A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

More generally, a function  is convex on an interval  if for any two points  and  in  and any  where ,

(Rudin 1976, p. 101; cf. Gradshteyn and Ryzhik 2000, p. 1132).

If  has a second derivative in , then a necessary and sufficient condition for it to be convex on that interval is that the second derivative  for all  in .

If the inequality above is strict for all  and , then  is called strictly convex.

Examples of convex functions include  for  or even ,  for , and  for all . If the sign of the inequality is reversed, the function is called concave.

REFERENCES:

Eggleton, R. B. and Guy, R. K. "Catalan Strikes Again! How Likely is a Function to be Convex?" Math. Mag. 61, 211-219, 1988.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1132, 2000.

Rudin, W. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, 1976.

Webster, R. Convexity. Oxford, England: Oxford University Press, 1995.