Prove or disprove this argument

Prove or disprove this argument

Let $L>0$ and let $\Omega$ be the set of all integrable functions from
$[0,L]$ to $]0,+\infty[$.
For all $\varphi, \psi \in \Omega$ define $\left \langle \varphi,\psi
\right \rangle:=\int_{0}^{L}\varphi(x)\psi(x)dx$.
Let $f,h\in \Omega$ such that $\left \langle f,h \right
\rangle=\frac{1}{2}L^{2}$. Also, consider the set
$\omega(k):=\{g\in \Omega\colon \left \langle \textbf 1,g \right
\rangle=1\wedge\left \langle h,g \right \rangle=k>0\}$,
where $\textbf 1\in \Omega$ is the function that maps everything to $1$.
Note that $f$ or $h$ are not $\textbf 1$.
Is it true that for every $k\in]0,+\infty[$:
$\displaystyle \frac{\max \limits_{\large g\in \omega(k)}\left \langle f,g
\right \rangle}{\min \limits_{\large g\in \omega(k)}\left \langle f,g
\right \rangle}$ is constant, that is, does there exist $\alpha \in \Bbb
R$ such that $\displaystyle \frac{\max \limits_{\large g\in
\omega(k)}\left \langle f,g \right \rangle}{\min \limits_{\large g\in
\omega(k)}\left \langle f,g \right \rangle}=\alpha$.