, G)-stabilizable, we then will be interested by bounds of l
in terms of B
It is worth noticing that, for given p, bounds of lp in the form (resp., αB
+ (1 − α)G) exist for some α ∈ [0, 1]. This follows from (1.2) with the relationships
valid for all a, b > 0 and p ≠ 0.
We begin by regarding bounds of l
in a convex-geometric form as well:
Theorem 4.1. Let α, β ∈ [0, 1] be such that
for some p. Then there holds
Proof. Since lp is (Bp, G)-stabilizable then Theorem 3.2 gives
According to Lemma 2.3 we have, for all a, b > 0,
For all a, b > 0, we can write and it is easy to verify that,
from which the desired double inequality (4.3) follows. □
Corollary 4.2. Let α, β ∈ [0, 1] be two real numbers such that
Then there holds
Proof. Taking p = 1 in the above theorem, with the fact that l1 = L and B1 = A, we immediately obtain the announced result. □
Let us now examine the following examples in the aim to illustrate the above theoretical results.
Example 4.1. It is not hard to verify that G < l
for every p > 0, with reversed double inequality for p < 0. Theorem 4.1 immediately gives (with α = 0, β = 1 for p > 0; α = 1, β = 0 for p < 0)
for each real number p ≠ 0. It is easy to verify that the double inequality (4.9) refines the initial one. In particular, we have
which refines the arithmetic-logarithmic-geometric mean inequality G < L < A.
Theorem 4.3. Let α ∈ [0, 1] be such that
for some p > (<)0, respectively. Then one has
If moreover α < (>)1/3 then (4.12) refines (4.11).
Proof. Assume that
for some p > 0. According to Theorem 4.1, the first inequality of (4.3) holds and the arithmetic-geometric mean inequality gives
The desired inequality follows after a simple reduction. Further, the inequality
for p > 0 is reduced to
which holds when α < 1/3. For the reversed inequalities, the same arguments as previous study, so completes the proof. □
If we get p = 1 in the above theorem, we immediately obtain the following result.
Corollary 4.4. Let α ∈ [0, 1] be a real number satisfying that
Then one has
If moreover α < 1/3 then (4.18) refines (4.17).
Theorem 4.3 tells us that every given bound of l
in a convex-geometric form yields another bound of l
in an analogs, but different, form. Illustrating this latter point, we will deduce a better bound of l
than the above ones. Precisely, we may state the next result.
Theorem 4.5. Let p be a real number. If p > 0 then one has
If p < 0 then the above inequality is reversed. In particular the following inequality holds true
Proof. Assume that p > 0. Starting from G < l
(see Example 4.1), we are in the situation of Theorem 4.3 with α = 0, and so we have . Let us iterate successively this procedure: if in the step n, we have
then in the step n + 1, we obtain
It is easy to see that the real sequences (α
converges to 1/3 for every given initial data α0 ∈ [0, 1]. The desired inequality follows by letting n → +∞ in the recursive inequality
The proof is similar for p < 0. Taking p = 1 in (4.19) we obtain (4.20), so completes the proof. □
To understand the interest of the above theorem, let us observe the following example.
Example 4.2. Let us apply Theorem 4.3 to the previous inequality . Then, the next inequality
holds true for each real number p (p ≠ 0). In particular, taking p = 1 we obtain
which refines A1/3G2/3 < L.
Remark 4.1. The inequality (4.20) was proved by Leach and Sholander , while (4.26) has been shown by Sāndor . These two inequalities were proved by different methods therein while together obtained here via the same approach. In the same sense, other examples will be seen later (see Remarks 4.4, 4.5, and 5.1).
Remark 4.2. As well known, inequality (4.20) is the best possible in the sense that the constant α = 1/3 cannot be improved in AαG1−α< L. This latter point rejoins the fact that if we apply Corollary 4.4 to (4.20) we obtain the same inequality.
Remark 4.3. By virtue of the relationships (4.1), it has been possible to begin by stating and proving the results of the above corollaries and then to deduce those of the corresponding theorems (with discussion on p). Details of this latter point are omitted for the reader.
Now, we will be interested by bounds of l
in a convex-arithmetic expression as well:
Theorem 4.6. Let α, β ∈ [0, 1] be two real numbers such that
for some real number p. Then there holds
Proof. By the same arguments as previous, we have
Again, thanks to Lemma 2.3, we obtain
By virtue of the identity (4.6), we obtain the desired result after simple manipulations. □
As in the above, taking p = 1 in the latter theorem we immediately obtain the following result.
Corollary 4.7. Let α, β ∈ [0, 1] be two real numbers such that
Then there holds
Theorem 4.6 has many interesting consequences. For instance, we give the two following corollaries.
Corollary 4.8. Let α ∈ [0, 1] be such that
Then we have
If α > 1=3 then (4.34) refines (4.33).
Proof. According to Theorem 4.6, we have
If we write
and we apply the arithmetic-geometric mean inequality, i.e.,
we obtain the announced result after substituting this latter inequality in (4.35). If α > 1/3, it is easy to see by similar manner as previous that (4.34) refines (4.33) and the proof is completed. □
The following inequality holds true
Proof. Similarly to the above, it is sufficient to see that the sequence (α
) defined by
converges to 1/3 and the desired result follows as previous. We omit the routine detail here. □
Remark 4.4. The inequality (4.38) was differently proved by Carlson  and here obtained by the same approach as (4.20) and (4.26).
Let us illustrate the above theoretical examples with the following examples.
Example 4.3. Consider the above mean-inequality L < (1/3)A + (2/3)G which corresponds to α = 1/3 in Corollary 4.8. With this, the obtained refinement is given by
Of course, we can combine some the above results to improve the lower and upper bounds of L. The following example explains this situation.
Example 4.4. Let us consider the following double inequality
Combining Theorems 4.1 and 4.6 we immediately obtain
The reader can easily verify that this latter double inequality refines the initial one, so proving our desired aim.
Theorem 4.10. Let α ∈ [0, 1] be such that
for some p ≤ 1. Then there holds
Proof. If (4.43) holds then Theorem 4.6 gives
This, with p ≤ 1 and the monotonicity of power means, yields
The arithmetic-geometric mean inequality gives
and the desired inequality follows by combining (4.46) and (4.47) with a simple reduction. □
Taking p = −1 in the above theorem, with the fact that l−1 = L* = G2/L and B−1 = H = G2/A, we immediately obtain the next result.
Corollary 4.11. Let α ∈ [0, 1] be such that
Then one has
If moreover α > 1/3 then (4.49) refines (4.48).
Theorem 4.12. For all real number p ≤ 1 with p ≠ 0, we have
In particular, the following inequality holds
Proof. We left it to the reader as an interesting exercise. □
We end this section by stating another result showing how to obtain a lower bound of the logarithmic mean L when we start from an upper bound of its dual L*. In fact, since L* is (A, H)-stabilizable then we search bounds of L* in terms of A and H. Precisely, we have the following.
Let α be a real number satisfying that
Then we have
If moreover α > 1/3 then (4.53) refines (4.52).
Proof. Since L* is (A, H)-stabilizable then we obtain, with Proposition 2.2,
Thanks to relationships (2.5) for obtaining
Due to the point-wise convexity of the mean-map m ↦ m*, with A* = H and H* = A, we obtain
which after reduction yields the desired result.
The following inequality holds true
Proof. Similarly to the same idea as in the above we have
is the sequence defined by the same recursive relation as in the proof of Corollary 4.9. Letting n → +∞ we obtain
The general relation m* = G2/m valid for all mean m, gives in particular, L* = G2/L, H = G2/A and A = G2/H. Substituting this in the latter inequality, we obtain the desired result. □
Remark 4.5. The inequality (4.57) was differently proved by Chen  and shown here by the same approach as (4.20), (4.26), and (4.38), so proving the interest of this study. Further, we notice that it is easy to verify that (4.57) is stronger than (4.51).