say $n_1,n_2$ respectively, are strictly less than $n$. Hence we get $$f_{G}(n=n_1+n_2)   \leq \max\{f_{G_1}(n_1),f_{G_1}(n_2), f_{G_2}(n_1), f_{G_2}(n_2)\}
   \leq \max\{ f_{G_1}(n),f_{G_2}(n)\}.\nonumber$$ Last inequality is coming from the fact that hyperbolicity function is non decreasing. When there is a cancellation $xyz=1$ in $G_1$ (or in $G_2$) then we are in the following setting, $T_1=T_1'x, T_2=y$ and $T_3=zT_2'$, where $xyz=1$ in $G_1$ (or in $G_2$). In this case we can also write the triangle $\Delta$ as union of triangle $\Delta$ formed by $x,y,z$ and the triangle $\Delta_2$ (bigon) formed by $T_1',T_2'$ with lengths of normal forms $n_1,n_2$. Then we get $$f_{G}(n=n_1+n_2)   \leq \max\{ f_{G_1}(n_1),f_{G_1}(n_2), f_{G_2}(n_1), f_{G_2}(n_2)\}
\leq \max\{ f_{G_1}(n),f_{G_2}(n)\}.\nonumber$$ Hence by induction we can see that $f_G(t)\leq \max\{ f_{G_1}(t),f_{G_2}(t)\}$ for all $t$. Which implies that, $$\begin{aligned}
\liminf_{t\rightarrow \infty}\frac{f_G(t)}{t}\leq \liminf_{t\rightarrow\infty}\frac{f_{G_1}(t)+f_{G_2}(t)}{t} \rightarrow 0\nonumber. \end{aligned}$$ Hence $G$ is a lacunary hyperbolic group.

For the other direction let $G=G_1* G_2$ is LHG. Note that we have $$\begin{aligned}
        f_{G_i}(t)\leq f_G(t)\ \textit{for all } t> 0\ \textit{and for all }i=1,2.
    \end{aligned}$$ since every geodesic triangles in $G_i$ is also a geodesic triangle in $G$ for all $i=1,2$ as the embedding $G_i\rightarrow G_1*G_2$ is a $(1,0)$ quasi isometric embedding for all $i=1,2$. There exists a sequence, say $\{ y_i\}$ such that, 