Abstract:
For a,b>0 with a\not=b, let T(a,b) denote the second Seiffert mean defined by T(a,b)=((a-b)/(2arctan((a-b)/(a+b)))) and A_{r}(a,b) denote the r-order power mean. We present the sharp bounds for the second Seiffert mean in terms of power means: A_{p_1}(a,b)

Abstract:
In this note we obtain sharp bounds for the Seiffert mean in terms of a two parameter family of means. Our results generalize and extend the recent bounds presented in the Journal of Inequalities and Applications (2012) and Abstract and Applied Analysis (2012).

Abstract:
We derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values , and the least values , such that the double inequalities and hold for all with . Here, , , , and denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers and , respectively.

Abstract:
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. 1. Introduction For , the power mean of order and the Seiffert mean of two positive numbers and are defined by The main properties of the power mean are given in [1]. It is well known that is strictly increasing with respect to for fixed with . Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities for the power mean can be found in the literature [2–16]. The Seiffert mean was introduced by Seiffert in [17], it can be rewritten in the following symmetric form (see [18, ]): Let , , and be the arithmetic, geometric, logarithmic, harmonic, and identric means of two positive numbers and , respectively. Then it is well known that for all with . In [9], Alzer and Janous presented the sharp power mean bounds for the sum as follows: for all with . In [17], Seiffert proved that for all with . The following power mean bounds for the Seiffert mean was given by Jagers [19]: for all with . In [20, 21], the authors presented the bounds for the Seiffert mean in terms of and as follows: for all with . The following sharp lower power mean bounds for , , and can be found in [4, 6]: for all with . The purpose of this paper is to answer the question: for , what are the greatest value and the least value such that the double inequality holds for all with . 2. Lemmas In order to prove our main result, we need several lemmas which we present in this section. Lemma 2.1. Let , and . Then there exists such that for , for and . Proof. Simple computations lead to Inequality (2.5) implies that is strictly increasing in . Then (2.3) and (2.4) lead to that there exists such that for and for . Hence, is strictly decreasing in and strictly increasing in . Therefore, Lemma 2.1 follows from (2.1) and (2.2) together with the monotonicity of . Lemma 2.2. If , then the following statements are true:(1) ;(2) ;(3) . Proof. Simple computations lead to + = ; + + = ; + + = . Lemma 2.3. If , then holds for all with . Proof. Without loss of generality, we assume that . Let and . Then Let Then simple computations lead to where ？/？ + , where + + , Let and . Then From (2.21)–(2.23) we clearly see that for , hence is strictly increasing in . Then (2.20) implies that for , hence is strictly increasing in . It follows from (2.18) and the monotonicity of that for , hence is strictly increasing in . Then

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

Abstract:
We find the greatest value α and least value β such that the double inequality αA(a,b)+(1-α)H(a,b)0 with a≠b. Here A(a,b), H(a,b), and P(a,b) denote the arithmetic, harmonic, and Seiffert's means of two positive numbers a and b, respectively.

Abstract:
In a pair of recent articles [PRL 105 (2010) 041302 - arXiv:1005.1132; JHEP 1103 (2011) 056 - arXiv:1012.2867] two of the current authors have developed an entropy bound for equilibrium uncollapsed matter using only classical general relativity, basic thermodynamics, and the Unruh effect. An odd feature of that bound, S <= A/2, was that the proportionality constant, 1/2, was weaker than that expected from black hole thermodynamics, 1/4. In the current article we strengthen the previous results by obtaining a bound involving the (suitably averaged) w parameter. Simple causality arguments restrict this averaged parameter to be <= 1. When equality holds, the entropy bound saturates at the value expected based on black hole thermodynamics. We also add some clarifying comments regarding the (net) positivity of the chemical potential. Overall, we find that even in the absence of any black hole region, we can nevertheless get arbitrarily close to the Bekenstein entropy.

Abstract:
In this paper, we find the greatest values $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $\alpha_{4}$, $\alpha_{5}$, $\alpha_{6}$, $\alpha_{7}$, $\alpha_{8}$ and the least values $\beta_{1}$, $\beta_{2}$, $\beta_{3}$, $\beta_{4}$, $\beta_{5}$, $\beta_{6}$, $\beta_{7}$, $\beta_{8}$ such that the double inequalities $$A^{\alpha_{1}}(a,b)G^{1-\alpha_{1}}(a,b)0$ with $a\neq b$, where $G$, $A$ and $Q$ are respectively the geometric, arithmetic and quadratic means, and $N_{GA}$, $N_{AG}$, $N_{AQ}$ and $N_{QA}$ are the Neuman means derived from the Schwab-Borchardt mean.