Let $f(x)$ be a continuous function over a compact set X, and $0<C_1\leq f(x)\leq C_2$ on this set. Suppose $C_2$ is much larger than $C_1$, i.e., $C_2\gg C_1$.
Let $\zeta$ be a reference measure supported on X. For example, we can take $\zeta$ as the Lebegue measure. Then $\mathbb{E}_{x\sim \zeta}f(x)=\int_{X} f(x) dx/|X|$.
I want to give a bound of $$ \frac{\mathrm{Var}_{x\sim \zeta}(f(x))}{(\mathbb{E}_{x\sim \zeta}f(x))^2}, $$ or equivalently, $$ \frac{\mathbb{E}_{x\sim \zeta}(f^2(x))}{(\mathbb{E}_{x\sim \zeta}f(x))^2}. $$
Simplicity, we can obtain the bound $\frac{(C_2-C_1)^2}{4C_1^2}$ of the first term by using some well-known inequalies.
My question is, can we obtain a bound in the complexity of $O(1)$, which is somehow irrelative to $C_1$ and $C_2$? For example, $(C_1-C_2)^2/(C_2+C_1)^2$ is $O(1)$, and $\frac{(C_2-C_1)^2}{4C_1^2}$ is $O(C_2^2/C_1^2)$.
Let $p = C_2/(C_1 + C_2) \in (0, 1)$. Taking $\zeta$ such that $f$ is $C_1$ on a set of measure $p$ and is $C_2$ on a set of measure $1- p$ (and $\zeta(X) = 1$), then $E(f^2)/E(f)^2 = (C_1 + C_2)^2/4C_1C_2$ (equivalently $Var(f)/E(f)^2 = (C_2 - C_1)^2/4C_1C_2$ as in the bound you mention), which is $\Theta(C_2/C_1)$ when $C_1/C_2 = O(1)$. So you cannot have a bound that is $O(1)$ unless $C_2 = O(C_1)$ as well.