TITLE: Bound on variance of function of a random variable
QUESTION [5 upvotes]: Suppose $0\leq X\leq 1.$ Suppose we are given that $\mathrm{Var}(X)\leq a$ where $a$ is some small constant.
What are the best upper bounds we can provide on $\mathrm{Var}(f(X))$ if 
a)  $f:[0,1]\mapsto\mathbb{R}$ is a Lipschitz function with Lipschitz constant $L$, such as say, $f(x) = x^2$ which has Lipschitz constant $2.$
b)  $f:[0,1]\mapsto\mathbb{R}$ is not Lipschitz but is a Hölder continuous function such as say $f(x) = \sqrt{x}.$
I am interested in upper bounds that go to zero as $a$ goes to zero.

REPLY [7 votes]: In any cases, we utilize the following inequality:
$$ \operatorname{Var}(f(X)) \leq \Bbb{E}[(f(X) - f(\Bbb{E}X))^2]. $$
This is easily proved from the inequality $\operatorname{Var}(Y) \leq \Bbb{E}[Y^2]$ with $Y = f(X) - f(\Bbb{E}X)$. Now assume that
$$|f(x) - f(y)| \leq C|x - y|^{\alpha} \qquad \forall x, y \in [0, 1]$$
for some $\alpha \in (0, 1]$ and $C \in (0, \infty)$. Then 
$$ \operatorname{Var}(f(X)) \leq C^2 \Bbb{E}[|X - \Bbb{E}X|^{2\alpha}]. $$
From the Jensen's inequality, we have
$$ \Bbb{E}[|X - \Bbb{E}X|^{2\alpha}] \leq \Bbb{E}[|X - \Bbb{E}X|^{2}]^{\alpha} = \operatorname{Var}(X)^{\alpha}. $$
Consequently we have
$$\operatorname{Var}(f(X)) \leq C^2 \operatorname{Var}(X)^{\alpha}. $$