In the theory of concentration inequalities, the function
$$ h_1(u) = 1+u - \sqrt{1+2u} $$
plays an important role as the Fenchel–Legendre conjugate of $\psi(t) = \tfrac{t^2}{2(1-t)}$. In their monograph, Boucheron, Lugosi, and Massart choose the suggestive name $h_1$ for this function. (Curiously, this function is not the limit as $p\uparrow 1$ of $h_p$ defined by the same authors, nor does it have any obvious relation to the function $h(u) = (1+u)\log (1+u) - u$ also defined.)
Unfortunately, it is hard for me to learn more about this function as I don't know if it has an accepted name for me to type into google. This leads to my question:
Does the function $h_1$ have an accepted name?
Beyond the name, I am also interested in a conceptual/operational interpretation of $h_1$. For instance, $h_p(x)$ for $0<x,p < 1$ is defined by Boucheron, Lugosi, and Massart to be the KL divergence between the distributions of Bernoulli variables with success probabilities $x$ and $p$. Does $h_1$ (which again I note is confusingly not $\lim_{p\uparrow 1} h_p(x)$) have a similar interpretation?