Can I minimize $x^a + y^a$ by minimizing $x+y$? Is there such a mathematical identity?

Question

I would like to minimize the following expression: $$x^a+y^a.$$

I wonder if a mathematical identity exists where a minimization of $x+y$ implies a minimization of the above.

Where:

• a is a positive constant,
• $x > 0$,
• $y > 0$

It's my first attempt at posting a question and appreciate any help! Cheers

Answer 1

For $a\geq 1$ you can use $$\frac{x^a+y^a}{2}\geq\left(\frac{x+y}{2}\right)^a,$$ which is Jensen for $f(x)=x^a$.