Q:

About a kind of real sequences $z_{k}$ making an implication true

What kind of real sequences $(z_{k})$ making the following implication true:
$x_{k}=O(y_{k}/z_{k})$ imply $y_{k}=O(x_{k})$
where $O$ is the big $O$ notation. Here all these sequences are never zero for all indices $k$.

A:

EDIT: New answer as the original question was modified so that the sequences could not be identically zero. 

There are no real sequences which make it true in general. For example, set
$$y_k=kx_kz_k$$
Then it is surely true that
$$x_k=O\left(\frac{y_k}{z_k}\right)$$
for any $z_k$. This is because
$$x_k\leq kx_k =1\cdot \left(\frac{kx_k z_k}{z_k}\right)=1\cdot\left(\frac{y_k}{z_k}\right)$$
is true for all $k\geq 1$. However, it is obviously false that
$$y_k=O(x_k)$$
as this would imply that there exists $c$ such that
$$y_k=kx_k\leq cx_k$$
$$k\leq c$$
for all but a finite number of $k$. This is obviously false. We conclude that there is no sequence $z_k$ such that the following implication is always true: $x_{k}=O(y_{k}/z_{k})$ implies $y_{k}=O(x_{k})$

