Using the Alzer integral **inequality** and the elementary properties of the gamma function, a **double** **inequality** for gamma function is established, which is an improvement of Merkle’s **inequality**. Copyright q 2009 X. Zhang and Y. Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. In the paper, a new and elegant lower bound in the second Ker- shaw’s **double** **inequality** is established, some alternative simple and polished proofs are given, several deduced functions involving the gamma and psi func- tions are proved to be decreasingly monotonic and logarithmically completely monotonic, and some remarks and comparisons are stated.

Abstract. In the article, the logarithmically complete monotonicity of a class of functions involving the Euler’s gamma function are proved, a class of the first Kershaw type **double** inequalities are established, and the first Kershaw’s **double** **inequality** and Wendel’s **inequality** are generalized, refined or extended. Moreover, an open problem is posed.

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Remark 1. Since I(a, b) < A(a, b) for positive numbers a and b with a 6= b, in- equality (9) refines the right hand side inequalities in (5) and (8). This means that Theorem 1 refines, extends and generalizes the right hand side **inequality** in the second Kershaw’s **double** **inequality** (5).

Very recently, Wang and Chu [8] found the greatest value a and the least value b such that the **double** **inequality** A a (a, b)H 1- a (a, b) < P(a, b) < A b (a, b)H 1- b (a, b) holds for a, b >0 with a ≠ b; For any a Î (0, 1), Chu et al. [10] presented the best-possible bounds for P a ( a , b ) G 1 -a ( a , b ) in terms of the power mean; In [2], the authors proved that the **double** **inequality** aA ( a , b ) + (1 - a ) H ( a , b ) < P ( a , b ) < bA ( a , b ) + (1 - b ) H ( a , b ) holds for all a , b > 0 with a ≠ b if and only if a ≤ 2/ π and b ≥ 5/6; Liu and Meng [5] proved that the inequalities

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These numerical values hint us the approximating accuracy of the lower and up- per bounds in the **double** **inequality** (1.3) for the ratio of two consecutive Bernoulli numbers with even indexes: as k becomes larger and larger, the **double** inequal- ity (1.3) becomes more and more accurate.

In this note, using only elementary techniques, we demonstrate that the sequence Sn is strictly increasing and that 1.1a holds; in addition, we establish a sharp estimate of the rate of [r]

Chen, “The monotonicity of the ratio between generalized logarithmic means,” Journal of Mathematical Analysis and Applications, vol.. Chen, “Refinements, extensions and generalizations o[r]

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Wang, “The optimal convex combination bounds of arithmetic and harmonic means for the Seiﬀert’s mean,” Journal of Inequalities and Applications, vol.. Chu, “An optimal inequality for pow[r]

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the exponential function, RGMIA Res. Qi, Three-log-convexity for a class of elementary functions involving exponential function, J. Chen, A complete monotonicity property of the gamma fu[r]

There have been a lot of literature about these two **double** inequalities and their history, background, refinements, extensions, generalizations and applications. For more information, please refer to [5, 7, 9, 10, 11, 12, 14, 19, 20, 21, 22, 23, 30, 32, 34, 35, 36, 38, 39, 40, 41, 49] and the references therein.

(a, b, c) ∈ {(a, b, c) : (b −a)(1−a−b +2c) ≤ 0}∩{(a, b, c) : (b−a)(|a−b|−a−b+2c) ≤ 0}\{(a, b, c) : b = c +1 = a+1}\{(a, b, c) : a = c +1 = b+1}. These conclusions can be used to extend, generalize, refine and sharpen [25, Theorem 1], **inequality** (2) and some other known results.

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This means that Wendel’s **double** **inequality** 2.8 and Gautschi’s first **double** **inequality** 2.22 are not included in each other but they all contain Gautschi’s second **double** **inequality** 2.23. Remark 2.17. By the convex property of ln Γx, Merkle recovered in 20, 59–63 inequalities in 2.22 and 2.23 once again. See Section 4.

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For possible applications of the double inequality 2.11 in the theory of approximations, the accuracy of bounds in 2.11 for the arc tangent function is described by Figures 1 and 2.... J[r]

[4] Necdet Batir, An Interesting Double Inequality for Euler’s Gamma Function, JIPAM , 5 (4) (2004), Article 97. (submitted for publication) [10] Melvyn B[r]

Especially, Zhu [] showed an upper bound for inequality . and proved that the following double inequality x + +.. This article is distributed under the terms of the Creative C[r]

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Cîrtoaje (J. Nonlinear Sci. Appl. 4(2):130-137, 2011) conjectured that the **inequality** a (2b) x + b (2a) x ≤ 1 with **double** power-exponential functions holds for all nonnegative real numbers a, b with a + b = 1 and all x ≥ 1. In this paper, we shall prove the conjecture aﬃrmatively.

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In this paper, we introduce a more general approach which consists in finding a par- ticular part of the solution set of a given fixed point problem, i.e., fixed points which solve a variational **inequality**. More precisely, the goal of this paper is to present a method for finding hierarchically a fixed point of a nonexpansive semigroup S = {T(s)} s ≥ 0 with respect to another monotone operator A, namely,

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Recently, many authors have studied on ˇ Cebysev **inequality** for **double** integrals, please see ([7], [8], [11] [15]). In [8], Guazene-Lakoud and Aissaoui established a weighted ˇ Cebysev type **inequality** for **double** integrals using the probability density function. In this paper, we obtain a generalized weighted ˇ Cebysev type **inequality** similar to this **inequality** for **double** integrals using the weighted funtions which are not necessarily the probability density functions. Moreover, we established an Ostrowski type **inequality** for **double** integral which is the generalization of the **inequality** given in [17].

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In [3], the **inequality** (1.2) is established by the use of integral identity involving Peano kernels. In [7], Pachpatte obtained an **inequality** in the view (1.2) by using elementary analysis. The interested reader is also refered to ( [3], [4], [6]- [13]) for Ostrowski type inequalities in several independent variables and for recent weighted version of these type inequalities see [1], [2], [9] and [11].