Inequalities for the double gamma function

We establish various new upper and lower bounds in terms of the classical gamma and digamma functions for the double gamma function (or Barnes G-function).     

On Ruijsenaars' asymptotic expansion of the logarithm of the double gamma function

We study the remainder RN(x) in an asymptotic expansion due to S.N.M. Ruijsenaars, for the logarithm of the double gamma function. We show that for any even number N the function is completely monotonic on (0,∞). This proves a recent conjecture of H.L. Pedersen. In addition, we give an estimate for the remainder RN(x) for N even.     

Two new asymptotic expansions of the ratio of two gamma functions

Formal expansions, giving as particular cases semiasymptotic expansions, of the ratio of two gamma functions are obtained.     

Completely monotone functions and the Wallis ratio

The aim of the paper is to improve known estimates of the Wallis ratio. Moreover, we show that these improvements are valid, because certain functions involving the continuous version of the Wallis ratio are completely monotone.     

Norm estimates for the Alexander transforms of convex functions of order alpha

The hyperbolic sup norm of the pre-Schwarzian derivative of a locally univalent function on the unit disk measures the deviation of the function from similarities. We present sharp norm estimates for the Alexander transforms of convex functions of order α, 0?α<1.     

一个包含Euler函数及k阶Smarandache ceil函数的方程及其正整数解

设k≥2为给定的整数．对任意正整数n,k阶Smarandache ceil函数Sk（n）定义为Sk（n）=min{x：x∈N,n｜x^k}．本文的主要目的是利用初等方法研究函数方程Sk（n）=Ф（n）的可解性,并给出该方程的所有正整数解,其中Ф（n）为Euler函数．     

The empirical distribution function and strong laws for functions of order statistics of uniform spacings

Let N points be arbitrarily chosen on the circle with unit circumference, and order them clockwise. The uniform mth order spacings are then defined as the clockwise distances between any pair of points having m − 1 other points in between. A Glivenko-Cantelli theorem and nonlinear almost sure bounds for the empirical distribution function based on these uniform spacings are derived. The parameter m is allowed to increase with N to infinity. Applications to linear combinations of functions of mth order spacings are given.     

On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation

(1)

where , , μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , μ(z)≥μ₀>0, μ₁(z)≥0, for all , and , , , a weak solution u_ε1,ε2(x,t) having an asymptotic expansion of order N+1 in two small parameters ε₁, ε₂ is established for the following equation associated to (1)_2,3:

(2)

     

Attainment and (sub)differentiability of the infimal convolution of a function and the square of the norm

Let X be a Banach space whose norm is simultaneously LUR and Gateaux (Fréchet) smooth. Under some assumptions, it is shown that the infimal convolution of a fairly general function on X and the square of the norm is generically strongly attained and hence is Gateaux (Fréchet) differentiable. This contains a result of S. Dutta on distance functions.