4. Given complex numbers $z_{1}$ and $z_{2}$ satisfy $\arg \frac{z_{1}}{z_{2}}=\frac{5 \pi}{6}$, and $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|$. Then the minimum value of $\frac{\left|2 z_{1}+t z_{2}\right|}{\left|z_{2}\right|}$ is $\qquad$
4. $\frac{\sqrt{3}}{3}$.

From Fig. 5, we can easily see that
$$
\begin{array}{l}
\left|\frac{z_{1}}{z_{2}}\right|=\frac{\sqrt{3}}{3} . \\
\text { Then } \frac{z_{1}}{z_{2}} \\
=\frac{\sqrt{3}}{3}\left(\cos \frac{5 \pi}{6}+\mathrm{i} \sin \frac{5 \pi}{6}\right) . \\
\text { Therefore, } \frac{\left|2 z_{1}+t z_{2}\right|}{\left|z_{2}\right|}=\left|2 \frac{z_{1}}{z_{2}}+t\right| \\
=\left|-1+t+\frac{\sqrt{3}}{3} \mathrm{i}\right| \geqslant \frac{\sqrt{3}}{3} .
\end{array}
$$