A double inequality for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean

Abstract

In the paper, the authors find the greatest value λ and the least value µ such that the double inequalityand T (a, b) = 2 π π/2 0 a 2 cos 2 θ + b 2 sin 2 θ d θ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers a and b.2010 Mathematics Subject Classification. 26E60, 33E05.

Sharp inequalities for Toader mean in terms of other bivariate means

“…There are many bounds for the Toader mean in terms of various elementary means, see for example, [6,7,[9][10][11][12][13][14][15]17,[22][23][24][25][26][27][28], and recent papers [16,[29][30][31][32]. In particular, we mention here several interesting results.…”

Sharp bounds for the Sándor–Yang means in terms of arithmetic and contra-harmonic means

“…Let with . Then the arithmetic mean [ 1 – 4 ], the quadratic mean [ 5 ], the contra-harmonic mean [ 6 – 9 ], the Neuman–Sándor mean [ 10 – 12 ], the second Seiffert mean [ 13 , 14 ], and the Schwab–Borchardt mean [ 15 , 16 ] of a and b are defined by respectively, where and are respectively the inverse hyperbolic sine and cosine functions. The Schwab–Borchardt mean is strictly increasing, non-symmetric and homogeneous of degree one with respect to its variables.…”

Quadratic transformation inequalities for Gaussian hypergeometric function

“…As is well known, making use of the hypergeometric function, Branges proved the famous Bieberbach conjecture in 1984. Since then, and its special cases and generalizations have attracted attention of many researchers, and was studied deeply in various fields [ 2 , 5 , 9 , 11 – 18 , 20 , 22 , 23 , 26 , 30 , 31 , 35 – 37 , 40 , 45 , 46 , 48 ]. A lot of geometrical and analytic properties, and inequalities of the Gaussian hypergeometric function have been obtained [ 3 , 6 – 8 , 19 , 29 , 32 , 34 , 38 , 49 ].…”