Abstract
We present the best possible power mean bounds for the product for any p > 0, α ∈ (0,1), and all a, b > 0 with a ≠ b. Here, Mp(a, b) is the pth power mean of two positive numbers a and b.
Highlights
For p ∈ R, the pth power mean Mp a, b of two positive numbers a and b is defined by ⎧ Mp a, b ⎪⎨ ap bp ⎪⎩√ab,2 1/p, p / 0, p 0.It is well known that Mp a, b is continuous and strictly increasing with respect to p ∈ R for fixed a, b > 0 with a / b
The power mean has been the subject of intensive research
Many remarkable inequalities and properties for the power mean can be found in literature 1–22
Summary
Mp a, b is the pth power mean of two positive numbers a and b
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