Suppose $f:\mathbb{D}\rightarrow\mathbb{C}$ is analytic such that $|f|<2$. Also, suppose that $f''(0)=4i$. Find the value of $f(i/2)$.

Question

Question: Suppose $f:\mathbb{D}\rightarrow\mathbb{C}$ is analytic such that $|f|<2$. Also, suppose that $f''(0)=4i$. Find the value of $f(i/2)$.

Thoughts: If this was the first derivative, I feel like I could just use Schwarz-Pick, but being given the second derivative is throwing me off a bit. I still feel like it is the "right track", because we are given the second derivative of $f$ at $0$, so it is (at least to me) suggesting something with Schwarz Lemma, but I can't quite get anything to come out nicely. Any thoughts? Thank you!

I wonder if C-R could be used since we know its analytic. Like working backwards and we know if $f=u+iv$ then $u_x=v_y$ and $u_y=-v_x$ — MyMathYourMath, Oct 12 '21 at 19:38
I wonder if Schwarz Lemma can be generalized to any maximum on the modulus of $f$ — MyMathYourMath, Oct 12 '21 at 19:39
I thought of that, but I wasn't quite able to make anything work using C-R equations.
Ahhh, I hadn't thought of trying to use maximum modulus principle here... — User7238, Oct 12 '21 at 19:41
wait, if its $\mathbb{C}-$linear, you can pill the $i$ out and maybe you can use the hypothesis to get a hold of the real and imaginary parts? like if $z=x+iy$ then $x^2+y^2<4$ — MyMathYourMath, Oct 12 '21 at 19:43

score 4 · Accepted Answer · answered Oct 12 '21 at 19:55

4

One possible solution is via Parseval's identity for holomorphic functions (see for example Proof of Parseval's identity):

Let $f(z) = \sum_{n=0}^\infty a_n z^n$ be the Taylor series of $f$ in $\Bbb D$, then $$ \sum_{n=0}^\infty |a_n|^2 r^{2n} = \frac{1}{2\pi}\int_0^{2\pi}|f(re^{it})|^2 \, dt \le 4 \, . $$ Taking the limit $r \to 1$ gives $$ \sum_{n=0}^\infty |a_n|^2 \le 4 \, . $$ But $|a_2|^2 = |f''(0)/2|^2 = 4$, so that all other Taylor coefficients must be zero.

It follows that $f(z) = 2i z^2$.

answered Oct 12 '21 at 19:55

Martin R

128,226

so you're using the fact that since $f$ is analytic it has a convergent power series who converges on the open unit disk? ahhhhh and made use of coefficients!! nicely done!!!! – MyMathYourMath Oct 12 '21 at 19:56
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@HossienSahebjame: Yes, the Taylor series of a holomorphic function $f$ converges in the largest open disk contained in the domain of $f$. – Martin R Oct 12 '21 at 19:59
This is pretty. Thank you, as always, @MartinR, I really appreciate it! – User7238 Oct 12 '21 at 22:13
@MartinR just out of curiosity, Hossein mentioned above the idea of using maximum modulus here. I haven't been able to get it to come out nicely. Do you see a way to use that?
On the other hand, using Parseval's identity here is very nice! Answer accepted! – User7238 Oct 18 '21 at 10:35
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@User7238: I don't know. I may be possible, but I haven't found a solution using the MMP yes. – Martin R Oct 18 '21 at 11:23
Interesting. This is an old qualifier problem from my school from years ago. I know folks now run into Parseval's identity, so I am familiar with it, but it is not formally discussed in the class now. So, I suppose I am just trying to figure out whether it 1) was dealt with more back then or 2) they see a different solution Nonetheless, I really appreciate, as always, all your input! – User7238 Oct 18 '21 at 13:16

Suppose $f:\mathbb{D}\rightarrow\mathbb{C}$ is analytic such that $|f|<2$. Also, suppose that $f''(0)=4i$. Find the value of $f(i/2)$.

1 Answers1