A third logarithmic functional equation and Pexider generalizations

Document Type

Article

Publication Date

9-2005

Department

Department of Mathematical Sciences

Abstract

Let f:]0, ∞[→ ℝ be a real valued function on the set of positive reals. Then the functional equations: f(x + y) - f(xy) = & f(1/x + 1/y) f(x + y) - f(x) - f(x) = f(1/x + 1/y) and f(xy) = f(x) + f(y) are equivalent to each other. If f,g,h:]0, ∞ [→ ℝ are real valued functions on the set of positive reals then f(x + y) - g(xy) = h(1/x + 1/y) is the Pexider generalization of f(x + y) - f(xy) = f(1/x + 1/y). We find the general solution to this Pexider equation. If f,g,h,k:]0,∞ [ → ℝ are real valued functions on the set of positive reals then f(x + y) - g(x) - h(y) = k(1/x + 1/y) is the Pexider generalization of f(x + y) - f(x) - f(y) = f(1/x + 1/y). We find the twice differentiable solution to this Pexider equation.

Publication Title

Aequationes Mathematicae

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