Stability of a Logarithmic Functional Equation in Distributions on a Restricted Domain

Let ℝ be the set of real numbers, ℝ+ = {x ∈ ℝ ∣ x > 0}, ϵ ∈ ℝ+, and f, g, h : ℝ+ → ℂ. As classical and L∞ versions of the Hyers‐Ulam stability of the logarithmic type functional equation in a restricted domain, we consider the following inequalities: |f(x + y) − g(xy) − h((1/x) + (1/y))| ≤ ϵ, and f(x+y)-g(xy)-h(1(1/x)+/y)L∞(Γd)≤ϵ in the sectors Γd = {(x, y) : x > 0, y > 0, (y/x) > d}. As consequences of the results, we obtain asymptotic behaviors of the previous inequalities. We also consider its distributional version u∘S-v∘Π-w∘RΓd≤ϵ, where u, v, w ∈ 𝒟′(ℝ+), S(x, y) = x + y, Π(x, y) = xy, R(x, y) = 1/x + 1/y, x, y ∈ ℝ+, and the inequality ·Γd≤ϵ means that |〈·,φ〉|≤ϵ∥φ∥L1 for all test functions φ∈Cc∞(Γd).