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Problem 4

Part of 2013 CIIM

Problems(1)

CIIM 2013 Problem 4

Source:

8/9/2016
Let a1,b1,c1,a2,b2,c2a_1,b_1,c_1,a_2,b_2,c_2 be positive real number and F,G:(0,∞)→(0,∞)F,G:(0,\infty)\to(0,\infty) be to differentiable and positive functions that satisfy the identities: xF=1+a1x+b1y+c1G\frac{x}{F} = 1 + a_1x+ b_1y + c_1G yG=1+a2x+b2y+c2F.\frac{y}{G} = 1 + a_2x+ b_2y + c_2F. Prove that if 0<x1≤x20 < x_1 \leq x_2 and 0<y2≤y10 < y_2 \leq y_1, then F(x1,x2)≤F(x2,y2)F(x_1,x_2) \leq F(x_2,y_2) and G(x1,y1)≥G(x2,y2).G(x_1,y_1) \geq G(x_2,y_2).
CIIM 2013CIIMundergraduate