A concavity property of the complete elliptic integral of the first kind. Issue 9 (1st September 2020) - Alzer, Horst; Richards, Kendall C.
ABSTRACT: We prove that the function G a ( x ) = a − log ⁡ ( 1 − x ) K ( x ) ( a ∈ R ) is strictly concave on ( 0, 1 ) if and only if a ≥ 8 / 5 . This solves a problem posed by Yang and Tian and complements their result that 1 / G a ( a ≥ 0 ) is strictly concave on ( 0, 1 ) if and only if a = 4 / 3 . Moreover, we apply our concavity theorem to present several functional inequalities involving K . Among others, we prove that if a ≥ 8 / 5, then 2 a π + 1 < a − log ⁡ ( r ′ ) K ( r ) + a − log ⁡ ( r ) K ( r ′ ) ≤ 2 a + log ⁡ ( 2 ) K ( 1 / 2 ) for all r ∈ ( 0, 1 ), where r ′ = 1 − r 2 . Both bounds are sharp and the sign of equality holds if and only if r = 1 / 2 .