Convexity and Monotonicity Involving the Complete Elliptic Integral of the First Kind - Tian, Jing-Feng; Yang, Zhen-Hang (Resultate der Mathematik, 2023)
Let K r be the complete elliptic integral of the first kind defined on 0 , 1 . By virtue of the auxiliary function H f , g = f ′ / g ′ g - f , we prove that the function Q p x = ln p / 1 - x K x is strictly convex on 0 , 1 if and only if 0 < p ≤ 4 , thus answering a conjecture. Moreover, we completely described the monotonicity of Q p x on 0 , 1 for different p ∈ 0 , ∞ .