The authors prove the following interesting inequality for convex functions: Suppose that positive numbers \(s_{i,j}\) \((i= 0,1,2; j= 1,\dots,n)\) satisfy \(s_{1,j}\leq s_{0,j}\leq s_{2,j}\) \((j= 1,\dots,n)\) and \(a_ j s^{-1}_{i,1}+ b_ j s^{-1}_{i,j}= 1\) \((i= 0,1,2; j= 2,\dots,n)\) for positive constants \(a_ j\), \(b_ j\) \((j= 2,\dots,n)\). If \(f_ j: (0,\infty)\to \mathbb{R}\) \((j= 1,\dots,n)\) are convex functions, then \[ \sum^ n_{j=1} f_ j(s_{0,j})/s_{0,j}\leq \max_{i=1,2} \left(\sum^ n_{j=1} f_ j(s_{i,j})/s_{i,j}\right). \] Some connected results are also given.