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 a, b > 0 with a = b. Then the arithmetic mean A(a, b) [1][2][3][4], the quadratic mean Q(a, b) [5], the contra-harmonic mean C(a, b) [6][7][8][9], the Neuman-Sándor mean NS(a, b) [10][11][12], the second Seiffert mean T(a, b) [13,14], and the Schwab-Borchardt mean SB(a, b) [15,16] of a and b are defined by respectively, where sinh -1 (x) = log(x + √ x 2 + 1) and cosh -1 (x) = log(x + √ x 2 -1) are respectively the inverse hyperbolic sine and cosine functions. The Schwab-Borchardt mean SB(a, b) 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 .
Let X(a, b) and Y (a, b) denote symmetric bivariate means of a and b. Then Yang [39] introduced the Sándor-Yang mean R XY (a, b) = Y (a, b)e X(a,b) SB[X(a,b),Y (a,b)] -1 and presented the explicit formulas for R QA (a, b) and R AQ (a, b) as follows: Very recently, the bounds involving the Sándor-Yang means have been the subject of intensive research. Numerous interesting results and inequalities for R QA (a, b) and R AQ (a, b) can be found in the literature [40][41][42].

Lemmas
In order to prove our main results, we need several lemmas, which we present in this section. .
is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2 (see [46]) Let A(t) = ∞ k=0 a k t k and B(t) = ∞ k=0 b k t k be two real power series converging on (-r, r) (r > 0) with b k > 0 for all k. If the non-constant sequence is increasing (decreasing) for all k, then the function t → A(t)/B(t) is strictly increasing (decreasing) on (0, r).

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

Results and discussion
In this paper, we provide the optimal upper and lower bounds for the Sándor-Yang means R QA (a, b) and R AQ (a, b) in terms of combinations of the arithmetic mean A(a, b) and the contra-harmonic mean C(a, b). 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 R QA (a, b) and R AQ (a, b). These results are improvements and refinements of the previous results.