What are some important papers that use complex analytic techniques to get good bounds? The motivation behind this question is somewhat similar to that of the tricky project launched by Gowers et al, but is certainly a specialization. My work tends to rely on both exact formulae and analytic techniques, most notably from complex analysis and Fourier analysis. It appears that search engines and fora like this is a great way to discover useful exact formulae that at least partially fit the bill, but analytic techniques, especially those magic ones like stationary phase approximation, are harder to come by, perhaps due to their subtle and amorphous nature. Thus there seems a genuine need to enumerate some of the most exemplary articles that exploit these techniques, from which I can draw inspirations. 
Note that I am specifically interested in getting bounds, as opposed to exact formulae. So for instance Riemann's use of complex analytic continuation to derive the functional equation would not count, even though one could argue exact formulae are a special case of bounds. An example that helped me move forward is Nazarov's extension of Remez's inequality. Another is Lucia's solution to this problem of mine, as well as the paper he cited. 
Not to give the wrong impression, I do think most ground-breaking results, including bounds, are proved by exploiting idiosyncratic properties pertaining to the problem itself. These could be some exact formulae, some symmetries, or some geometric interpretation. However it has become clear to me that analytic number theorists often have an edge in situations involving bounds of algebraic formulae.
 A: Not quite sure if this is what you're looking for, but the following paper by Beals, Gaveau and Greiner used what is essentially the method of stationary phase to get uniform upper bounds on the subelliptic heat kernel for Heisenberg groups.  In this case an exact formula for the heat kernel was already known (as, essentially, the Fourier transform of the Mehler kernel), but since it involves an oscillatory integral, it is not so easy to read off bounds from the formula.
Many other authors have used similar techniques to get either bounds or asymptotics for heat kernels of various kinds.  There is a paper of Gaveau from 1977, studying asymptotics for the subelliptic heat kernel on the 3-dimensional Heisenberg group, that I think sort of kicked it off.

Beals, Richard; Gaveau, Bernard; Greiner, Peter C.:
  Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. 
  J. Math. Pures Appl. (9) 79 (2000), no. 7, 633–689. MR 1776501, DOI 10.1016/S0021-7824(00)00169-0

A: 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.
