A concavity property of generalized complete elliptic integrals.

We prove that, for p ∈ (1 , ∞) and β ∈ R , the function x ↦ β − log ⁡ 1 − x p K p (x p) is strictly concave on (0 , 1) if and only if β ≥ λ (p) := 2 p (p 2 − 2 p + 2) (p − 1) (2 p 2 − 3 p + 3) , where K p represents the generalized complete p-elliptic integrals of the first kind defined by K p (r) := ∫ 0 π p / 2 d θ (1 − r p sin p p ⁡ θ) 1 − 1 / p , where π p := 2 p B (1 / p , 1 − 1 / p) , π 2 = π , and sin p is the generalized sine function, with sin 2 = sin. This extends the recently obtained corresponding result for the case that p = 2. We then apply this concavity property to obtain the following functional inequality (likewise extending the previously established result for the case that p = 2): For all r ∈ (0 , 1) , we have 2 β π p + 1 < β − log ⁡ (r ′) K p (r) + β − log ⁡ (r) K p (r ′) ≤ 2 β + 2 log ⁡ (2 p) K p (1 / 2 p) , where r ′ = 1 − r p p , p ∈ (1 , ∞) , and β ≥ λ (p). Both bounds are sharp. The sign of equality holds if and only if r = 1 / 2 p . [ABSTRACT FROM AUTHOR]