Corollary VIII.1 on pg. 549 of Flajolet and Sedgewick's book "Analytic Combinatorics" states:
(Saddle-point bounds for generating functions). Let $G(z)$, not a polynomial, be analytic at $0$ with non-negative coefficients and radius of convergence $R \leq + \infty$. Assume that $G(R^-)=+\infty$. Then one has \begin{equation} [z^n]G(z) \leq \frac{G(\zeta)}{\zeta^n}, \quad \text{with } \zeta \in (0,R) \text{ the unique root of} \quad \zeta \frac{G'(\zeta)}{G(\zeta)}=n+1. \end{equation} They prove this by Cauchy's integral formula, using a contour that passes through a saddle point of the surface $\left |\frac{G(z)}{z^{n+1}} \right |$, being the modulus of the integrand in that formula. This makes sense to me.
My question is about the following more naive analysis, built with ingredients borrowed from the proof of Corollary VIII.1, that seems to avoid consideration of saddle points, and yet to give an upper bound on $[z^n]G(z)$ that cannot be worse than that given by Corollary VIII.1:
I believe that it follows from Cauchy's Integral Formula that \begin{equation} [z^n]G(z) = \frac{1}{2 \pi i} \int_{\gamma} \frac{G(z)}{z^{n+1}} dz, \end{equation} where $\gamma$ is any circle of radius $r$ centred on the origin, with $r$ less than the radius of convergence of $G(z)$.
Since $G(z)$ has non-negative real coefficients, I believe that \begin{equation} \sup_{z \in \gamma} |G(z)| = G(r), \end{equation} because equality is achieved in the triangle inequality, when all the summands have the same argument.
Now by the usual bound on the modulus of the integral, (which is another consequence of the triangle inequality) we have \begin{equation} \begin{aligned} % hspace inserted because otherwise [z^n] did not appear. \hspace{1mm} [z^n]G(z) &= |[z^n]G(z)| \\ &\leq \frac{1}{2 \pi} L(\gamma) \sup_{z \in \gamma} \left |\frac{G(z)}{z^{n+1}} \right | \\ &= r G(r)/r^{n+1} \\ &= G(r) / r^n. \end{aligned} \end{equation} If we minimise this upper bound with respect to $r$, we find that $r$ should solve the equation \begin{equation} r G'(r)/G(r) = n, \end{equation} not the equation in Corollary VIII.1, which follows from considering stationary points of the integrand, namely \begin{equation} r G'(r)/G(r) = n+1. \end{equation} It seems that we avoided considering saddle points, and that the bound we obtained cannot be worse than any found by considering saddle points, because it is given by the minimum of the function used as the upper bound in the Corollary, that is, $G(r)/r^n$. It is proved in Note VIII.4 on pg. 550 that the function $G(r)/r^n$ is upwardly convex, hence this stationary point is a minimum, and is unique.
Have I made a mistake ? Are the two upper bounds in fact equal ? If not, which is less ?
I am sure that I must have missed the point, made a mistake in my calculation, failed to notice a typo in the book, or some combination of those errors.
I have read a bit further in the book, and have come to Note VIII.5 on pg. 550, which describes the simpler proof (that avoids discussion of saddle points) as a "minor optimization".