monotone convex function

Question

g(x) is a monotone convex function, g(0) = 0.

prove that 2g(x) < g(2x)

Answer 1

By the definition of convexity we have that for all a and b the function g(x) satisfies the inequality g((a+b)/2)<=g(a)/2+g(b)/2. For b=0 we obtain that g(a/2)<=g(a)/2 since g(0)=0. Now it we set a/2=x we arrive at g(x)<=g(2x)/2 which proves the claim.