On the absolute monotonicity related to Alzer–Qiu's conjecture

Let K(r) and arctanh⁡r be the complete elliptic integral of the first kind and inverse hyperbolic tangent function, respectively. In 2004, Alzer and Qiu conjectured that r↦G(r)=ln⁡(2/π)+ln⁡K(r)ln⁡arctanh⁡r−ln⁡ris strictly increasing and convex from (0,1) onto (3/4,1). In this paper, we prove the stronger results than Alzer–Qiu's conjecture, that is, the functions −(1/G(x))′, (ln⁡G(x))′ and Gp(x) for all p>0 are all absolutely monotonic on (0,1). This leads to a series of new sharp bounds for K(r), which greatly improve and extend existing results. The significance of our findings lies not only in solving the Alzer–Qiu's conjecture and further establishing several related absolutely monotonic functions, but more importantly, in providing a new and effective approach to deal with the absolute monotonicity of fractional functions.