For $x_i \sim N(\mu, \sigma_i^2) (i = 1, 2,\cdots, n)$ and the $x_i$'s independent, this paper gives necessary and sufficient conditions under which the weighted average of the $x_i$'s, with weights proportional to inverses of the sample variances, has uniformly smaller variance than any of the $x_i$'s.
"Estimating the Common Mean of Several Normal Populations." Ann. Statist. 5 (5) 1047 - 1050, September, 1977. https://doi.org/10.1214/aos/1176343959