Abstract
We find the greatest value and least value such that the double inequality holds for all with . Here , , and denote the arithmetic, harmonic, and Seiffert's means of two positive numbers and , respectively.
Highlights
B > 0 with a / b the Seiffert’s mean P a, b was introduced by Seiffert 1 as follows: P a, b a−b 4 arctan a/b − πRecently, the inequalities for means have been the subject of intensive research 2– 11
The purpose of this paper is to find the greatest value α and the least value β such that the double inequality αA a, b 1 − α H a, b < P a, b < βA a, b 1 − β H a, b holds for all a, b > 0 with a / b
The double inequality αA a, b 1−α H a, b < P a, b < βA a, b 1−β H a, b holds for all a, b > 0 with a / b if and only if α ≤ 2/π and β ≥ 5/6
Summary
B > 0 with a / b the Seiffert’s mean P a, b was introduced by Seiffert 1 as follows: P a, b a−b 4 arctan a/b − πRecently, the inequalities for means have been the subject of intensive research 2– 11. B > 0 with a / b the Seiffert’s mean P a, b was introduced by Seiffert 1 as follows: P a, b a−b Many remarkable inequalities for the Seiffert’s mean can be found in the literature 12–17 .
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