Abstract
We derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values , and the least values , such that the double inequalities and hold for all with . Here, , , , and denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers and , respectively.
Highlights
B > 0 with a / b, the Seiffert’t 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
We prove that 2/9C a, b 7/9G a, b is the best possible lower convex combination bound of the contraharmonic and geometric means for Seiffert’s mean
We prove that 1/πC a, b 1 − 1/π G a, b is the best possible upper convex combination bound of the contraharmonic and geometric means for Seiffert’s mean
Summary
B > 0 with a / b, the Seiffert’t 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. We present the optimal convex combination bounds of contraharmonic and geometric means for Seiffert’s mean as follows.
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