What is the justification for this inequality?

[quasar987]

What is the justification for this inequality?

$$\int_0^1(|u(s)|+|v(s)|)(|u(s)|-|v(s)|)ds\leq \left(\int_0^1(|u(s)|^2+|v(s)|^2)ds\right)^{1/2}\left(\int_0^1(|u(s)-v(s)|^2)ds\right)^{1/2}$$

where u and v are complex-valued square-integrable Riemann integrable functions on [0,1].

Thx.

 

[bham10246]

This reminds me of Holder's inequality, and the right hand side of the inequality is the L^2 norm. Go to Wikipedia and given p and q, your p and q equal 2.