# Asymptotic behavior of logarithm and some univalent functions

For $$z$$ near infinity (sufficiently large $$z$$), I want to verify the following claims :

1. If $$f$$ is univalent (an analytic injective function) such that $$f(z)=z+O(1/z)$$, then $$f^{-1}(z)=z+O(1/z)$$.

2. $$\log|z+O(1)|=\log|z|+o(1)$$ (How does this identity involve both Big-Oh and little-oh?)

Well, I try to using power series expansion for $$\log(1+x)$$ but it does not seem work.

For 1, let $$|z|\ge R>0$$ with $$R$$ large tbd and $$w=f(z)$$

$$|w-z| \le C/|z|$$ means $$|w| \le |z|+ C/|z| \le 2|z|$$ as well as $$|w| \ge |z|- C/|z| \ge |z|/2$$ if we choose $$R^2 \ge 2C$$ hence we have $$|w-z| \le 2C/|w|, |z| >R$$ as above and $$O(1/w)=O(1/z)$$

Since $$z=f^{-1}(w)$$ we get $$|w-f^{-1}(w)|=O(1/w)$$ or $$f^{-1}(w)=w+O(1/w)$$ Changing variables back to $$z$$ we are done!

For 2 we have $$\log|z+O(1)|=\log |z|+\log |(1+O(1)/z)|$$

But $$O(1)/z=o(1)$$ for $$|z|$$ large and $$\log |1+x| \le 2|x|, |x| \le 1/2$$, so $$\log |(1+O(1)/z)|=\log |(1+o(1)|=o(1)$$ and putting all together we get

$$\log|z+O(1)|=\log |z|+o(1)$$ so we are done!

• Thanks, I have a few things to clarify. First, |w|>|z|-C/|z|=(|z|^2-C)/|z|>C/|z| so how can you end up with |z|/2 on the second inequality. Well, I think you are using the reverse triangle inequality to get |w-z|<2C/|w| but I cannot see how. Finally, for #2 I guess you use log|1+x|<2|x|, |x|<1/2 (I trust this from the graph). Does it matter to check that |o(1)|<1/2? Well, from the definition we cannot tell, so I'm curious how do we properly apply the inequality. – Nothingone May 20 at 4:03
• if $|z|>R, R^2>2C$ then $|z|- C/|z| \ge |z|/2$ by a simple check (pass $|z|/2$ to the left, $-C/|z|$ to the right, simplify etc); for the last $o(1)$ means a small quantity that goes to zero in absolute value) so it is automatically less than any fixed $\delta >0$ so in particular $1/2$; the logarithmic inequality follows by the Taylor series immediately by majorizing that with a geometric series- if you are not sure about these notations, try and redo this with limits and inequalities; it's messier and longer but it's worth doing it at least once to undertsand the usage of $O,o$; – Conrad May 20 at 11:31