Show that $3\int_{[0,1)} |f|d\lambda+4\int_{[1,2]} |f|d\lambda\leq 5\bigg( \int_{[0,2]} |f|^2d\lambda \bigg)^{1/2}$

Question

In some homework I have to show that for $f \in \mathcal{L}^2([0,2],\lambda)$ then: $$3\int_{[0,1)} |f|d\lambda+4\int_{[1,2]} |f|d\lambda\leq 5\bigg( \int_{[0,2]} |f|^2d\lambda \bigg)^{1/2}$$

But I have run out of good ideas. I tried applying Pythagoras and Minkowski but with no luck

Any hint would be appreciated

Answer 1

First, by Hoelder inequality $$ 3 \int_{[0,1)} |f| + 4 \int_1^2 |f| \le 3 (\int_{[0,1)} |f|^2 ) ^{1/2} + 4 (\int_{[1,2]} |f| ^2)^{1/2}. $$ Then by Cauchy-Schwarz on $\mathbb R^2$ $$ 3 (\int_{[0,1)} |f|^2 ) ^{1/2} + 4 (\int_{[1,2]} |f| ^2)^{1/2} = \pmatrix{3 \\ 4}^T\cdot \pmatrix{ ( \int \dots)^{1/2} \\ (\int \dots)^{1/2}}\\ \le \sqrt{3^2 + 4^2} \sqrt{ \int_{[0,1)} |f|^2 + \int_{[1,2]} |f| ^2} = 5 \sqrt{\int_{[0,2)} |f|^2} $$