Yu-Ming Chu

The Optimal Convex Combination Bounds of Arithmetic and Harmonic Means for the Seiffert's Mean

We find the greatest value and least value such that the double inequality holds for all with . Here , , and denote the arithmetic, harmonic, and Seifferts means of two positive numbers and , respectively.

Sharp Generalized Seiffert Mean Bounds for Toader Mean

For 𝑝∈[0,1], the generalized Seiffert mean of two positive numbers 𝑎 and 𝑏 is defined by 𝑆𝑝(𝑎,𝑏)=𝑝(𝑎−𝑏)/arctan[2𝑝(𝑎−𝑏)/(𝑎

Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means

We answer the question: for , what are the greatest value and the least value such that the double inequality holds for all with . Here, , , and denote the power of order , Seiffert, and geometric means of two positive numbers and , respectively.

An Optimal Double Inequality between Power-Type Heron and Seiffert Means

For , the power-type Heron mean and the Seiffert mean of two positive real numbers and are defined by , ; , and , ; , , respectively. In this paper, we find the greatest value and the least value such that the double inequality holds for all with .

Some Comparison Inequalities for Generalized Muirhead and Identric Means

For , with , the generalized Muirhead mean with parameters and and the identric mean are defined by and , , , , respectively. In this paper, the following results are established: (1) for all with and ; (2) for all with and ; (3) if , then there exist such that and .

On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

We prove that the double inequality ( 𝜋 / 2 ) ( a r t h 𝑟 / 𝑟 ) 3 / 4 + 𝛼 ∗ 𝑟 𝒦 ( 𝑟 ) ( 𝜋 / 2 ) ( a r t h 𝑟 / 𝑟 ) 3 / 4 + 𝛽 ∗ 𝑟 holds for all 𝑟 ∈ ( 0 , 1 ) with the best possible constants 𝛼 ∗ = 0 and 𝛽 ∗ = 1 / 4 , which answer to an open problem proposed by Alzer and Qiu. Here, 𝒦 ( 𝑟 ) is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.