A simple but effective upper bound technique is to have a quantity expressed  by a Cauchy integral, choose an optimal contour $\gamma$, and proceed by inserting the absolute values along $\gamma$. For example, a clean  estimate $\log{\binom{a}{b}} \leq b \log(b/a) + (a-b)\log(1-b/a)$ on the binomial coefficients follows from expressing 
$$
\binom{a}{b} = \Big| \int_{|z| = R} \frac{(1-z)^a}{z^b} \, d\theta \Big| \leq R^{-b}
(1+R)^a
$$
 and noting that the bound is optimized for the choice $R = b/(a-b)$. This is the same idea as in Lucia's solution to your problem, with the added point that the choice of the contour is on  disposal. Many papers on diophantine approximations involve this idea in extrapolation arguments, e.g. Gelfond's solution to Hilbert's 7th problem.