Bo-Yong Long

Optimal Inequalities for Generalized Logarithmic, Arithmetic, and Geometric Means

For , the generalized logarithmic mean , arithmetic mean , and geometric mean of two positive numbers and are defined by , for , , for , , and , , for , and , , for , and , , and , respectively. In this paper, we find the greatest value (or least value , resp.) such that the inequality (or , resp.) holds for (or , resp.) and all with .

Bounds of the Neuman-Sándor Mean Using Power and Identric Means

In this paper we find the best possible lower power mean bounds for the Neuman-Sandor mean and present the sharp bounds for the ratio of the Neuman-Sandor and identric means.

Sharp bounds for Seiffert and Neuman-Sándor means in terms of generalized logarithmic means

In this paper, we prove three sharp inequalities as follows: P(a,b)>L2(a,b), T(a,b)>L5(a,b) and M(a,b)>L4(a,b) for all a,b>0 with a≠b. Here, Lr(a,b), M(a,b), P(a,b) and T(a,b) are the r th generalized logarithmic, Neuman-Sándor, first and second Seiffert means of a and b, respectively.MSC:26E60.

Optimal Power Mean Bounds for the Weighted Geometric Mean of Classical Means

For , the power mean of order of two positive numbers and is defined by , for , and , for . In this paper, we answer the question: what are the greatest value and the least value such that the double inequality holds for all and with ? Here , , and denote the classical arithmetic, geometric, and harmonic means, respectively.

Optimal generalized Heronian mean bounds for the logarithmic mean

In this article, we establish a double inequality between the generalized Heronian and logarithmic means. The achieved result is inspired by the articles of Lin and Shi et al., and the methods from Janous. The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal.2010 Mathematics Subject Classification: 26E60.

Sharp bounds by the power mean for the generalized Heronian mean

In this article, we answer the question: For p, ω ∈ ℝ with ω > 0 and p(ω - 2) ≠ 0, what are the greatest value r1 = r1(p, ω) and the least value r2 = r2(p, ω) such that the double inequality Mr1a,b 0 with a ≠ b? Here Hp,ω(a, b) and Mr (a, b) denote the generalized Heronian mean and r th power mean of two positive numbers a and b, respectively.2010 Mathematics Subject Classification: 26E60.

Optimal Inequalities for Power Means

We present the best possible power mean bounds for the product for any , , and all with . Here, is the th power mean of two positive numbers and .

A Best Possible Double Inequality for Power Mean

We answer the question: for any with and , what are the greatest value and the least value , such that the double inequality holds for all with ? Where is the th power mean of two positive numbers and .

Optimal bounds for the first Seiffert mean in terms of the convex combination of the logarithmic and Neuman-Sándor mean

In this paper, we find the least value α and the greatest value β such that the double inequality αL(a,b)+(1−α)M(a,b) < P(a,b) < βL(a,b)+(1−β)M(a,b) holds for all a,b > 0 with a = b , where L(a,b),M(a,b) and P(a,b) are the logarithmic, the Neuman-Sándor, and the first Seiffert means of two positive numbers a and b , respectively. Mathematics subject classification (2010): 26E60.

Several sharp inequalities about the first Seiffert mean

In this paper, we deal with the problem of finding the best possible bounds for the first Seiffert mean in terms of the geometric combination of logarithmic and the Neuman–Sándor means, and in terms of the geometric combination of logarithmic and the second Seiffert means.