  The estimation of a multivariate mean $\theta$ is considered under natural
modifications of balanced loss function of the form: (i) $\omega \,
\rho(\|\delta-\delta_0\|^2) + (1-\omega) \, \rho(\|\delta-\theta\|^2) $, and
(ii) $\ell \left( \omega \, \|\delta-\delta_0\|^2 + (1-\omega) \,
\|\delta-\theta\|^2 \right)\,$, where $\delta_0$ is a target estimator of
$\gamma(\theta)$. After briefly reviewing known results for original balanced
loss with identity $\rho$ or $\ell$, we provide, for increasing and concave
$\rho$ and $\ell$ which also satisfy a completely monotone property,
Baranchik-type estimators of $\theta$ which dominate the benchmark
$\delta_0(X)=X$ for $X$ either distributed as multivariate normal or as a scale
mixture of normals. Implications are given with respect to model robustness and
simultaneous dominance with respect to either $\rho$ or $\ell
