Upper bound for $\mathbb{E}[|X-\mu|^3]$ Let $X$ be a real-valued random variable on some probability space $(\Omega,\mathscr{F},\mathbb{P})$, and let $\mu := \mathbb{E}[X]$, where $\mathbb{E}$ is the expectation. Is it possible to find an expression or at least an upper bound for $\mathbb{E}\bigl[|X-\mu|^3\bigr]$ in terms of the mean $\mu$ and/or of the variance of $X$?
 A: The following Inequality might be a useful tool to upper bound the third absolute moment.

(Jensen's Inequality)
  If $\varphi$ is a convex function on an open interval of $L$ and $X$ is a random variable whose support is contained in $L$ with a finite expectation value, then
\begin{equation*}
    \varphi(\mathbb{E}[X])\leq \mathbb{E}[\varphi(X)].
\end{equation*}

Using the Jensen's inequality for the convex function $\varphi(x)
=x^{\frac{4}{3}}$, we get
\begin{eqnarray*}
    \mathbb{E}[|X-\mu|^3]^{\frac{4}{3}}&\leq& \mathbb{E}[(|X-\mu|^3)^{\frac{4}{3}}]\\
    \mathbb{E}[|X-\mu|^3]&\leq& \mathbb{E}[(X-\mu)^4]^{\frac{3}{4}}
\end{eqnarray*}
A: 
I don't know anything useful here, I think. I just know that $X$ is a weighted sum of binary random variables which are either $0$ or $1$.

This condition might turn out to be useful. It means that $X$ takes values in $[0,1]$ interval!
The  first bound is not in terms of $\mu$ and variance, but might be useful, depending on Your purpose.  

For any interval $[a,A]\subset \mathbb{R}$ and real number $p\ge1$, we have that
  $$\sup_{X \in \mathcal{X}[a,A]}\mathbb{E}\Bigg[\Bigg| \frac{X-\mu}{A-a} \Bigg|^p\Bigg]=K_{p}$$
  where $\mathcal{X}[a,A]$ is the set of all random variables $X$ with values in the interval $[a,A]$, and
  $$K_{p}=\sup_{x\in[0,1]}\kappa_{p}(x),$$
  where $\kappa_{p}(x) = x(1 -x)^p + (1 - x)x^p$.

For $p=3$ we get upper bound of $\frac{1}{8}$. Link to paper:
https://www.researchgate.net/publication/236032347_The_smallest_upper_bound_for_the_pth_absolute_central_moment_of_a_class_of_random_variables.
You might be also interested in bounds $(1.9)$ and $(1.10)$ from
https://arxiv.org/abs/1503.03786.
They actually use variance and smaller moments.
