Sharp bounds for the Sándor-Yang means in terms of arithmetic and contra-harmonic means.

In the article, we provide several sharp upper and lower bounds for two Sándor-Yang means in terms of combinations of arithmetic and contra-harmonic means.

Preliminaries

Let with . Then the arithmetic mean [1-4], the quadratic mean [5], the contra-harmonic mean [6-9], the Neuman–Sándor mean [10-12], the second Seiffert mean [13, 14], and the Schwab–Borchardt mean [15, 16] of a and b are defined by respectively, where and are respectively the inverse hyperbolic sine and cosine functions. The Schwab–Borchardt mean is strictly increasing, non-symmetric and homogeneous of degree one with respect to its variables. It can be expressed by the degenerated completely symmetric elliptic integral of the first kind [17]. Recently, the Schwab–Borchardt mean has attracted the attention of many researchers. In particular, many remarkable inequalities for the Schwab–Borchardt mean and its generated means can be found in the literature [18-38].

Let and denote symmetric bivariate means of a and b. Then Yang [39] introduced the Sándor–Yang mean and presented the explicit formulas for and as follows:

Very recently, the bounds involving the Sándor–Yang means have been the subject of intensive research. Numerous interesting results and inequalities for and can be found in the literature [40-42].

Neuman [43] established the inequality for with .

In [44], Xu proved that the double inequalities hold for all with if and only if  , , and .

From (1.6) and (1.7), together the well-known inequalities we clearly see that for all with .

The main purpose of this paper is to find the best possible parameters such that the double inequalities hold for all with .

Lemmas

In order to prove our main results, we need several lemmas, which we present in this section.

Lemma 2.1

(see [45])

Let with , be continuous on and differentiable on , and on . If is increasing (decreasing) on , then so are the functions If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2.2

(see [46])

Let and be two real power series converging on () with for all k. If the non-constant sequence is increasing (decreasing) for all k, then the function is strictly increasing (decreasing) on .

Lemma 2.3

The function is strictly increasing from onto .

Proof

Let , . Then elaborate computations lead to Let Then and for all .

It follows from Lemma 2.2 and (2.2)–(2.5) that is strictly increasing on .

Note that Therefore, Lemma 2.3 follows from Lemma 2.1, (2.1), and (2.6) together with the monotonicity of . □

Lemma 2.4

The function is strictly increasing from onto .

Let , , , and . Then elaborate computations lead to and

It is well known that the function is strictly decreasing on , hence equation (2.9) leads to the conclusion that the function is strictly increasing on .

Note that

Therefore, Lemma 2.4 follows from Lemma 2.1 and (2.7)–(2.9) together with the monotonicity of . □

Lemma 2.5

Let and Then the following statements are true:

If , then for all ;

If  , then there exists such that for and for .

Part follows easily from for all if .

For part , if , then numerical computations lead to

It follows from (2.11) and (2.14) that for all .

Therefore, part follows easily from (2.12), (2.13), (2.15), and the numerical results and . □

Lemma 2.6

Let and Then the following statements are true:

If , then for all ;

If  , then there exists such that for and for .

Part follows easily from for all if .

For part , if  , then numerical computations lead to

It follows from (2.16) and (2.19) that for .

Therefore, part follows easily from (2.17), (2.18), and (2.20) together with the numerical results and . □

Main results

We are now in a position to state and prove our main results.

Theorem 3.1

The double inequality holds for all with if and only if and .

Clearly, inequality (3.1) can be rewritten as

Since , , and are symmetric and homogenous of degree one, we assume that . Let . Then from (1.1), (1.2), and (1.4) we know that inequality (3.2) is equivalent to

Let . Then and

Therefore, inequality (3.1) holds for all with if and only if and follows from (3.2)–(3.4) and Lemma 2.3. □

Theorem 3.2

The double inequality holds for all with if and only if and  .

Clearly, inequality (3.5) can be rewritten as

Since , , and are symmetric and homogenous of degree one, we assume that . Let . Then from (1.1), (1.3), and (1.5) we see that inequality (3.6) is equivalent to Let . Then and

Therefore, inequality (3.5) holds for all with if and only if and follows from (3.6)–(3.8) and Lemma 2.4. □

Theorem 3.3

The double inequality holds for all with if and only if and .

Since , , and are symmetric and homogenous of degree one, without loss generality, we assume that . Let , , and . Then , , and (1.1), (1.2), and (1.4) lead to Let Then simple computations lead to where where is defined as in Lemma 2.5.

We divide the proof into four cases.

Case 1 . Then it follows from (3.9)–(3.14) and Lemma 2.5(1) that

Case 2 . Let and . Then power series expansion leads to

Equations (3.9), (3.10), and (3.15) lead to the conclusion that there exists such that for all with .

Case 3 . Then (3.13) leads to

Let be the number given in Lemma 2.5(2). Then we divide the discussion into two subcases.

Subcase 1 . Then for follows easily from (3.13) and (3.14) together with Lemma 2.5(2).

Subcase 2 . Then Lemma 2.5(2) and (3.14) lead to the conclusion that is strictly decreasing on the interval . Then, from (3.16) and Subcase 1, we know that there exists such that for and for .

It follows from Subcases 1 and 2 together with (3.12) that is strictly increasing on and strictly decreasing on . Therefore, follows from (3.9)–(3.11) and (3.16) together with the piecewise monotonicity of .

Case 4 . Then (3.11) leads to

Equations (3.9) and (3.10) together with inequality (3.17) imply that there exists such that for all with . □

Theorem 3.4

The double inequality holds for all with if and only if and .

Since , , and are symmetric and homogenous of degree one, without loss generality, we assume that . Let , , and . Then , and (1.1), (1.3), and (1.5) lead to Let Then simple computations lead to where where is defined as in Lemma 2.6.

We divide the proof into four cases.

Case 1 . Then it follows from (3.18)–(3.23) and Lemma 2.6(1) that

Case 2 . Let and , then power series expansion leads to

Equations (3.18), (3.19), and (3.24) lead to the conclusion that there exists such that for all with .

Case 3 . Then, from (3.20) and (3.22) together with numerical computations, we get

Let be the number given in Lemma 2.6(2). Then we divide the discussion into two subcases.

Subcase 1 . Then for follows easily from (3.22) and (3.23) together with Lemma 2.6(2).

Subcase 2 . Then Lemma 2.6(2) and (3.23) lead to the conclusion that is strictly decreasing on the interval . Then, from (3.25) and Subcase 1, we know that there exists such that for and for .

It follows from Subcases 1 and 2 together with (3.21) that is strictly increasing on and strictly decreasing on . Therefore, follows from (3.18)–(3.20) and (3.25) together with the piecewise monotonicity of .

Case 4 . Then (3.21) leads to

Equations (3.18) and (3.19) together with inequality (3.26) imply that there exists such that for all with . □

Results and discussion

In this paper, we provide the optimal upper and lower bounds for the Sándor–Yang means and in terms of combinations of the arithmetic mean and the contra-harmonic mean . Our approach may have further applications in the theory of bivariate means.

Conclusion

In the article, we find several best possible bounds for the Sándor–Yang means and . These results are improvements and refinements of the previous results.