Problem
of the
Week

PROBLEM 217

A real valued function is sublinear if it satisfies the property:  f (x + y) £ f (x) + f (y), for any 0 £ x £ y.  Suppose that (a) f (0) = 0 and (b) f ''(x) < 0.  Show that the function f must be sublinear.   

Note: In an early version of this problem I neglected to restrict the variables to be non-negative. Can you come up with a counterexample which shows this restriction is necessary?

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ã2005 Alberto L. Delgado