Sharp upper and lower bounds for the gamma function

2009 ◽  
Vol 139 (4) ◽  
pp. 709-718 ◽  
Author(s):  
Horst Alzer
Keyword(s):  
Lower Bounds ◽  
Gamma Function ◽  
Upper And Lower Bounds ◽  
Image Position

We prove that for all x > 0, we havewith the best possible constants α = 0 and $\beta=\tfrac{1}{1620}$.

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

scholarly journals ON THE ORDER OF MAGNITUDE OF SUMS OF NEGATIVE POWERS OF INTEGRATED PROCESSES

2012 ◽  
Vol 29 (3) ◽  
pp. 642-658 ◽  
Author(s):  
Benedikt M. Pötscher
Keyword(s):  
Lower Bounds ◽  
Random Variables ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Integrated Process ◽  
Order Of Magnitude ◽  
Image Position ◽  
Integrated Processes

Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.

On the Zeros of Power Series with Exponential Logarithmic Coefficients

1981 ◽  
Vol 24 (3) ◽  
pp. 257-271 ◽  
Author(s):  
W. Gawronski ◽  
U. Stadtmüller
Keyword(s):  
Power Series ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Number Of Zeros ◽  
Logarithmic Coefficients ◽  
Image Position

In this paper we investigate the zeros of power series1for some functions of coefficients A. In particular, we derive upper and lower bounds for the number of zeros of f in its domain of analyticity.

Multifractal dimensions of product measures

1996 ◽  
Vol 120 (4) ◽  
pp. 709-734 ◽  
Author(s):  
L. Olsen
Keyword(s):  
Probability Measure ◽  
Lower Bounds ◽  
Borel Probability Measure ◽  
Multifractal Spectrum ◽  
Upper And Lower Bounds ◽  
Packing Measure ◽  
Multifractal Spectra ◽  
Product Measures ◽  
Borel Probability ◽  
Image Position

AbstractWe study the multifractal structure of product measures. for a Borel probability measure μ and q, t Є , let and denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [O11] Let μ be a Borel probability merasure on k and let v be a Borel probability measure on t. Fix q, s, t Є . We prove that there exists a number c > 0 such that for E ⊆k, F ⊆l and Hk+l provided that μ and ν satisfy the so-called Federer condition.Using these inequalities we give upper and lower bounds for the multifractal spectrum of μ × ν in terms of the multifractal spectra of μ and ν

scholarly journals COUNTING THE NUMBER OF SOLUTIONS TO THE ERDŐS–STRAUS EQUATION ON UNIT FRACTIONS

2013 ◽  
Vol 94 (1) ◽  
pp. 50-105 ◽  
Author(s):  
CHRISTIAN ELSHOLTZ ◽  
TERENCE TAO
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Diophantine Equation ◽  
Upper And Lower Bounds ◽  
Natural Numbers ◽  
Number Of Solutions ◽  
Waring’S Problem ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

AbstractFor any positive integer $n$, let $f(n)$ denote the number of solutions to the Diophantine equation $$\begin{eqnarray*}\frac{4}{n} = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}\end{eqnarray*}$$ with $x, y, z$ positive integers. The Erdős–Straus conjecture asserts that $f(n)\gt 0$ for every $n\geq 2$. In this paper we obtain a number of upper and lower bounds for $f(n)$ or $f(p)$ for typical values of natural numbers $n$ and primes $p$. For instance, we establish that $$\begin{eqnarray*}N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\ll \displaystyle \sum _{p\leq N}f(p)\ll N\hspace{0.167em} {\mathop{\log }\nolimits }^{2} N\log \log N.\end{eqnarray*}$$ These upper and lower bounds show that a typical prime has a small number of solutions to the Erdős–Straus Diophantine equation; small, when compared with other additive problems, like Waring’s problem.

Some inequalities for arithmetic and geometric means

1999 ◽  
Vol 129 (2) ◽  
pp. 221-228 ◽  
Author(s):  
Horst Alzer
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Geometric Means ◽  
Weighted Arithmetic ◽  
Image Position

Let An and Gn (respectively, A′n and G′n) be the weighted arithmetic and geometric means of x1, …, xn (respectively, 1 – x1, …, 1 – xn). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such thatholds for all xi ∈ [a, b] (i = 1, …, n; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author.

scholarly journals Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Faris Alzahrani ◽  
Ahmed Salem ◽  
Moustafa El-Shahed
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Gamma Function ◽  
Open Problem ◽  
Upper And Lower Bounds ◽  
Digamma Function ◽  
Sharp Bounds ◽  
Gamma Functions ◽  
Numer Math

AbstractIn the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$ Γ q ( x + 1 ) / Γ q ( x + s ) for all real number s and $0< q\neq1$ 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$ 0 < s < 1 when letting $q\to 1$ q → 1 and a new inequality for all $s>1$ s > 1 .

scholarly journals On Some Complete Monotonicity of Functions Related to Generalized k − Gamma Function

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Hesham Moustafa ◽  
Hanan Almuashi ◽  
Mansour Mahmoud
Keyword(s):  
Lower Bounds ◽  
Gamma Function ◽  
Logarithmic Derivative ◽  
Complete Monotonicity ◽  
Upper And Lower Bounds ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

In this paper, we presented two completely monotonic functions involving the generalized k − gamma function Γ k x and its logarithmic derivative ψ k x , and established some upper and lower bounds for Γ k x in terms of ψ k x .

An Asymptotic Formula for a Sum of Products of Powers

1978 ◽  
Vol 21 (4) ◽  
pp. 427-433
Author(s):  
Ronald J. Evans
Keyword(s):  
Asymptotic Formula ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Sum Of Products ◽  
Image Position

Fix an integer r ≥ 2 and positive numbers b1, …, br. Write σ = b1+ …+br Let . In this note we evaluate the constant A (when it exists) for which1where the sum is over all vectors2We also obtain upper and lower bounds for the sum in (1).

Estimates for the state density for ordinary differential operators with white gaussian noise potential

1983 ◽  
Vol 95 (3-4) ◽  
pp. 263-274
Author(s):  
V. B. Moscatelli ◽  
M. Thompson
Keyword(s):  
Wiener Process ◽  
Lower Bounds ◽  
Gaussian Noise ◽  
Differential Operators ◽  
Asymptotic Properties ◽  
State Density ◽  
The State ◽  
Upper And Lower Bounds ◽  
Ordinary Differential Operators ◽  
Image Position

SynopsisThe present paper is concerned with developing the existence and asymptotic properties of the state density N(λ) associated with certain higher order random ordinary differential operators A of the formwhere Ao has homogeneous and ergodic coefficients with respect to the σ-algebra generated by the Wiener process q(ω, x). The analysis uses the Weyl min-max principle to determine rough upper and lower bounds for N(λ).

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