On the superstability of Lobačevskiǐ's functional equations with involution (Q288757)

Summary: Let \(G\) be a uniquely \(2\)-divisible commutative group and let \(f, g : G \to \mathbb{C}\) and \(\sigma : G \to G\) be an involution. In this paper, generalizing the superstability of Lobačevskiǐ's functional equation, we consider \(\left|f \left((x + \sigma y) / 2\right)^2 - g(x) f(y)\right| \leq \psi(x)\) or \(\psi(y)\) for all \(x, y \in G\), where \(\psi : G \to \mathbb{R}^+\). As a direct consequence, we find a weaker condition for the functions \(f\) satisfying the Lobačevskiǐ functional inequality to be unbounded, which refines the result of Găvrută and shows the behaviors of bounded functions satisfying the inequality. We also give various examples with explicit involutions on Euclidean space.