Fix a $T\in\mathbb{R}$ and a $\sigma\in\mathbb(0,\infty)$ consider the set $$B=\{t\in[-T,T]:|f(\sigma+it)|>\ell\}$$ where $$f:\{s\in\mathbb{C} : \Re (s)>0\}\to\mathbb{C}$$ is a uniformly bounded and holomorphic function over the right half-plane. Set $$\sup_{s\in\{w\in\mathbb{C} : \Re (w)>0\}}|f(s)|=\lambda$$ and $\ell<\lambda$ and$$|B|=\int_{-T}^{T}\chi_B(t)\,\mathrm{d}t,$$ where $\chi_B$ is the characteristic function of B. It's easy to see that $|B|<2T$. Can we say that, for every $T\in\mathbb{R}$, there is an $\epsilon>0$, such that $$|B|>\epsilon 2T.$$
Or we need to know more about f, I know that this is true when $f$ is almost periodic.