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monotonous exponential functions

Source: RMO District 2005, 10th Grade, Problem 1

March 5, 2005
functionalgebra proposedalgebra

Problem Statement

Let a,b>1a,b>1 be two real numbers. Prove that a>ba>b if and only if there exists a function f:(0,∞)→Rf: (0,\infty)\to\mathbb{R} such that i) the function g:R→Rg:\mathbb{R}\to\mathbb{R}, g(x)=f(ax)−xg(x)=f(a^x)-x is increasing; ii) the function h:R→Rh:\mathbb{R}\to\mathbb{R}, h(x)=f(bx)−xh(x)=f(b^x)-x is decreasing.
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