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Assuming $x,y\in\mathbb C^n$,consider:

$$f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ\in\mathbb R\}$$

Which of the following is/are correct?
1.$ f(x,y)$≤$||x||^2+||y||^2+2|(x,y)|$
2. $f(x,y)$= $||x||^2+||y||^2+2Re(x,y)$
3. $f(x,y)$= $||x||^2+||y||^2+2|(x,y)|$
4. $f(x,y)$> $||x||^2+||y||^2+2|(x,y)|$

How can I solve this problem ,I am completely stuck on it . can anyone help me?thanks

1 Answers 1

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Let $x = re^{i\alpha},y=se^{i\beta}$ $$|x\exp(i\theta)+y\exp(i\phi)|^2 = |r\exp(i(\theta+\alpha))+s\exp(i(\phi+\beta))|^2$$ A little bit of manipulations give $$r^2+s^2+2rs\cos(\theta+\alpha-\phi-\beta)$$ So, the supremum is $r^2+s^2+2rs$

Can you solve it from here?