ILL 1992 Problem 57

For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$ , $G = \sqrt[3]{abc}$ , $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that $$\left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.$$