Sharp bounds for the Toader mean in terms of arithmetic and geometric means (Q2025792)

The authors offer sharp lower and upper bounds for a mean of two arguments considered by \textit{G. Toader} [J. Math. Anal. Appl. 218, No. 2, 358--368 (1998; Zbl 0892.26015)], in terms of the arithmetic and geometric mean. In the proofs, the well-known connection between the Toader mean and the complete elliptic integral of second type is used. A basic method of proof is the recurrence method combined with the classical result of \textit{M. Biernacki} and \textit{J. Krzyz} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 9, 135--147 (1957; Zbl 0078.26402)] on the monotonicity of the ratio of power series. Another tool is a monotonicity criteria due to \textit{Z.-H. Yang} et al. [J. Math. Anal. Appl. 428, No. 1, 587--604 (2015; Zbl 1321.26019)].