# Application of Liouville's Theorem

Let $f(z)$ be an entire function such that $$|f(z)|<\frac{1}{|\text{Im}(z)|},\qquad z\in\Bbb C-\Bbb R.$$ The question asked me to prove that $f(z)=0$. At least looking at it, it really seems to have an application of Liouville's theorem lurking around somewhere, but I haven't found it.

My thoughts first led me to think about doing this by contradiction and using Picard's little theorem. So, I've considered a strip containing the real axis (say of width $2$ for simplicity). This will imply the complement maps into the unit disk, which implies that this strip has to map almost everywhere else on the complex plane. Now, on the imaginary axis, I know that $f(z)$ will vanish as $z\to\infty$, and this seems like it might be useful. I'm sort of under the impression $f(z)$ might have an essential singularity at $\infty$, which would mean $f(1/z)$ has an essential singularity at $0$. Either way, it of course can't have a pole at $\infty$ because of $f(z)$ vanishing on the imaginary axis, and if it is a removable singularity, it must be $0$, which still gives the solution. Therefore, I think it must have something to do with $f(z)$ having an essential singularity at $\infty$.

I was hoping to combine this with the above inequality and deduce a contradiction, but I haven't thought of one yet.

Any hints or suggestions? I'm studying for a prelim. exam, so this isn't homework.

Update: So, in the spirit of searching methods using the essential singularity at $z=0$, we have $$\left|f\left(\frac{1}{z}\right)\right|\leq\frac{1}{\left|\text{Im}\left(\frac{1}{z}\right)\right|}=\frac{|z|^2}{|y|}.$$ Thus, for $|y|>1$, we have $$\left|f\left(\frac{1}{z}\right)\right|<|y|\left|f\left(\frac{1}{z}\right)\right|<|z|^2.$$ Thus, if I can show that $|f(1/z)|$ is a polynomial, we'll know it can't have an essential singularity at the origin, concluding my proof.

• Do you know Jensen's inequality, the one for holomorphic functions (not for convex ones)? It's a corollary of Jensen's formula if you drop the sum over zeros. $$\log |f(a)|\le \frac{1}{2\pi}\int_0^{2\pi} \log|f(a+re^{it})|\,dt$$ May 1, 2013 at 3:52
• @user75064: Yes I know the inequality, but I am not quite sure how to apply it here. May 1, 2013 at 3:57
• Fix a real number $a$ (arbitrary) and let $r\to \infty$. May 1, 2013 at 4:06
• @user75064, no matter how large your circle is, there is always an arc with small imaginary part and there is no uniform bound on that arc.
– user27126
May 1, 2013 at 4:14
• @Sanchez Correct, but I don't need a uniform bound. See my answer below. May 1, 2013 at 4:23

Let $\sum_{n=0}^\infty a_n z^n$ be the Taylor expansion of $f(z)$. Also denote $a_{-1}=a_{-2}=0$. By definition, $$a_n=\frac{1}{2\pi i}\int_{|z|=R}\frac{f(z)}{z^{n+1}}dz,\quad\forall n\ge -2,\, \forall R>0.\tag{1}$$

Note that $\mathrm{Im} (z)=\frac{z-\bar{z}}{2i}$, so when $|z|=R$, $$\mathrm{Im} (z)=\frac{z-\frac{R^2}{z}}{2i}.\tag{2}$$ Substituting $(2)$ into $(1)$, we have $$a_{n-1}-R^2 a_{n+1}=\frac{1}{\pi}\int_{|z|=R}\frac{\mathrm{Im} (z)\cdot f(z)}{z^{n+1}}dz,\quad\forall n\ge -1,\, \forall R>0.\tag{3}$$ By the assumption of $f$ and continuity, $|\mathrm{Im} (z)\cdot f(z)|\le 1$ on $\mathbb{C}$. Then from $(3)$ we know $$|a_{n-1}-R^2a_{n+1}|\le \frac{2}{R^n},\quad\forall n\ge -1,\, \forall R>0.\tag{4}$$

Given $n\ge -1$, dividing both sides of $(4)$ by $R^2$ and letting $R\to \infty$, we have $a_{n+1}=0$. Therefore, all the coefficients of the Taylor expansion of $f$ are $0$, i.e. $f\equiv 0$.

• Nice, but do you really need induction? Letting $R\to\infty$ in (4) immediately gives $a_{n+1}=0$ and $a_{n-1}=0$. May 1, 2013 at 4:16
• @user75064: Yes, you are right. Thank you!
– 23rd
May 1, 2013 at 4:19
• @user75064: I have edited my answer based on your suggestion. Thank you again.
– 23rd
May 1, 2013 at 4:24
• I'm really not sure I follow the steps to $(3)$. Could you elaborate just a little? Thanks. May 1, 2013 at 4:26
• @Landscape: I know your proof didn't use my ideas exactly, but I didn't see any reason for the bounty to go to waste. Your approach seemed most direct, so I liked it. I'm still thinking on how to solve it using my original thoughts. I went to a professor and his thoughts matched mine identically up to this point, at which he got stuck as well. May 11, 2013 at 17:52

"Me too!" Fix an arbitrary $a\in \mathbb R$. By Jensen's inequality for every $r>0$ we have $$\begin{split} 2\pi \log|f(a)| &\le \int_0^{2\pi} \log|f(a+re^{it})|\,dt \le \int_0^{2\pi} \log(|r\sin t|^{-1})\,dt \\ &= -2\pi\log r- \int_0^{2\pi} \log(|\sin t|)\,dt \end{split}$$ The integral on the right is finite, because $\log |x|$ is integrable near $0$. Let $r\to \infty$ to conclude $f(a)=0$. Since $f$ vanishes on $\mathbb R$, it vanishes identically.

By the way, this works equally well for bounds like $|f(z)|\le 1/|\mathrm{Im}\,z|^{N}$ or even $|f(z)|\le \exp( 1/|\mathrm{Im}\,z|^{1-\epsilon})-1$. Fails with the assumption $|f(z)|\le \exp( 1/|\mathrm{Im}\,z|)-1$, probably for a good reason.

• Probably the next thing I'll attempt will be showing it for the exponential bound. Thanks! May 1, 2013 at 4:34
• This is nice, yet the "...because $\,\log|x|\,$ is integrable near zero" part is a little shaky, unless the OP knows this: $$\lim_{\epsilon\to 0^+}\int\limits_e^a\log x\,dx=\lim_{\epsilon\to 0}\left.\left(x\log x-x\right)\right|_\epsilon^a=\text{finite}$$ since $\,x\log x\xrightarrow[x\to 0^+]{}0\,$ May 1, 2013 at 11:55
• @DonAntonio: Thanks, but it seems pretty obvious. I think most calculus II students should be able to show something like that. May 1, 2013 at 14:29
• I think you're right, @Clayton . Thanks. May 1, 2013 at 18:16
• Nice application of Jensen's. Two small remarks. The first integral should have $a + r e^{it}$ instead of $re^{it}$ and in the last expression it should be $-2\pi \log(r)$.
– WimC
May 1, 2013 at 19:08

Let $w \in \mathbb{C}$ and let $R > |w|$. We first bound the analytic function $(z^2 - R^2)f(z)$ on the circle $|z| = R$.

If $|z| = R$ and $\operatorname{Re}z \ge 0$, then

$$|(z- R)f(z)| \le \frac{|z - R|}{\operatorname{Im}z} = \sec \theta$$

where $\theta \in [0, \dfrac{\pi}{4}]$ so $|(z - R)f(z)| \le \sqrt{2}$.

For $\operatorname{Re} z \le 0$ and $|z| = R$, we have

$$|(z + R)f(z)| \le \sqrt{2}$$

Hence,

$$|(z^2 - R^2)f(z)| \le 3R$$

on $|z| = R$. By the maximum modulus principle the same bound holds in the interior of the disk $|z| \le R$ and so

$$|f(w)| \le \frac{3R}{|w^2 - R^2|}$$ Letting $R \to \infty$, we get the desired result.

• Where does the $3R$ come from? May 1, 2013 at 4:38
• @Clayton, I was just wondering that myself. A sharper bound is $2\sqrt{2} R$, since the point on $|z| = R$ farthest from $\pm R$ is $\mp R$ at a distance of $2R$. It looks like this was rounded up to $3R$. May 1, 2013 at 4:41
• @Clayton $|z - R||z + R||f(z)| \le 2\sqrt{2}R$
– Ink
May 1, 2013 at 4:42
• Okay, I was wondering about that. Last question: Why is $\theta\in[0,\pi/4]$? I think I have a vague idea, just not 100% sure. May 1, 2013 at 4:50
• Nice answer! A little simplification of your argument: when $|z|=R$, $|z^2-R^2|=|z^2-z\bar{z}|=|z||z-\bar{z}|=2R|\mathrm{Im}(z)|$. @AntonioVargas, so $2R$ is a better bound. May 1, 2013 at 5:57