A concavity property of generalized complete elliptic integrals
Kendall C. Richards, Jordan N. Smith
Abstract
We prove that, for p∈(1,∞) and β∈R, the function x↦β−log1−xpKp(xp) is strictly concave on (0,1) if and only if β≥λ(p):=2p(p2−2p+2)(p−1)(2p2−3p+3), where Kp represents the generalized complete p-elliptic integrals of the first kind defined by Kp(r):=∫0πp/2dθ(1−rpsinppθ)1−1/p, where πp:=2pB(1/p,1−1/p), π2=π, and sinp is the generalized sine function, with sin2=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 2βπp+1<β−log(r′)Kp(r)+β−log(r)Kp(r′)≤2β+2log(2p)Kp(1/2p), where r′=1−rpp, p∈(1,∞), and β≥λ(p). Both bounds are sharp. The sign of equality holds if and only if r=1/2p.