与Gamma函数有关的对数完全单调函数及其应用

为了完善函数G_(α,β)(x)(其中参数α∈R,β≥0)及函数1/G_(α,β)(x)在区间(0,∞)上的对数完全单调性和相关不等式,利用Taylor展开式、Gamma函数、Psi函数的级数表达式和积分表达式研究了函数G_(α,β)(x)和函数1/G_(α,β)(x)数的对数完全单调性,将函数G_(α,β)(x)和函数1/G_(α,β)(x)对数完全单调的充分条件扩大;利用对数完全单调性得到新的不等式,并通过对特殊情形的研究,得到一个形式简单对称的双边不等式,该不等式对阶乘数之乘积与∏nk=1k~k的商做出估计.     

包含函数Γ（x）的对数完全单调函数及不等式

基于欧拉Gamma函数的奇特性质,利用函数的单调性理论以及一些简单函数的积分表达式与级数展开式证明了函数f_α(x),α∈R和函数s(x)的对数完全单调性,并利用该性质得出了一个比原有结论更精确的不等式以及一个双边不等式.     

两个广义伽玛分布之间新的Kullback-Leibler距离

在Kullback-Leibler距离的基础上,对Kullback-Leibler距离进行改进,给出了新的Kullback-Leibler距离,并讨论了它的性质.计算了两个不同广义伽玛分布之间新的Kullback-Leibler距离.推导出伽玛分布、Weibull分布、Rayleigh分布、正态分布、指数分布新的Kullback-Leibler距离.另外在新的KullbackLeibler距离下,还得到digamma函数Ψ(x)=(Γ'(x)/(Γ(x))为单调递增函数.     

关于伽玛函数的单调性质

伽玛函数的单调性质和对数完全单调性质被获得了.     

一个著名特殊函数恒等式的多元推广及其概率证明

将贝塔函数和伽玛函数推广到多元情形,利用随机变量的独立性和概率密度变换公式,得到广义贝塔函数和广义伽玛函数之间的一个重要关系式,它是贝塔函数与伽玛函数关系式:B(a,b)=(Γ(a)Γ(b))/(Γ(a+b))(a0,b0)的推广,还得到一些更加一般的积分恒等式.     

利用函数的单调性解竞赛题

函数的单调性在竞赛中应用十分广泛 ,它对于研究图像的特征、确定函数的值域 ,有着重要作用 .同时 ,因为单调函数y =f(x)中x与y是一一对应的 ,所以就可把复杂的高次方程 f(x) =f(a)化为简单的方程x =a ,高次不等式 f(x)≥f(a)化为简单的不等式x≥a或x≤a ,从而使问题驭繁为简 .本文就此特性 ,举例加以说明 .例 1　求不等式 12 x+5 x≤ 13 x 的解集 .(第 13届“希望杯”竞赛试题 )分析　若将不等式 12 x+5 x≤ 13 x 变形为( 1213 ) x+( 513 ) x≤ 1,则可引进函数f(x) =( 1213 ) x+( 513 ) x,利用函数 y =f(x)的单调性帮助解决问题 .解　原不…     

利用单调函数的性质证明一类不等式

文[1]用单调函数的性质,变更定义中的表达形式,非常简单地证明了一类不等式,读后深受启发．如果变更定义中的表达形式为f(x1)-f(x2)＜0(或＞0),f(x2)/f(x1)＞1(或＜1),解决我们常用数学归纳法证明的一类数列不等式,将收到较好的效果．     

单调函数及其应用

设 f为定义在D上的函数 ,若对于D中任意两个数x1,x2 ,当x1f(x2 )时 ,称 f为D上严格递减函数 .递增函数和递减函数统称为单调函数 ,函数的单调性是函数的重要性质之一 ,利用函数的单调性 ,可以比较函数值的大小 ,证明一些不等式以及解决某些方程问题和函数极值问题 .例 1　证明 |x1+x2 +… +xn|1+|x1+x2 +… +xn|≤ |x1|1+|x1|+|x2 …     

一类函数的对数完全单调性

通过对参数λ,μ的讨论,主要利用函数的单调性理论,已有对数完全单调函数的性质以及幂函数的积分表达式研究了函数Gλ,μ(x)及函数[Gλ,μ(x)]-1的对数完全单调性,并在此基础上得到了一定条件下函数Gλ,μ(x)及[Gλ,μ(x)]-1对数完全单调的充要条件.     

如何用导数判断函数单调性

函数的单调性是函数的重要性质,也是高考的热点问题,若利用函数定义求解,一般较为复杂.但是利用导数求函数的单调就有效地解决了这一难题.一般地,设函数y=f(x)在某个区间内可导,如果f′(x)>0,则f(x)为增函数;如果f′(x)<0,则f(x)为减函数.下面对利用导数判断函数的单调性的几个注意点加以说明.一、f′(x)>0(<0)是f(x)为增(减)函数的充分不必要条件例1用导数来判断函数f(x)=x3(x∈     

Gamma triads

The Ramanujan Journal - Although values of the gamma function have been evaluated only at the integers and the half-integers, some products of three gamma functions, evaluated at rational numbers,...     

Gamma function inequalities

We prove various new inequalities for Euler’s gamma function. One of our theorems states that the double-inequality $$\alpha \cdot \Bigl(\frac{1}{\Gamma\,(\sqrt{x})}+\frac{1}{\Gamma\,(\sqrt{y})}\Bigr) {\kern-1pt}<{\kern-1pt} \frac{1}{\Gamma\,( \sqrt{x+y-xy})}+ \frac{1}{\Gamma\,( \sqrt{xy})} {\kern-1pt} <{\kern-1pt} \beta \cdot \Bigl( \frac{1}{\Gamma\,(\sqrt{x})}+\frac{1}{\Gamma\,(\sqrt{y})}\Bigr)$$ is valid for all real numbers x,y?∈?(0,1) with the best possible constant factors $\alpha=1/\sqrt{2}=0.707...$ and β?=?1.     

Group Graded Gamma Rings

ＧｒｏｕｐＧｒａｄｅｄＧａｍｍａＲｉｎｇｓＦａｎＦｕｓｈｅｎｇ（樊复生）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮｏｒｔｈｅａｓｔｅｒｎＵｎｉｖｅｒｓｉｔｙ，Ｓｈｅｎｙａｎｇ，１１０００６）ＧｕｏＬｉｎｇｚｈｏｎｇ（郭令忠）（Ｄｅｐａｒｔｍ...     

Approximation by Gamma type operators

In this study, we introduce newly defined Gamma operators which preserve constants and e^2μ·, μ>0 functions. In accordance with this purpose, we focus on their approximation properties such as uniform convergence, rate of convergence, asymptotic formula, and saturation results. Superior properties of introduced operators have been tested both theoretically and numerically in certain senses to highlight the performance of the new constructions of Gamma operators.     

Fractional Gamma and Gamma-Subordinated Processes

We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order ν > 0. In the case ν > 1, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index 1/ν. For ν less than one an analogous result holds, with the subordinator replaced by the inverse. In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value. Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see i.e., [42 Van Noortwijk, J.M., 2009. A survey of the application of Gamma processes in maintenance. Reliability Engineering System Safety 94: 2–21.[Crossref], [Web of Science ®] , [Google Scholar]] and the references therein).

As a consequence of the previous results, the fractional generalizations of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained. These processes are particularly relevant for a wide range of financial and technological applications.     

利用Gamma函数求积分的几种形式

借助幂函数与对数函数的变量替换对Gamma函数从形式上加以推广,使Gamma函数中指数函数部分为指数函数与幂函数或对数函数与幂函数的复合函数时仍可求值,以扩大Gamma函数的使用范围.     

Gamma invariants for dense lattices

A subclass of the class of dense lattices, called strongly dense lattices, is defined and its properties investigated. In particular, an invariant of strongly dense lattices is defined and shown to take all possible values. Connections to module theory are explored, including some constructions of von Neumann regular rings with strongly dense lattices of two-sided ideals. Received September 16, 1997; accepted in final form July 22, 1998.     

Characterization of the Gamma semigroup

The Gamma semigroup with parameter \(b>0\) on \(L^p(\mathbb R^+)\) is defined by

$$\begin{aligned} W_b(t)f(x)=\frac{1}{\Gamma (t)}\int _0^x(x-y)^{t-1}e^{-b(x-y)}f(y)\,dy. \end{aligned}$$

Let S denote the multiplication operator \(f(x)\rightarrow xf(x)\) with maximal domain D(S) in \(L^p(\mathbb R^+)\). The bounded operator V on \(L^p(\mathbb R^+)\) is S-Volterra if D(S) is V-invariant and \([S,V]=V^2\) on D(S). For \(1<p<\infty \), we characterize the Gamma semigroup as the unique regular semigroup \(V(\cdot )\) on \(L^p(\mathbb R^+)\) with imaginary type less than \(\pi \), such that V(1) is S-Volterra and \(V(1)u^b=Su^b\), where \(u^b(x):=e^{-bx}\).

     

Mixtures of Gamma Distributions With Applications

This article proposes a Bayesian density estimation method based upon mixtures of gamma distributions. It considers both the cases of known mixture size, using a Gibbs sampling scheme with a Metropolis step, and unknown mixture size, using a reversible jump technique that allows us to move from one mixture size to another. We illustrate our methods using a number of simulated datasets, generated from distributions covering a wide range of cases: single distributions, mixtures of distributions with equal means and different variances, mixtures of distributions with different means and small variances and, finally, a distribution contaminated by low-weighted distributions with different means and equal, small variances. An application to estimation of some quantities for a M/G/1 queue is given, using real E-mail data from CNR-IAMI.