# Maximal possible value of $|f(z)|$ given an upper bound of its value on the unit circle $|f(e^{i\theta})|\leq g(\theta)$

Let $$f(z)$$ be a function that is complex analytic in an open region containing the unit disk. Suppose we are given an upper bound of $$|f(z)|$$ on the unit circle, i.e. $$|f(e^{i\theta})|\leq g(\theta), 0\leq\theta\leq 2\pi$$. Then what is the maximal possible value of $$|f(0)|$$?

I feel this should be related to Cauchy's integral technique, since we have $$f(0)=\frac{1}{2\pi i}\oint_{|z|=1}f(z)\frac{h(z)}{z}dz,$$ where $$h(z)$$ is analytic in the closed unit disk and $$h(0)=1$$. Therefore, $$|f(0)|\leq \max_{0\leq\theta\leq2\pi}g(\theta)|h(e^{i\theta})|.$$ Hopefully by choosing an optimal $$h(z)$$ we could find a tight upper bound of $$|f(0)|$$.

Is this a well-researched problem? More generally, if we are given an upper bound of the absolute value of an analytic function $$f(z)$$ in a region or a closed curve, how could we get an optimal upper bound on $$|f(z)|$$ in the interior of this region or curve? Any useful hints or references are highly welcomed. Thanks in advance!

Assuming that $$g$$ is continuous and strictly positive, you can explicitly construct an extremal function $$f$$ maximizing $$|f(0)|$$ as follows: Find a harmonic function $$u$$ in the unit disk with boundary values $$\log g$$, let $$v$$ be its harmonic conjugate, and define $$f=e^{u+iv}$$. Then $$f$$ is analytic in the unit disk, with $$|f|=e^u = g$$ on the boundary, and $$|f(0)| = e^{u(0)} = \exp\left[\frac{1}{2\pi} \int_0^{2\pi} \log g(\theta) \, d\theta\right].$$ On the other hand, if you have any complex analytic function $$\tilde{f}$$ satisfying the given estimate, then $$\tilde{u} = \log |\tilde{f}|$$ is subharmonic in the unit disk, so you get $$\log |\tilde{f}(0)| = \tilde{u}(0) \le \frac{1}{2\pi} \int_0^{2\pi} \log g(\theta) \, d\theta = \log |f(0)|.$$ I am pretty sure you can weaken the assumptions significantly to allow $$g$$ to have zeros and discontinuities, as long as it does not get too wild.
P.S.: I just realized that you wanted $$f$$ to extend as a complex analytic function to a larger disk. The extremal function constructed above does not even necessarily extend continuously to the closed unit disk, so you will have to rescale and look at functions $$(1-\epsilon)f((1-\delta)z)$$ with small $$\delta,\epsilon > 0$$ to get arbitrarily close to the bound given above.
• Dear Prof. Geyer, thanks a lot for your answer. In my research I met a problem where $f(z)$ is analytic everywhere except on real line $(-\infty,-1]\cup [1,+\infty)$, and I know an upper bound for $|f(z)|$ when $\mathrm{Im}(z)\neq 0$. The question is how to find the Max possible value of $|f(0)|$. What is the general method/idea to solve this type of problems? What reference should I look into? Thanks in advance. – Lagrenge Feb 23 '20 at 22:24