double inequality

Top PDF double inequality:

A Double Inequality for Gamma Function

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.

7 Read more

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

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.

8 Read more

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

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.

10 Read more

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

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).

8 Read more

A best possible double inequality between Seiffert and harmonic means

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
Show more

7 Read more

A Double Inequality for the Ratio of Two Consecutive Bernoulli Numbers

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.

7 Read more

A Sharp Double Inequality for Sums of Powers

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]

7 Read more

An Optimal Double Inequality for Means

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]

11 Read more

An Optimal Double Inequality between Power Type Heron and Seiffert Means

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]

11 Read more

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

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]

5 Read more

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

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.

8 Read more

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

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.

14 Read more

Bounds for the Ratio of Two Gamma Functions

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.

84 Read more

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

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]

9 Read more

On a Result of Hardy and Ramanujan

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]

6 Read more

Refined quadratic estimations of Shafer’s inequality

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]

11 Read more

Proof of an open inequality with double power exponential functions

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.

11 Read more

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

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,

10 Read more

GENERALIZED WEIGHTED CEBYSEV AND OSTROWSKI TYPE INEQUALITIES FOR DOUBLE INTEGRALS

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].
Show more

10 Read more

On the Generalized Ostrowski Type Integral Inequality for Double Integrals

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].

6 Read more

Show all 10000 documents...