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In mathematics , Jensen 's inequality , named after the Danish mathematician Johan Jensen , relates the value of a convex function of an integral to the integral of the convex function .. mathematics. mathematics. Johan Jensen. Johan Jensen ( mathematician ). convex function. convex function. integral. integral. It was proven by Jensen in 1906 .. Given its generality , the inequality appears in many forms depending on the context , some of which are presented below .. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation ; it is a simple corollary that the opposite is true of concave transformations .. Jensen 's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function , which is Jensen 's inequality for two points  : the secant line consists of weighted means of the convex function -LRB- where t ∈ -LSB- 0,1 -RSB- -RRB- ,. convex function. convex function. secant line. secant line. while the graph of the function is the convex function of the weighted means ,. convex function. convex function. Thus , Jensen 's inequality is. In the context of probability theory , it is generally stated in the following form  : if X is a random variable and is a convex function , then. convex function. convex function. probability theory. probability theory. random variable. random variable