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.

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.

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Weisstein, Eric W. "Convex Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConvexFunction.html