Refinements of bounds for the arithmetic mean by new Seiffert-like means

In the article, we present the sharp upper and lower bounds for the arithmetic mean in terms of new Seiffert-like means, which give some refinements of the results obtained in [1]. As applications, two new inequalities for the sine and hyperbolic sine functions will be established.

Sharp bounds for the Seiffert-like means and their related special functions have attracted the attention of several researchers [24][25][26]. In particular, the following chain of inequalities had been established in [2] for all a, b > 0 with a b, where A(a, b) = (a + b)/2 is the arithmetic mean.
Very recently, Nowicka and Witkowski [1] proved that the double inequalities

Lemmas
To prove our main results we need several lemmas, which we present in this section.
, then so is f /g .
The following lemma is a useful tool for dealing with the monotonicity of the ratio of two power series. The first part of Lemma 2.2 is first established by Biernacki and Krzyz [28], while the second part comes from Yang et al. [ [30]) Suppose that the power series f (x) = ∞ n=0 a n x n and g(x) = ∞ n=0 b n x n have the radius of convergence r > 0 with b n > 0 for all n ∈ N 0 = N ∪ {0}. Let h(x) = f (x)/g(x) and H f,g = ( f /g )g − f . Then the following statements hold true: (1) If the non-constant sequences {a n /b n } ∞ n=0 is increasing (decreasing) for all n ≥ 0, then h(x) is strictly increasing (decreasing) on (0, r); (2) If for certain m ∈ N, the sequence {a k /b k } 0≤k≤m and {a k /b k } k≥m both are non-constant, and they are increasing (decreasing), respectively. Then h(x) is strictly increasing (decreasing) on (0, r) if and is strictly increasing (decreasing) on (0, x 0 ) and strictly decreasing (increasing) on (x 0 , r).
Let us recall the Taylor series expansions for cot x and csc x, which can be found in [31].
Lemma 2.3. For |x| < π, then we have the Taylor series formulas where B 2n is the even-index Bernoulli numbers for n ∈ N.
For the readers' convenience, recall from [31, p.804, 23.1.1] that the Bernoulli numbers B n may be defined by the power series expansion The first few Bernoulli numbers B 2k are Some other Taylor series formulas for the functions involving cot x and csc x can be obtained from Lemma 2.3 by differentiation.
Lemma 2.5. Let B 2n be the even-index Bernoulli numbers for n ∈ N. Then Proof. Differentiation yields which in conjunction with Lemma 2.3 gives the desired results.
It can be easily seen from (2.1) and Lemma 2.2(1) that Lemma 2.6 will be proved if we can show that the sequence a n b n = 2 2n−1 + 1 2(2 2n − 1) is strictly increasing for n ≥ 1.

Main results
Theorem 3.1. The double inequality Proof. Since M sin (a, b), M tan (a, b) and A(a, b) are symmetric and homogeneous of degree one, without loss of generality, we may assume that a > b > 0.