Weighted integral and integro-differential inequalities

A general weighted integral inequality for two continuous functions on an interval [a,b] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson's integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals.     

指数凸函数的积分不等式及其在Gamma函数中的应用

仿对数凸函数的概念,给出指数凸函数的定义,并证明有关指数凸函数的几个积分不等式,作为应用,得到一个新的Kershaw型双向不等式.     

A monotonicity property of the composition of regularized and inverted-regularized gamma functions with applications

The gamma function and its various modifications such as the (upper) incomplete, regularized and inverted-regularized incomplete gamma functions are of importance in both theory and applications. In this note we observe an ‘if and only if ’ relationship between a certain axiom of insurance risk management and a monotonicity property of the composition of regularized and inverted-regularized incomplete gamma functions, assuming that the risks follow gamma distributions. We derive the monotonicity property by utilizing the above noted relationship and a probabilistic technique. The aforementioned insurance axiom, called consistent no-undercut, is explained in detail and related to several techniques of analysis.     

Inequalities for Cyclic Functions

The nth cyclic function is defined by

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We prove that if k is an integer with 1kn−1, then

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holds for all positive real numbers x with the best possible constantsα=1 and β= 2n-k over n.     

The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form.     

An extension of the incomplete beta function for negative integers

The incomplete beta function Bx(a,b) is defined for a,b>0 and 0<x<1. Its definition can be extended, by regularization, to negative non-integer values of a and b. In this paper we define the incomplete beta function Bx(a,b) for negative integer values of a and b. Further we prove that the function exists for m,n=0,1,2,… and all a and b.     

妙用指数函数求解几类积分问题

擒要在本文中笔者针对几类积分问题通过指数函数来对积分进行转换,从而达到简化计算的目的.进一步地笔者探讨了指数函数在求解某些微分方程中的作用.     

Precise and fast computation of a general incomplete elliptic integral of second kind by half and double argument transformations

We developed a method to compute simultaneously two associate incomplete elliptic integrals of the second kind, B(φ|m) and D(φ|m), by the half argument formulas of Jacobian elliptic functions and the double argument transformations of the integrals. The relative errors of the new method are sufficiently small as 5-10 machine epsilons. Meanwhile, the new method runs 3-6 times faster than that using Carlson’s R_D. As a result, it enables a precise and fast computation of arbitrary linear combination of the incomplete elliptic integrals of the first and the second kind, F(φ|m) and E(φ|m).     

一类非线性时滞Volterra-Fredholm型积分不等式及其应用

在文献马庆华和J.Pecǎri,2008的基础上,建立了一个新的VolterraFredholm型非线性时滞积分不等式.把参考文献中不等式右端被积因子w(u)推广成w_1(u)u和w_1(u)w_2(u)的非线性函数.运用放大技巧、积分微分技巧、变量替换技巧、反函数技巧、常量与变量的辩证关系,给出了不等式中未知函数的估计.推广了文献中相应不等式的结果.最后,用所得结果给出了Volterra-Fredholm积分方程解的估计.     

Two remarkable doubly exponential series transformations of Ramanujan

The purpose of this note is to prove two doubly exponential series transformations found in Ramanujan’s second notebook. Dedicated to the memory of Professor K G Ramanathan     

The Igusa Zeta Function Associated with a Composite Power Function on the Space of Rectangular Matrices

On the space of real rectangular n × m matrices, we introduce a composite power function and study the zeta integral associated with it. We describe the properties of the Igusa zeta function on the basis of the properties of a generalized composite power function and establish a functional relation for the zeta integral. As a result, the Fourier transform of a generalized composite power function is found in explicit form.     

SOME GENERALIZED GRONWALL-LIKE INEQUALITIES WITH DELAY ON TIME SCALES

In this paper, we establish some new Gronwall-like inequalities which can be used as tools in the theory of integral equations with delay on time scales.