On inequality for convex function (Q2730648)

The author considers function \( f:\mathbb R\to\mathbb R \) which is convex on \([ a;b ] \). He proves that for \( x_{i}\in [ a,b ] ,m_{i}\geq 0(i=1,\ldots ,n),\sum ^{n}_{i=1}m_{i}\neq 0: \) NEWLINE\[NEWLINEf\left( \frac{\sum ^{n}_{i=1}m_{i}x_{i}}{\sum ^{n}_{i=1}m_{i}}\right) \leq \frac{\sum ^{n}_{i=1}m_{i}f\left( x_{i}\right) }{\sum ^{n}_{i=1}m_{i}}\leq \frac{f(a)-f(b)}{a-b}\cdot \frac{\sum ^{n}_{i=1}m_{i}x_{i}}{\sum ^{n}_{i=1}m_{i}}+\frac{af(b)-bf(a)}{a-b}NEWLINE\]NEWLINE It is shown that there exists a function such that these inequalities became identities.