A concavity property of generalized complete elliptic integrals. Issue 3 (4th March 2021) - Richards, Kendall C.; Smith, Jordan N.
Abstract : 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 .