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Convex Function

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ConvexFunction

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 f(x) is convex on an interval [a,b] if for any two points x_1 and x_2 in [a,b] and any lambda where 0<lambda<1,

f[lambdax_1+(1-lambda)x_2]<=lambdaf(x_1)+(1-lambda)f(x_2)

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

If f(x) has a second derivative in [a,b], then a necessary and sufficient condition for it to be convex on that interval is that the second derivative f^('')(x)>=0 for all x in [a,b].

If the inequality above is strict for all x_1 and x_2, then f(x) is called strictly convex.

Examples of convex functions include x^p for p=1 or even p>=2, xlnx for x>0, and |x| for all x. If the sign of the inequality is reversed, the function is called concave.

See also

Convex, Concave Function, Interval, Logarithmically Convex Function

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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.

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Convex Function

Cite this as:

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

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