Abstract
For , the generalized logarithmic mean , arithmetic mean and geometric mean of two positive numbers and are defined by , ; , , , ; , , ; , , ; and , respectively. In this paper, we give an answer to the open problem: for , what are the greatest value and the least value , such that the double inequality holds for all ?
Highlights
For p ∈ R, the generalized logarithmic mean Lp a, b of two positive numbers a and b is defined by ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨a, ap p 1 − bp 1 1 a−b 1/p, a b, p / 0, p / − 1, a / b, Lp a, b ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩1 e ln bb aa b−a b − ln 1/ a, b−a, p 0, a / b, p −1, a / b
1.12 holds for all a, b > 0? The purpose of this paper is to give the solution to this open problem
In order to establish our main result, we need two lemmas, which we present
Summary
For p ∈ R, the generalized logarithmic mean Lp a, b of two positive numbers a and b is defined by. The generalized logarithmic mean has been the subject of Journal of Inequalities and Applications intensive research. In , Alzer and Janous established the following sharp double inequality see , Page 350 : Mlog 2/ log 3 a, b. 1.8 hold for all positive real numbers a and b with a / b if and only if α ≤ 2/3 and β ≥ 2/e 0.73575 . The following problem is still open: for α ∈ 0, 1 , what are the greatest value p and the least value q, such that the double inequality. 1.12 holds for all a, b > 0? The purpose of this paper is to give the solution to this open problem
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