Let \(f\) be a continuous function from \(I\) (an interval in \(\mathbb{R})\) to an interval in \(\mathbb{R}_ +\) possessing the property \(f((x_ 1+x_ 2)/2)\leq f^{1/2}(x_ 1)f^{1/2}(x_ 2)\) for all \(x_ 1,x_ 2\in I\), with equality when \(x_ 1=x_ 2\). Then the functional inequality \(f((1/\alpha)\sum^ n_{j=1}\alpha_ jx_ j)\leq(\prod^ n_{j=1}f^{\alpha_ j} (x_ j))^{(1/ \alpha)}\), where \(\alpha=\sum^ n_{j=1}\alpha_ j\), holds for \(f\), with equality when \(x_ 1=\cdots=x_ n\). Furthermore an analogous result holds if the signs of above inequalities are reversed. After proving this theorem the author gives many of its applications.