Maximum density linear combination chi squares

Question

I have a positive linear combination of chi square variables \begin{equation*} X=\sum_{i=1}^k \lambda_i \chi^2(r_i) \end{equation*} the degrees of freedom satisfy $r_i>1$. I need an upperbound of the maximum density of $X$. I searched for results on the maximum of convolution functions etc. It is also important to know that the closed form expression of density of $X$ is not known (and may not exist). I have bounds of the density of $\chi^2(r_i)$ when $r_i>1$, namely $1$.