Theorem 2.1. For fixed ω ≥ 0 and all a, b > 0 with a ≠ b we have
(2.1)
and and H0,ω(a, b) are the best possible upper and lower generalized Heronian mean bounds of the logarithmic mean L(a, b), respectively.
Proof. Without loss of generality, we assume a > b and put . Then from (1.1) and (1.4) we get
(2.2)
Let
(2.3)
Then simple computations lead to
(2.4)
(2.5)
where .
(2.7)
(2.9)
where .
(2.11)
where
(2.13)
(2.15)
where
(2.16)
(2.17)
(2.18)
(2.19)
where .
(2.20)
(2.21)
where
(2.22)
(2.23)
From (2.22) we clearly see that f6(t) is strictly decreasing in [1, +∞), then (2.23) leads to that
for t ∈ (1, + ∞).
It easily follows from (2.5)-(2.21) and (2.24) that
for t ∈ (1, + ∞).
Therefore, follows from (2.2)-(2.4) and (2.25).
On the other hand, H0,ω(a, b) = M0(a, b) < L(a, b) follows from (1.3).
Next, we prove that H0,ω(a, b) and are the optimal lower and upper generalized Heronian mean bounds of the logarithmic mean L(a, b).
For any , ω ≥ 0 and x > 0, from (1.1) and (1.4) we have
(2.26)
(2.27)
Equations (2.26) and (2.27) imply that for any ω ≥ 0 and there exist sufficiently large X = X(ε, ω) > 1 and sufficiently small δ = δ (ε, ω) > 0; such that H
ε,ω
(1, x) > L(1, x) for x ∈ (X, +∞) and for x ∈ (0, δ).
Remark 2.1. If we take ω = 0, then inequality (2.1) reduce to inequality (1.3).
Remark 2.2. If we take ω = 4, then the upper bound in inequality (2.1) becomes the upper bound in inequality (1.5).