On certain inequalities for means, III

Companion inequalities for certain bivariate means

“…The goal of this paper is to obtain companion inequalities of type (12) for the inequalities (8)- (10). One of our results will give an improvement of (13) . Also, new proofs of the inequalities (8)-(10) are offered.…”

“…For more details the interested reader is referred to [1]- [6] and [13]- [16]. Many remarkable inequalities and identities have been established.…”

A best-possible double inequality between Seiffert and harmonic means

“…10) Let f 2 (t) = f 1 (t)/2, f 3 (t) = f 2 (t)/3, f 4 (t) = f 3 (t)/4, f 5 (t) = f 4 (t)/5, f 6 (t) = f 5 (t)/6 and f 7 (t) = f 6 …”

“…From equations (2.11) and (2.12), together with the piecewise monotonicity of f 1 (t), we know that there exists t 6 > t 5 >1 such that f 1 (t) is strictly decreasing in [1, t 6 ] and strictly increasing in [t 6 , +∞). Then, equations (2.7)-(2.10) lead to the conclusion that there exists t 7 > t 6 >1 such that f(t) is strictly decreasing in [1, t 7 ] and strictly increasing in [t 7 , +∞).…”

“…In particular, many remarkable inequalities for the Seiffert mean can be found in the literature [1][2][3][4][5][6][7][8][9]. (p ≠ 0) and M 0 (a, b) = √ ab be the harmonic, geometric, logarithmic, identric, arithmetic, contraharmonic, and p-th power means of two different positive numbers a and b, respectively.…”