A Pick function related to the sequence of volumes of the unit ball in n-space∗
نویسندگان
چکیده
We show that Fa(x) = lnΓ(x + 1) x ln(ax) is a Pick function for a ≥ 1 and find its integral representation. We also consider the function f(x) = � πx/2 Γ(1 + x/2) �1/(x ln x) and show that ln f(x + 1) is a Stieltjes function and that f(x + 1) is completely monotonic on ]0,∞[. In particular f(n) = Ω ln n) n , n ≥ 2 is a Hausdorff moment sequence. Here Ωn is the volume of the unit ball in Euclidean n-space. 2010 Mathematics Subject Classification: primary 33B15; secondary 30E20, 30E15.
منابع مشابه
Parabolic starlike mappings of the unit ball $B^n$
Let $f$ be a locally univalent function on the unit disk $U$. We consider the normalized extensions of $f$ to the Euclidean unit ball $B^nsubseteqmathbb{C}^n$ given by $$Phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),$$ where $gammain[0,1/2]$, $z=(z_1,hat{z})in B^n$ and $$Psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$ in which $betain[0,1]$, $f(z_1)neq 0$ a...
متن کاملA Continuous Plane Model to Machine Layout Problems Considering Pick-Up and Drop-Off Points: An Evolutionary Algorithm
One of the well-known evolutionary algorithms inspired by biological evolution is genetic algorithm (GA) that is employed as a robust and global optimization tool to search for the best or near-optimal solution with the search space. In this paper, this algorithm is used to solve unequalsized machines (or intra-cell) layout problems considering pick-up and drop-off (input/output) points. Such p...
متن کاملFrom the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups
The Lorentz transformation of order $(m=1,n)$, $ninNb$, is the well-known Lorentz transformation of special relativity theory. It is a transformation of time-space coordinates of the pseudo-Euclidean space $Rb^{m=1,n}$ of one time dimension and $n$ space dimensions ($n=3$ in physical applications). A Lorentz transformation without rotations is called a {it boost}. Commonly, the ...
متن کاملOn a sequence related to the coprime integers
The asymptotic behaviour of the sequence with general term $P_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $P_n-6/pi^2$ is found.
متن کاملAn extension theorem for finite positive measures on surfaces of finite dimensional unit balls in Hilbert spaces
A consistency criteria is given for a certain class of finite positive measures on the surfaces of the finite dimensional unit balls in a real separable Hilbert space. It is proved, through a Kolmogorov type existence theorem, that the class induces a unique positive measure on the surface of the unit ball in the Hilbert space. As an application, this will naturally accomplish the work of Kante...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010