n, m \geq 1$.