On a functional equation related to the Ptolemaic inequality
type: Item
instance of: scholarly article
title: On a functional equation related to the Ptolemaic inequality
zbMATH Open document ID: 0658.39003
DOI: 10.1007/BF01830679
author: Claudi Alsina
author: J.-L. García Roig
published in: Aequationes Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137247
review text: Let R denote the set of real numbers and \(R^+\) denote the subset of its nonnegative elements. The authors prove the following interesting theorems. Theorem 1: Let f be a function from R into \(R^+\), strictly increasing on \(R^+\) and satisfying the following conditions: (i) \(f(x-y)=f(x)f(y)f(x/f(x)^ 2-y/f(y)^ 2)\), for all x,y in \(R\setminus \{0\};\) (ii) \(f(x)\cdot f(x/f(x)^ 2)=1\), for all x in \(R\setminus \{0\};\) (iii) \(f(x)=f(-x),\) for all x in R. Then there exists a positive constant a, \(a>0\), such that \(f(x)=a| x|\) for all x in R. Theorem 2: Let f be a function from \(R^ n\) into \(R^+\) satisfying the following conditions: (i) \(f(x-y)=f(x)f(y)f(x/f(x)^ 2-y/f(y)^ 2)\), for all x,y in \(R^ n\setminus \{0\};\) (ii) \(f(x)f(x/f(x)^ 2)=1\), for all x in \(R^ n\setminus \{0\};\) (iii) \(f(-x)=f(x)\), for all x in \(R^ n;\) (iv) For any fixed \(x\neq 0\) the function f(\(\lambda\) x) is strictly increasing when \(\lambda\) varies on \(R^+;\) (v) \(f(x+y)\leq f(x)+f(y)\), for all x,y in \(R^ n.\) Then \((R^ n,f)\) is a normed space which is an inner product space.
Mathematics Subject Classification ID: 39B99
zbMATH DE Number: 4075701
zbMATH Keywords: Ptolemaic inequality
zbMATH Keywords: normed linear space
zbMATH Keywords: inner product space
zbMATH Keywords: involution
zbMATH Keywords: symmetric bilinear functional
reviewed by: Michael Th. Rassias
MaRDI profile type: Publication
cites work: Q3213662
cites work: Q3854084
cites work: Q3232347
cites work: Eine Charakterisierung der positiv definiten quadratischen Formen
OpenAlex ID: W2023254414
description: scientific article; zbMATH DE number 4075701