A concavity property of generalized complete elliptic integrals. Issue 3 (4th March 2021)

• Record Type: Journal Article
• Authors: Richards, Kendall C.; Smith, Jordan N.
• Is Part Of: Integral transforms and special functions. Volume 32:Issue 3(2021)
• Page Start: 240
• Page End: 252
• Publication Date: 2021-03-04
• Subjects: Generalized complete elliptic integrals of the first kind -- absolute monotonicity -- concavity -- hypergeometric function -- inequalities; 26D07 -- 33C05 -- 33E05; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2020.1815726
• Languages: English
• ISSNs: 1065-2469

Abstract

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 .