Is there any continuous function that satisfies the following conditions? Is there any continuous function  $y=f(x)$ on $[0,1]$ that satisfies the following conditions?
$$
f(0) = f(1) = 0,
$$
and
$$ 
f(a)^2-f(2a)f(b)+f(b)^2<0,
$$
for some $0<a<b<1$.
I tried to test with several functions (with different $a,b$) but non of them satisfied. 
Any help is appreciated. Thank you in advance.

All you have to do is choose numbers $a,b$ so that $0\lt a\lt 2a\lt b\lt1$ (or $0\lt a\lt b\lt2a\lt1$), then find numbers $y_1,y_2,y_3$ satisfying $y_1^2-y_2y_3+y_3^2\lt0,$ and then construct a continuous function (e.g. a fourth degree polynomial) $f(x)$ with $f(0)=f(1)=0,\ f(a)=y_1,\ f(2a)=y_2,\ f(b)=y_3.$ For example:
$$f(x)=x(x-1)(4x-1)(8x-7),\ a=\frac14,\ b=\frac34$$
