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

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

Added by Tom Leinster As a check, and to insure against any error in computing the 2nd derivative, I computed this: $$ \frac{1}{2}\bigl( f(0) + f(0.6) \bigr) - f(0.3) = -0.00018332\ldots < 0, $$ again proving non-convexity. (I chose $0.6$ because that's roughly the value that illustrates the non-convexity most vividly. But even so, notice how close to zero this is.)