On a stability of logarithmic-type functional equation in Schwartz distributions (Q1938199)

Summary: We prove the Hyers-Ulam stability of the logarithmic functional equation \[ f(x + y) - g(xy) - h(1/x + 1/y) = 0, \] \(x, y > 0\), of Heuvers and Kannappan in both classical and distributional senses. As a classical sense, the Hyers-Ulam stability of the inequality \(|f(x + y) - g(xy) - h(1/x + 1/y)| \leq \epsilon\), \(x, y > 0\), is proved, where \(f, g, h : \mathbb R_+ \rightarrow \mathbb C\). As a distributional analogue of the above inequality, the stability of the inequality \(||u \circ (x + y) - v \circ (1/x + 1/y)|| \leq \epsilon\) is proved, where \(u, v, w \in \mathcal D' (\mathbb R_+)\) and \(\circ\) denotes the pullback of distributions.