I have some trouble with the following question:
Let
$$f(z)=\frac{\sin z}{z},\quad\text{for }z\in\mathbb{C}.$$
What is the maximum of $f$ in the closed unit disc
$$D:=\{z\in\mathbb{C}:|z|\leq 1\}?$$
Of course, we can apply the Maximum Modulus Principle to get that
$$\max_{z\in D}|f(z)|=\max_{|z|=1}\frac{|\sin z|}{|z|}=\max_{|z|=1}|\sin z|$$
But then I don't see any easy way to find the maximum of $\sin z$ on the unit circle. I ploted the graph and I found that it is
$$|\sin i|=\frac{e-e^{-1}}{2}$$
Here is my attempt to prove it:
Let $F(\theta)=|f(e^{i\theta})|^2$. Then,
\begin{align}
F(\theta) &= |\sin(\cos\theta+i\sin\theta)|^2 \\
&= |\sin(\cos\theta)\cosh(\sin\theta)+i\cos(\cos\theta)\sinh(\sin\theta)|^2 \\
&= \sin^2(\cos\theta)\cosh^2(\sin\theta)+\cos^2(\cos\theta)\sinh^2(\sin\theta) \\
&=\sin^2(\cos\theta)\cosh^2(\sin\theta)+(1-\sin^2(\cos\theta))\sinh^2(\sin\theta) \\
&=\sin^2(\cos\theta)\big(\cosh^2(\sin\theta)-\sinh^2(\sin\theta)\big)+\sinh^2(\sin\theta) \\
&= \sin^2(\cos\theta)+\sinh^2(\sin\theta)
\end{align}
Hence,
\begin{align}
F'(\theta) &= -2\sin(\cos\theta)\cos(\cos\theta)\sin\theta+2\sinh(\sin\theta)\cosh(\sin\theta)\cos\theta \\
&=\sinh(2\sin\theta)\cos\theta-\sin(2\cos\theta)\sin\theta
\end{align}
which vanishes whenever
$$\theta=\frac{n\pi}{2},\quad n\in\mathbb{Z}.$$
Then, we try and find
$$F(0)=\sin^2 1,\quad F(\pi/2)=\sinh^2 1,\quad F(\pi)=\sin^21,\quad F(3\pi/2)=\sinh^21$$
But $\sin^21<\sinh^21$ (which I am not sure how to justify with only pen and paper) so the maximum is with $\theta=\pi/2$ or $3\pi/2$. This gives the desired result.
This seems overly complicated for this kind of problem. Is there a better way?
Best Answer
The Taylor series for $\frac{\sin(z)}{z}$ has all its terms real and positive when $z=i$, and the corresponding terms are the same in magnitude for any $z$ on the unit circle, so the maximum has to occur for $z=i$.