Definition Let f be a function of a single variable defined on the interval I. Proposition Let U be a concave function of a single variable and g a nondecreasing and concave function of a single variable. Then f is concave. Source hide The result is a special case of a result for functions of many variables. Proof hide I first show that if f is concave on I then the first inequality in the result holds.
For a direct proof, see Rockafellar , Theorem 4. Assume that U and g are twice-differentiable. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-Hill, Webster, R. Oxford, England: Oxford University Press, Weisstein, Eric W.
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