Let $I_k$ be the systematic columns and let two codes be defined over the same systematic columns $I_k$ as below.
Let $[n_1,k]$ be the first linear systematic code with rate $\frac{k}{n_1} \leq r_1$, with generator matrix
\begin{equation} G_1 = [I_k | P_1]_{k \times n_1} \end{equation}
and $[n_2,k]$ be the second linear systematic code over the same $k$ information symbols with rate $\frac{k}{n_2} \leq r_2$. Its generator matrix is
\begin{equation} G_2 = [I_k | P_2]_{k \times n_2} \end{equation}
Consider a code with the generator matrix
\begin{equation} G = G_1 \cup G_2 = [I_k|P_1|P_2]_{k \times n}, \ \ \ where \ \ n=n_1+n_2-k \end{equation}.
How to obtain a tighter bound on the rate of this code $\frac{k}{n}$ in terms of both $r_1$ and $r_2 \ ?$
The rate of the code with generator matrix $[I_k \mid P_1 \mid P_2]_{k\times {(n_1+n_2-k)}}$ is exactly $$R = \frac{k}{n_1+n_2-k} = \frac{1}{\frac{1}{r_1}+\frac{1}{r_2}-1}.$$ So, what kind of "tighter bound" in terms of $r_1$ and $r_2$ are you looking for?