MathDB

Let

a_1,b_1,c_1,a_2,b_2,c_2

be positive real number and

F,G:(0,\infty)\to(0,\infty)

be to differentiable and positive functions that satisfy the identities:

\frac{x}{F} = 1 + a_1x+ b_1y + c_1G

\frac{y}{G} = 1 + a_2x+ b_2y + c_2F.

Prove that if

0 < x_1 \leq x_2

and

0 < y_2 \leq y_1

, then

F(x_1,x_2) \leq F(x_2,y_2)

and

G(x_1,y_1) \geq G(x_2,y_2).

Problem Statement