7
$\begingroup$

I am trying to find a simple method that does not use the tools of advanced differential calculus to find following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function $f:z \mapsto |z^3 + 2iz|$ from $\mathbb C$ to $\mathbb C$

$ \large { \displaystyle \max_{z \in {\mathbb C},|z| \leq 1} |z^3 +2i z |} $ Since : $(\forall z \in \Delta) \quad f(z) \leq 3 $ is obtained using triangular inequality, we can yet try to find some $z_0 \in {\mathbb C}$ such that $f(z_0)=3$

Does anybody have an idea?

Thanks.

  • 0
    A very, very simple method is asking [Wolfram](http://tinyurl.com/cp7jsor).2012-06-09

3 Answers 3

5

Hint: when do you get equality in the triangle inequality?

  • 0
    Well done! ${}$2012-06-09
5

If you've begun a study of complex functions, you may have seen the Maximum Modulus Principle. Since $z^3 + 2iz$ is a polynomial and entire (analytic in the complex plane), the maximum of $|z^3 + 2iz|$ you seek must occur on the boundary of the unit disk. Gerry's Hint then quickly points you in the right direction!

  • 0
    Ah, drat, you type faster +12012-06-09
3

Hint: $z^3 + 2iz$ is differentiable on $\mathbb{C}$ (i.e. holomorphic) so you can apply the maximum modulus principle, and deduce that the maximum of $f$ lies on the boundary of $\Delta$, which has a simple parametrisation, so you can use standard techniques from real one-variable calculus to find the maximum.

(A slightly different approach to Gerry Myerson's, much more complicated in this case but also far more general.)

  • 0
    @ hardmath and dbaupp : Thank you for the idea of maximum module principle.I was looking for a suitable method for beginning students. It turns out that the Pricipe gives the idea already! Thank you!2012-06-09