Functional Analysis, $L^p$ spaces

Question

Can anyone help me finding for which $p$ the function $$f(x)=\frac{1}{x^{\alpha}+x^{\beta}} \in L^{p}(0,\infty)$$ where $0<\alpha \le \beta<\infty$ are given.

Thanks a lot.

Answer 1

Hint 1

I believe $$f \in L^p(0,\infty) \Leftrightarrow \int_0^\infty |f(x)|^p dx < \infty.$$

So you need to find such a $p$ that $$ \int_0^\infty \frac{dx}{\left( x^\alpha + x^\beta \right)^p} < \infty. $$

How would you proceed to do that? Can you get an idea about how to evaluate that integral?

Hint 2 A possibly useful transformation is $$ \int_0^\infty \frac{dx}{\left( x^\alpha + x^\beta \right)^p} = \int_0^\infty \frac{x^{-\alpha p}dx}{\left( 1 + x^{\beta-\alpha} \right)^p}, $$ and you are given that $\beta - \alpha \ge 0$.