Related papers: The geometric mean is a Bernstein function
Let $a=(a_1,a_2,...c,a_n)$ for $n\in\mathbb{N}$ be a given sequence of positive numbers. In the paper, the authors establish, by using Cauchy's integral formula in the theory of complex functions, an integral representation of the principal…
In the paper, the authors show that the weighted geometric mean and the logarithmic mean are Bernstein functions and establish integral representations of these means by Cauchy's integral theorem in the theory of complex functions.
It is general knowledge that the harmonic mean $H(x,y)=\frac2{\frac1x+\frac1y}$ and that the geometric mean $G(x,y)=\sqrt{xy}\,$, where $x$ and $y$ are two positive numbers. In the paper, the authors show by several approaches that the…
In this note we revisit the classical geometric-arithmetic mean inequality and find a formula for the difference of the arithmetic and the geometric means of given $n\in\mathbb N$ nonnegative numbers $x_1,x_2,\dots,x_n$. The formula yields…
By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such…
The classical AM-GM inequality has been generalized in a number of ways. Generalizations which incorporate variance appear to be the most useful in economics and finance, as well as mathematically natural. Previous work leaves unanswered…
A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables…
In the 1980s, Kubo and Ando introduced operator means on $\mathbb{P}$, the open convex cone of positive definite operators. One significant example is the weighted geometric mean $$ A \sharp_{t} B = A^{1/2} (A^{-1/2} B A^{-1/2})^{t}…
The geometric mean of two matrices is considered and analyzed from a computational viewpoint. Some useful theoretical properties are derived and an analysis of the conditioning is performed. Several numerical algorithms based on different…
We give an upper bound for the weighted geometric mean using the weighted arithmetic mean and the weighted harmonic mean. We also give a lower bound for the weighted geometric mean. These inequalities are proven for two invertible positive…
An upper bound of the variation of argument of a holomorphic function along a curve on a Riemann surface is given. This bound is expressed through the Bernstein index of the function multiplied by a geometric constant. The Bernstein index…
The arithmetic mean is the mean for addition and the geometric mean is that for multiplication. Then what kind of binary operation is associated with the arithmetic-geometric mean (AGM) due to C. F. Gauss? If it is possible to construct an…
In this paper the geometric mean of partial positive definite matrices with missing entries is considered. The weighted geometric mean of two sets of positive matrices is defined, and we show whether such a geometric mean holds certain…
By a non-Gaussian integral we mean integral of the product of an arbitrary function and exponent of a polynomial. We develop a theory of such integrals, which generalizes and simplifies the theory of general hypergeometric functions in the…
This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective. We prove a formula for Whittaker functions of a real semisimple group as an…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
We introduce a new operation between nonnegative integrable functions on $\mathbb{R} ^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature…
We establish sharp inequalities involving the incomplete Beta and Gamma functions. These inequalities arise in the approximation of generalized Bernstein functions by higher order Thorin-Bernstein functions. Furthermore, new properties of a…
The Hirsch function of a given continuous function is a new function depending on a parameter. It exists provided some assumptions are satisfied. If this parameter takes the value one, we obtain the well-known h-index. We prove some…
In the paper, the authors establish an inequality involving exponential functions and sums, introduce a ratio of many gamma functions, discuss properties, including monotonicity, logarithmic convexity, (logarithmically) complete…