Monotonicity and convexity involving generalized elliptic integral of the first kind - Zhao, Tie-Hong; Wang, Miao-Kun; Chu, Yu-Ming (Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas, 2021)
In this paper, we present the monotonicity properties of the ratio between generalized elliptic integral of the first kind K a ( r ) and its approximation log [ 1 + 2 / ( a r ′ ) ] , and also the convexity (concavity) of their difference for a ∈ ( 0 , 1 / 2 ] . As an application, we give new bounds for generalized Grötzsch ring function μ a ( r ) and a upper bound for K a ( r ) .