Some new inequalities for means of two arguments

Abstract

Abstract. We prove certain new inequalities for special means of two arguments, including the identric, arithmetic, and geometric means.

Sharp two-parameter bounds for the identric mean

“…However, studying the family of functions defined by using similar arguments to those used in Lemma 3.1, we can prove the following exact version of Corollary 4.1, which extends the results of Seiffert [11] and those of Sándor and Trif [10]. …”

“…In the next corollary, the lower bound is an inequality due to Seiffert [11], it appears also in [10], while the upper bound is new and to be compared with the results of Sándor and Trif in [10]. …”

Companion inequalities for certain bivariate means

“…The second inequality in (15) and the following one √ 2A 2/3 < A + G, which is easy to prove, provide a refinement of inequality (13) Remark 1.…”

“…For the proof of (17) we use (15) and (16) Taking into account that the product of two positive strictly decreasing functions is also strictly decreasing we conclude that the constants 1 and 3/e are best possible. For the proof of (17) we use (15) and (16) Taking into account that the product of two positive strictly decreasing functions is also strictly decreasing we conclude that the constants 1 and 3/e are best possible.…”

Inequalities for differences of power means in two variables

“…Recently, the inequalities for differences of certain special means (such as the arithmetic, geometric, identric, logarithmic and power means) have attracted remarkable interest of many mathematicians and have motivated a large number of research papers (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein). To make the paper self-contained, we state these means given by …”