A New Bound on the Capacity of the Binary Deletion Channel with High Deletion Probabilities

Let $C(d)$ be the capacity of the binary deletion channel with deletion probability $d$. It was proved by
Drinea and Mitzenmacher that, for all $d$,
$C(d)/(1-d)\geq 0.1185
$. Fertonani and Duman recently showed that $\limsup_{d\to 1}C(d)/(1-d)\leq 0.49$. In this paper, it is proved that
$\lim_{d\to 1}C(d)/(1-d)$ exists and is equal to $\inf_{d}C(d)/(1-d)$. This result suggests the conjecture that the curve $C(d)$ my be convex in the interval $d\in [0,1]$. Furthermore,
using currently known bounds for $C(d)$, it leads to the upper bound $\lim_{d\to 1}C(d)/(1-d)\leq 0.4143$.