Two sharp inequalities for the norm of a factor of a polynomial

Let f ( x ) be a monic polynomial of degree n with complex coefficients, which factors as f ( x ) = g ( x ) h ( x ), where g and h are monic. Let‖ f ‖be the maximum of| f ( x ) |on the unit circle. We prove that‖ g ‖ ⩽ β n ‖ f ‖   and   ‖ g ‖ ‖ h ‖ ⩽ δ n ‖ f ‖ , where β = M ( P 0 ) = 1 38135 …, where P 0 is the polynomial P 0( x, y ) = 1 + x + y and δ = M ( P 1 ) = 1 79162…, where P 1 ( x, y ) = 1 + x + y ‐ xy , and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n → ∞.