Let \(S\) be a connected subset of the circle \(\{v \in \mathbb{C}: |v| = 1 \}\) such that \(1 \in S \neq \{1 \}\). Suppose that \(X\) is a complex algebra, \(Y\) is a complex Banach algebra and \(\alpha, \beta \in \mathbb{R} \setminus \{0 \}\). Assume that \(f: X \to Y\) is a mapping for which there exist a constant \(L < 1\) and a function \(\varphi: X \times X \to [0, \infty)\) such that \[ \|\mu f(\alpha x + \beta y) + \mu f(\alpha x - \beta y) - 2 \alpha f(\mu x)\| \leq \varphi(x,y) \] and \[ \varphi(x,y) \leq \frac{L}{|\alpha|} \varphi(\alpha x, \alpha y) \] for all \(\mu \in S\), \(x,y \in X\).
If \( \| f(x y) - f(x) f(y) \| \leq \varphi(x, y), \;x,y \in X, \) then there exists a unique homomorphism \(h: X \to Y\) such that \[ \|f(x) - h(x) \| \leq \frac{L}{2 |\alpha|(1 - L)} \varphi(x, 0), \;x \in X. \tag{1} \]
If \(Y = X\) and \( \| f(x y) - f(x)y - xf(y) \| \leq \varphi(x, y),\, \;x, y \in X, \) then there exists a unique derivation \(h: X \to X\) fulfilling inequality (1).
Now let \([x,y] := x y - y x\) for \(x, y \in X\). If \( \| f([x,y]) - [f(x),f(y)] \| \leq \varphi(x, y),\, x, y \in X, \) then there exists a unique \(\mathbb{C}\)-linear \(h: X \to Y\) such that \(h([x, y]) = [h(x), h(y)]\) for \(x, y \in X\) (i.e., \(h\) is a Lie homomorphism) and inequality (1) is fulfilled.
If \(Y = X\) and \[ \| f([x,y]) - [f(x),y] - [x,f(y)] \| \leq \varphi(x, y),\, x, y \in X, \] then there exists a unique \(\mathbb{C}\)-linear \(h: X \to X\) such that \(h([x,y]) = [h(x),y] + [x,h(y)]\) for \(x,y \in X\) (i.e., \(h\) is a Lie derivation) and inequality (1) is fulfilled.
In the last part of the paper the authors prove the hyperstability of homomorphisms in complex Banach algebras.