A monotonicity property involving the generalized elliptic integral of the first kind

In this paper, we prove that the function r → Y (r) = Ka(r) sin(πa)r′2 log(eR(a)/2/r′) − 1 r′2 is strictly increasing from (0,1) onto (π/[R(a)sin(πa)]−1,a(1−a)) for all a∈ (0,1/2] , where r′ = √ 1− r2 , Ka(r) is the generalized elliptic integral of the first kind, R(a) = −2γ −ψ(a)− ψ(1− a) , ψ is the classical psi function and γ = 0.57721566 · · · is the Euler-Mascheroni constant. Mathematics subject classification (2010): 33C05, 33B15, 26D15.