Lie $^*$-double derivations on Lie $C^*$-algebras

A unital $C^*$-algebra $\mathcal{A}$ endowed with the Lie product $[x,y]=xy- yx$ on $\mathcal{A}$ is called a Lie $C^*$-algebra. Let $\mathcal{A}$ be a Lie $C^*$-algebra and $g,h:\mathcal{A}\to \mathcal{A}$ be $\mathbb{C}$-linear mappings. A $\mathbb{C}$-linear mapping $f:\mathcal{A}\to \mathcal{A}$ is called a Lie $(g,h)$--double derivation if $f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all $a,b\in \mathcal{A}$. In this paper, our main purpose is to prove the generalized Hyers–Ulam–Rassias stability  of Lie $*$-double derivations on Lie $C^*$-algebras associated with the
following additive mapping:
$$
\sum^{n}_{k=2}(\sum^{k}_{i_{1}=2} \sum^{k+1}_{i_{2}=i_{1}+1}...
\sum^{n}_{i_{n-k+1}=i_{n-k}+1}) f(\sum^{n}_{i=1, i\neq
i_{1},..,i_{n-k+1} }
 x_{i}-\sum^{n-k+1}_{ r=1}x_{i_{r}})+f(\sum^{n}_{ i=1} x_{i})
=2^{n-1} f(x_{1})
$$
 for a fixed positive integer $n$ with $n \geq 2.$