  In this article, we present a new subadditivity behavior of convex and
concave functions, when applied to Hilbert space operators. For example, under
suitable assumptions on the spectrum of the positive operators $A$ and $B$, we
prove that \[{{2}^{1-r}}{{\left( A+B \right)}^{r}}\le
{{A}^{r}}+{{B}^{r}}\quad\text{ for }r>1\text{ and }r<0,\] and
\[{{A}^{r}}+{{B}^{r}}\le {{2}^{1-r}}{{\left( A+B \right)}^{r}}\quad\text{ for
}r\in \left[ 0,1 \right].\] These results provide considerable generalization
of earlier results by Aujla and Silva.
  Further, we present several extensions of the subadditivity idea initiated by
Ando and Zhan, then extended by Bourin and Uchiyama.
