A short proof of a variant of Jensen's inequality

In this paper we shall give a very short proof of the Jensen-Boas inequality. In the proof we shall use only Jensen’s inequality for sums, i.e.. (2) where P, = C;=, pi, pi > 0 (i = l,..., n), xi E I (i = l,..., n), and the Jensen- Steffensen inequality. Inequality (2) can be easily obtained from (1). If A(a) < A(v,) < n(y*) < ... < A( y,- ,) < I(b), then from the Jensen- Steffensen inequality we have the inequalities i.e.,