Relationship between $Var(X)$, $Var(Y)$ and $Cov(X,Y)$ for random variables with zero mean I have two correlated random variables $X$ and $Y$, both with zero mean.
Are there any relationship/constraints between $Var(X)$, $Var(Y)$ and $Cov(X,Y)$, apart from the obvious $Var(X) > 0$ and $Var(Y) > 0$, or can they assume any value, provided that the two obvious constraints are met?
And if I go to higher order moments, e.g. $E[X^3]$, $E[Y^3]$, $E[XY^2]$, $E[X^2Y]$, are there non obvious relationships/constraints between those moments, the variances and the covariance?
 A: When $\mathbb E[X]=\mathbb E[Y]=0$,
$$\text{Var}(X)=\mathbb E[X^2]\quad\text{Var}(Y)=\mathbb E[Y^2]\quad\text{Cov}(X,Y)=\mathbb E[XY]$$
and
$$2\vert\text{Cov}(X,Y)\vert\le\text{Var}(X)+\text{Var}(Y)$$
(since $\mathbb E[(X\pm Y)^2]>0$) and
$$\text{Cov}(X,Y)^2\le\text{Var}(X)\text{Var}(Y)$$
(by Jensen or Cauchy-Schwarz inequality).
When the moment generating function $\mathfrak m(t)=\mathbb E[e^{tX}]$ exists (in a neighbourhood of $0$), it expands as
$$\mathfrak m(t)=\sum_{i=0}^\infty \frac{\mathbb E[X^i]t^i}{i!}$$
which specifies constraints on the moments (as, e.g., $\mathfrak m(t)$ being positive near zero).
The most general result on moments is that

A sequence of numbers $(m_n)$ is the sequence of moments of a measure μ
if and only if a certain positivity condition is fulfilled; namely,
the Hankel matrices $H_n$, $${\displaystyle (H_{n})_{ij}=m_{i+j}}\quad 1\le i,j\le n$$
should be positive semi-definite.

A: They should be able to take any reasonable values, unless you introduce more constraints. For example consider
\begin{equation}
\begin{pmatrix} X \\ Y
\end{pmatrix} \sim N \left\{\begin{pmatrix} 0 \\ 0
\end{pmatrix}, \begin{pmatrix} \sigma_X^2 & \rho \sigma_X, \sigma_Y \\ \rho \sigma_X \sigma_Y & \sigma_Y^2
\end{pmatrix} \right\}.
\end{equation}
In this distribution, we can specify any $\sigma_X = \text{Var}(X) \geq 0$ and any $\sigma_Y = \text{Var}(Y) \geq 0$. We can also specify any $\text{Cov}(X, Y)$ such that $-\sigma_X \sigma_Y \leq \text{Cov}(X, Y) \leq \sigma_X \sigma_Y$ which is one the mathematical ("obvious"?) constraints of covariance; you can specify any valid value for the covariance by specifying an appropriate correlation coefficient, $\rho$.
For constraints on higher order moments, there are in general no "obvious" unless perhaps you have applied additional constraints to your problem.
Finally, the word "obvious" is cursed in mathematics; what is obvious to you may not be obvious to me, and vice-versa.
