From definition of $o$ and $\omega$
one states that $0 < c_1\cdot g(n) < f(n)$ for $n > n_0$ and some $c_1$ and another states that $0 < f(n) < c_2\cdot g(n)$ for $n > n_1$ and some $c_2$
It seems like there is no such function that satisfies condition. But i can't show this rigorously.
Note, that the definition of $\omega(g)$ says (assuming you refer to ordinary landau notation, in which case you did not look up the definitions correctly): For every $c_1>0$, there is a $n_1$, such that for all $n > n_1$ we have $0 < c_1|g(n)| < |f(n)|$. $o(g)$ is defined similar.
For $c_1 = 2$ and $c_2 = 1$ a function $f \in \omega(g) \cap o(g)$ must satisfy $2|g(n)| < |f(n)| < |g(n)|$ for all $n > n_0$ and some $n_0$, which is clearly impossible.