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Find all entire functions such that $|f(z)|\leq |\sin(z)|$ for all $z\in\mathbb C$.

Half an hour ago I asked the same question for $|f(z)|\leq |z^2-1|$ here but this time I cannot say something like $$|f(z)|\leq M|z|^m$$ so I don't know how to start. The problem is that $$ \sin(z) = \frac{e^{iz}-e^{-iz}}{2i} $$ gets really large if $z$ gets large.

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Jolien
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1 Answers1

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Hint: Look at the function

$$g(z) = \frac{f(z)}{\sin z}.$$

It is entire and - what?

Bounded. Hence constant. Thus $f(z) = c\cdot\sin z$ for some $c\in\mathbb{C}$ with $\lvert c\rvert \leqslant 1$.

Generally, if you have an inequality $\lvert f(z)\rvert \leqslant C\cdot\lvert g(z)\rvert$ between two meromorphic functions $f$ and $g\not\equiv 0$ on a domain $U\subset\mathbb{C}$, the quotient $\frac{f}{g}$ is a bounded holomorphic function on $U\setminus S$, where $S$ is the union of the pole sets of $f$ and $g$ and the zero set of $g$. As the union of finitely many closed discrete subsets of $U$, it is again a closed discrete subset of $U$, and hence all points of $S$ are removable singularities of $\frac{f}{g}$. It is customary to consider these removed without changing the notation, and so $\frac{f}{g}$ is a bounded holomorphic function on $U$. If $U = \mathbb{C}$ (or if $\mathbb{C}\setminus U$ is "thin enough" - has analytic capacity $0$), it follows that $\frac{f}{g}$ is constant, i.e. $f(z) = c\cdot g(z)$ for some $c$ with $\lvert c\rvert \leqslant C$.

Daniel Fischer
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  • I thought about that too, but why is that entire? What is $g(0)$? – Jolien May 27 '14 at 15:30
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    Riemann's removable singularity theorem says it's entire. What $g(0)$ is depends on $f$, but the inequality gives a bound on the modulus. – Daniel Fischer May 27 '14 at 15:34
  • I understand, thanks. – Jolien May 27 '14 at 15:36
  • Yes, but $f(0) = 0$ alone would not be sufficient if $\sin$ had a multiple zero in $0$. But the inequality $\lvert f(z)\rvert\leqslant \lvert \sin z\rvert$ implies that $f$ vanishes of at least the same order as $\sin$ at all zeros of $\sin$. – Daniel Fischer May 27 '14 at 15:39