On a functional equation related to the Ptolemaic inequality
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.
scientific article; zbMATH DE number 4075701