From Mid-Point to Full Convexity

Abstract

Let f be a continuous real function defined on an interval. After applying appropriate transformations, we may assume without loss of generality that 0 is in I and that f(0) =0. If f satisfies the Jensen inequality

$$ \begin{array}{*{20}c} {{\text{f}}\left( {\frac{{{\text{x + y}}}} {2}} \right) \leqslant \frac{{{\text{f}}\left( {\text{x}} \right){\text{ + f}}\left( {\text{y}} \right)}} {2}} & {{\text{for}}\,{\text{all}}\,{\text{x,}}\,{\text{y}}} \\ \end{array} \in {\text{I,}} $$

((1))

then, taking y = 0 in (1), we obtain f(x/2) ≤ f(x)/2. It follows by induction that, for all natural numbers n and x₁, x₂,..., x_n in dom f,

$$ {\text{f}}\left( {\sum\limits_{{\text{i}} = 1}^{\text{n}} {\frac{{{\text{x}}_{\text{i}} }} {{2^{\text{i}} }}} } \right) \leqslant \sum\limits_{{\text{i = }}1}^{\text{n}} {\frac{{{\text{f}}\left( {{\text{x}}_{\text{i}} } \right)}} {{2^{\text{i}} }}} . $$

((2))