A Turán-type inequality for the gamma function

We prove that the following Turán-type inequality holds for Euler's gamma function. For all odd integers n⩾1n⩾1 and real numbers x>0x>0 we haveα⩽Γ(n−1)(x)Γ(n+1)(x)−Γ(n)(x)2,α⩽Γ(n−1)(x)Γ(n+1)(x)−Γ(n)(x)2, with the best possible constantα=min1.5⩽x⩽2Γ(x)2ψ′(x)=0.6359….