Let $y,z\in(0,1)^n$ satisfy $||y||_1 = ||z||_1=1$.

Then $$ \frac{||z||_3}{||z||_2} \le K_n ||z/y||_\infty \frac{||y||_3}{||y||_2} $$ where $z/y\in\mathbb R^n$ is the coordinate-wise quotient of $z$ and $y$ and $K_n>0$ is some constant depending only on $n$. What's the best $K_n$ that makes this true? What if $2,3$ are replaced by general $p,q$?