Does the logarithm inequality extend to the complex plane? For estimates, the inequality log(y)<=y−1, y>0 is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, |log(z)|<= something for all z in bbbC Perhaps this would work?: log(z)<=\sqrt(\log^2|z|+arg(z)^2)
Does the logarithm inequality extend to the complex plane?
For estimates, the inequality is often helpful. Is there any sort of upper bound for the logarithm function in the complex plane? Specifically, something for all
Perhaps this would work?:
Answer & Explanation
For the principal branch you get the somewhat trivial inequality
Another restricted one
If one is seeking a global bound, then we have on the principal branch , where , and for
For , we have
while for , we have
A USEFUL LOCAL BOUND:
If we restrict such that for , then we can find a useful upper bound. We write such that
For example, if we take , then for
One application of the inequality is to prove that the infinite product representation for the sinc function, given by converges uniformly on compact sets.
Equipped with , we have for and
For any , there exists a number such that whenever , . And hence, the convergence of the infinite product representation of the sinc function is uniform.