What would give tight upper and lower bounds to the two $\alpha$ ($\alpha\in\mathbb{R}^+$) solutions of the following equation?
$$a\alpha-b\ln{\Gamma(\alpha)}-c=0$$
Background
I am working with a parameter, $\alpha$, whose (log) marginal posterior is in the form
$$a\alpha-b\ln{\Gamma(\alpha)}$$
The other parameters are gamma-distributed, but I am unable to marginalize more than one at a time, so I am using a slice-within-Gibbs approach. This is working well, but I am trying to speed it up for a general-use application. I have a couple ideas, one of which is to improve the sampling on the horizontal segment in the final step of the slice sampling algorithm. A lot of computation goes into checking that the sampling endpoints are outside of the horizontal segment. If I had a good upper and lower bound for the endpoints, it would provide a significant speedup.
Due to the nature of the slice sampler, the values of $a$, $b$, and $c$ guarantee two real values of $\alpha$ that satisfy the equation.