Sharp bounds for Heinz mean by Heron mean and other means

: In this paper, some sharp bounds for Heinz mean by Heron mean and other means are presented. Further, we point out a mistake in [1] and correct it. Finally, we extend the results to the corresponding operator means.


Introduction
The ν-weighted arithmetic mean−geometric mean inequality or ν-weighted AM − GM inequality is the following statement: If a, b ≥ 0 and 0 ≤ ν ≤ 1, then holds with equality if and only if a = b. The inequality (1.1) for ν = 1/2 reduces to the arithmetic mean-geometric mean inequality or AM − GM inequality (1. 2) The Heinz mean, introduced in Bhatia and Davis [2], is defined by for 0 ≤ ν ≤ 1 and a, b ≥ 0. We know that (1. 4) In [3], Bhatia defined the weighted mean of arithmetic mean and geometric mean as Heron means, wrote it as F α (a, b) = (1 − α)G(a, b) + αA(a, b) = (1 − α) √ ab + α a + b 2 , (1.5) and got the following result about Heinz and the Heron means.
Kittaneh, Moslehian and Sababheh come to the above conclusion from the main one (see [4,Theorem 2.1]), which is handled skillfully and is also used by Sababheh [5] on other means. The first objective of this paper is to sharpen the above inequality and obtain the result.
holds, where θ 2 can not be replaced by any smaller number.
We can find that (1.7) contains (1.6) and indicates that θ 2 is the best constant in (1.7). In addition, recently, in [1] Shi gave a lower bound for Heinz mean. holds.
We find that there are two issues in above result. First, there should be no √ ab on the right hand side of (1.8). This judgement is verified by calculation. Second, under the condition β 2 > α 2 /3, [1] draw the following conclusion: In fact, the coefficients of the same power in the left-hand side series are not greater than or equal to the right one, for example, the relationship β 4 /4! ≥ α 4 /5! is not true. The second objective of this paper is to reconstruct relevant results about (1.8) as follows. (1.10)
Proof Because the functions involved in this lemma are all even functions, we can assume that Then c n := a n b n = θ 2n .
Since {c n } n≥1 is decreasing for |θ| < 1 and increasing for |θ| > 1, by Lemma 2.1 we have that the the proof of Lemma 2.2 is complete.
Proof Using the power series expansions For the reason In fact, since

The proof of Theorem 1.1
Since when letting √ b/a = e t the last inequality is equivalent to Letting θ = 1 − 2ν in Lemma 2.2 we can obtain the above inequality. This completes the proof of Theorem 1.1.

With transformations
we may come to the conclusions Via

Corollary
At the beginning of this section, we want to draw some useful inferences of Theorem 1.2.
Then we obtain Similarly, letting √ 3β = −α in (1.10) gives the above inequality too. In this way, from Theorem 1.2, we can get the following corollary. Then

Comparation
In fact, we see that (4.2) gives a lower bound for H ν (a, b), while (1.6) or (1. H ν (a, b). Here, we get the following result by transforming t = a/b : (1.6) or (1.7) and (4.3) are equivalent to

7) and (4.3) show different upper bounds for
and respectively, where κ 1 and κ 2 are defined as (4.1). Since r i (ν, t) = r i (1 − ν, t) for i = 1, 2, we want to compare the advantages and disadvantages of these two inequalities above as long as we discuss them in this case ν ∈ (0, 1/2). Numerical experiments show that holds for all t > 0 and 0 < ν < 1/2. So the inequality (4.6) holds for all t > 0 and 0 < ν < 1. Then we have the following note.

Inequalities related to the Heinz operator mean
Now that we have the fact above, we can apply our conclusions to Heinz operator mean on Hilbert spaces.
Let B + denotes the set of all positive invertible operators on a Hilbert space H. For A, B ∈ B + and ν ∈ [0, 1], the weighted arithmetic operator mean A∇ ν B, geometric mean A ν B, and the Heinz operator mean H ν (A, B) are defined as Let ν ∈ [0, 1] and define the function K ν : R + −→ R by The function above was first introduced by Kittaneh and Krnic in [7]. Kittaneh and Krnic [7], Zhao, Wu, Cao, and Liao [8] obtained the following result. holds, where θ 2 can not be replaced by any smaller number.
As for the application of Theorem 1.2 on the same topic, it is not difficult to get the following results.  Remark 4.4. For further results of Heinz operator inequalities, interested readers can refer to [9][10][11][12][13].