Logarithmic Functional Equations

Soon-Mo Jung ()
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Soon-Mo Jung: Hongik University

Chapter Chapter 11 in Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, 2011, pp 253-266 from Springer

Abstract: Abstract It is not difficult to demonstrate the Hyers–Ulam stability of the logarithmic functional equation $$f(xy)=f(x)+f(y)\ {\rm for\ functions}\ f:(0,\infty)\rightarrow E$$ , where E is a Banach space. More precisely, if a function $$f:(0,\infty)\rightarrow E$$ satisfies the functional inequality $$\| f(xy)-f(x)-f(y)\| \leq \delta\ {\rm for\ some}\ \delta > 0\ {\rm and\ for\ all}\ x,y > 0$$ , then there exists a unique logarithmic function $$L:(0,\infty)\rightarrow E$$ (this means that $$L(xy)=L(x)+L(y)\ {\rm for\ all}\ x,y >0$$ ) such that $$\| f(x)-L(x)\| \leq \delta\ {\rm for\ any}\ x >0$$ . In this chapter, we will introduce a new functional equation $$f(x^y)=yf(x)$$ which has the logarithmic property in the sense that the logarithmic function $$f(x)={\rm ln}x (x > 0)$$ is a solution of the equation. Moreover, the functional equation of Heuvers $$f(x+y)=f(x)+f(y)+f(1/x+1/y)$$ will be discussed.

Date: 2011
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DOI: 10.1007/978-1-4419-9637-4_11

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