$f(x)$ and $g(x)$ are functions $\mathbb{R} \mapsto \mathbb{R}$. Both functions are non-zero between $x \in [a, b]$ apart from at most a finite number of points.
Is it always possible to find $\alpha$ such that $$\int_{a}^{b} (f(x) - \alpha g(x))^n \,dx < \int_{a}^{b} f(x)^n \,dx$$?
$$\alpha \in \mathbb{R} , \quad n \in \mathbb Z \setminus \{0\}$$
If the answer is not "yes" what about case of $n = 2$?
Edit: the question originally used $\mathbb{C}$ in place of now $\mathbb{R}$. Also integrals are Riemann integrals.
A standard example. On $[0,1]$, let $f(x) = \cos(2\pi x)$ and $g(x) = \sin(2\pi x)$. Take $n=2$. Then for all $\alpha$ we have $$ \int_0^1\big|f(x)-\alpha g(x)\big|^2\;dx \ge \int_0^1 |f(x)|^2\;dx $$ I used absolute values in there to allow for $\alpha$ to be complex.
This is because $f$ and $g$ are orthogonal in $L^2$.