To expand on Dirk's example: for $n=3$ and $p = [1/4, 1/4, 1/2]$ we have 
$$ f''(0) = 3\, \left( \ln  \left( 3 \right)  \right) ^{2}+6\,\ln  \left( 3
 \right) -10\,\ln  \left( 2 \right) \ln  \left( 3 \right) +9\, \left( 
\ln  \left( 2 \right)  \right) ^{2}-10\,\ln  \left( 2 \right) < 0$$