double inequality

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A Double Inequality for Gamma Function

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.

A New Lower Bound in the Second Kershaw's Double Inequality

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.

A Class of Logarithmically Completely Monotonic Functions and the Best Bounds in the First Kershaw's Double Inequality

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.

New Upper Bounds in the Second Kershaw's Double Inequality and its Generalizations

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).

A best possible double inequality between Seiffert and harmonic means

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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A Double Inequality for the Ratio of Two Consecutive Bernoulli Numbers

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.

A Sharp Double Inequality for Sums of Powers

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]

An Optimal Double Inequality for Means

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

An Optimal Double Inequality between Power Type Heron and Seiffert Means

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]

A Double Inequality for Divided Differences and Some Identities of Psi and Polygamma Functions

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]

Refinements, Extensions and Generalizations of the Second Kershaw's Double Inequality

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 Class of Logarithmically Completely Monotonic Functions and the Best Bounds in the Second Kershaw's Double Inequality

(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.

Bounds for the Ratio of Two Gamma Functions

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.

Sharpening and Generalizations of Shafer's Inequality for the Arc Tangent Function

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]

On a Result of Hardy and Ramanujan

[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]

Refined quadratic estimations of Shafer’s inequality

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]

Proof of an open inequality with double power exponential functions

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.

Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems

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,

GENERALIZED WEIGHTED CEBYSEV AND OSTROWSKI TYPE INEQUALITIES FOR DOUBLE INTEGRALS

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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On the Generalized Ostrowski Type Integral Inequality for Double Integrals

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].