
Convex functions
Suppose f and g are two increasing convex functions on [0,1] with $\displaystyle f(0) \leq g(0)$ and $\displaystyle f(1)=g(1)=1$. Show that
$\displaystyle (g'(x)f'(x))(1x) + g(x)  f(x) \geq 0$.
I know that $\displaystyle f(x)+(1x)f'(x) \leq 1$, with the same for g. But that doesn't look that helpfull.

Re: Convex functions
I forgot to mention that f and g are polynomials with nonnegative coefficients. I think it is sufficient that they are convex functions, but this maybe makes the problem easier?