Is Wirtinger's inequality valid on the space if $f$ is non-zero on the boundary?

Question

we know that $\pi^2 \int_0^a |f|^2 dx \leq a^2 \int_0^a |f'|^2 dx$ if $f$ is $C^1$ and $f(0)=f(a)=0$. I am interested in is this inequality also valid if $f(0) \neq 0$?

Answer 1

No. Let $f(x)=1-x$ and $a=1$. Then:

$$\pi^2 \int_0^1 (1-x)^2\ dx =\frac{\pi^2}{3}> \int_0^1 (-1)^2\ dx=1$$

Answer 2

No. Take $a=1$ and $f(x)=1-x$

Then $\pi^2 \int_0^1 |f|^2 dx =\frac{\pi^2}{3}$ but $ \int_0^1 |f'|^2 dx=1$