A Turلn-type inequality for the gamma function

We prove that the following Turán-type inequality holds for Eulerʹs gamma function. For all odd integers n ⩾ 1 and real numbers x > 0 we have α ⩽ Γ ( n − 1 ) ( x ) Γ ( n + 1 ) ( x ) − Γ ( n ) ( x ) 2 , with the best possible constant α = min 1.5 ⩽ x ⩽ 2 Γ ( x ) 2 ψ ′ ( x ) = 0.6359 … .