Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$.

I am looking for the expectation of the Mahalanobis norm $E[\sqrt{\sum_{i=1}^d \lambda_i g_i^2}]$.

I know it in the special case when all $\lambda_i=1$, which is the expectation of a $\chi$-distribution with $d$ degrees of freedom. I also know that $\sqrt{\sum_{i=1}^d\lambda_i}$ is an upper bound by Jensen's inequality. But I would need a better estimate than that.