Flanders' inequality

The inequality sin A sin B sin C<=((3sqrt(3))/(2π))^3 A B C, where A, B, and C are the vertex angles of a triangle. The maximum is reached for an equilateral triangle (and therefore at A = B = C = π/3) and has numerical value 0.56559562463... (OEIS A127205). The inequality was proven by Abi-Khuzam, also considered by Klamkin, and mentioned by Flanders as "an interesting related result" for the product sin A sin B sin C of the three angles A, B, and C of a triangle.