In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory. See more Let a function $${\displaystyle f}$$ be analytic on a closed disc of radius R centered at the origin. Suppose that r < R. Then, we have the following inequality: Here, the norm on … See more Define A by $${\displaystyle A=\sup _{ z \leq R}\operatorname {Re} f(z).}$$ If f is constant, the inequality is trivial since $${\displaystyle (R+r)/(R-r)>1}$$, so we may assume f is nonconstant. First let f(0) = 0. Since Re f is … See more WebAug 5, 2012 · This type of theorem is closely related to the Maximum Principle but characteristic of the class of Borel–Caratheodory Theorems is that one assumes only a …
[1410.4300] Carathéodory theorems for Slice Regular Functions
WebNov 19, 2024 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of … WebDec 2, 2014 · In this paper a sharp quaternionic version of the Carathéodory theorem is established for slice regular functions with positive real part, which strengthes a weaken version recently established by Alpay et al. using the Herglotz integral formula. Moreover, the restriction of positive real part can be relaxed so that the theorem becomes the … formula game companion download
Lecture 7: Borel Sets and Lebesgue Measure
WebIn mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial. It is named for Jacques Hadamard.. The theorem may be viewed as an extension of the fundamental theorem of algebra, … Web1.8. Borel sets and measures. Let E be a topological space. The ˙-algebra gen-erated by the set of open sets is E is called the Borel ˙-algebra of E and is denoted B(E). The Borel ˙-algebra of R is denoted simply by B. A measure on (E;B(E)) is called a Borel measure on E. If moreover (K) < 1 for all compact sets K, then is called a Radon ... WebGeometry Unit 4 Answers PHS. 4.6 (35 reviews) Term. 1 / 129. (L1) A (n) _____ is a closed plane figure formed by three or more line segments, such that each segment intersects … difficulty rating for superior hiking trail