Optimal inequalities for bounding Toader mean by arithmetic and quadratic means

Abstract

In this paper, we present the best possible parameters and such that the double inequality holds for all and with , and we provide new bounds for the complete elliptic integral of the second kind, where , and are the Toader, arithmetic, and quadratic means of a and b, respectively.

Optimal bounds for Toader mean in terms of general means

“…Therefore, the result agrees well with Theorem 3.3 in [11]. Remark 3 The same result can be found in [24].…”

“…Recently, there were published numerous articles which focus on the bounds for the Toader mean [12][13][14][15][16][17][18][19][20][21][22][23]. For example, Zhao, Chu and Zhang [24] presented the best possible parameters α(r) and β(r) such that the double inequality…”

Optimal Convex Combination Bounds for Toader Mean

“…Recently, the Toader mean has been the subject of intensive research. In particular, many remarkable inequalities for Toader mean and its generating can be found in the literature [7], [8], [9], [10], [11], [12], [13].…”

Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters

“…It is well known that many important means are the special cases of the quasi-arithmetic mean. For example, is the arithmetic-geometric mean of Gauss [ 54 – 60 ], is the Toader mean [ 61 – 70 ], and is the Toader-Qi mean [ 71 – 74 ].…”