= Adjustment coefficient for shape-one-half gamma claims
{title2=$R=\beta\frac{\rho-3+\sqrt{(\rho+9)(\rho+1)}}{2(1+\rho)},\quad\beta=1/(2\mu)$}
The <gamma distribution> of shape one half and rate $\beta$ has <moment-generating function> $(1-r/\beta)^{-1/2}$. With mean $\mu=1/(2\beta)$ and positive <relative safety loading> $\rho$, write $x=R/\beta$. Dividing out the zero root after squaring the adjustment equation gives $(1+\rho)^2x^2-(1+\rho)(\rho-3)x-4\rho=0$. Its unique positive root lies between zero and one, so it belongs to the transform domain and gives the display.
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