For a positive function $f$ on the interval $[0,1]$, the power mean of order $pinmathbb R$ is defined by smallskipcenterline{$displaystyle|, f,|_p=left(int_0^1 f^p(x),dxright)^{1/p}quad(pne0),qquad|, f,|_0=expleft(int_0^1ln f(x),dxright).$} Assume that $0