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.

Update: The following lower bound would be good enough for what I need. Numerics indicate that it holds, now I'm trying to come up with a proof:

$E[\sqrt{\sum_{i=1}^d \lambda_i g_i^2}] \ge E[\sqrt{\sum_{i=1}^{T} u_i^2}]$, where $T=\lfloor \sum_{i=1}^d \lambda_i \rfloor$, and $(u_i)_{i=1,...,T}$ are i.i.d sampled from a standard Gaussian. If this holds then the LHS is known.

(My intuition on why it might hold is that the sum under the square root on the LHS has more independent terms than the one on the RHS, which makes it closer to its expectation than the other one is. Both have the same expectation and so a Jensen inequality would err less on the LHS term than on the RHS term. Not a clearly formed proof yet though.)