Let Ω be a domain in the complex plane such that Ω ⊆ D, the open unit disc in the complex plane, and let K = Ω. We define A(D,K) = {f ∈ C(D) : f|K ∈ A(K)}, where A(K) ={g ∈ C(K) : g is analytic on int(K)} . For α ∈ (0, 1], we define LipA(D,K, α) = {f ∈ C(D) : pα,K(f) = sup{|f(z)−f(w)| |z−w|α : z,w ∈ K, z = w} < ∞}, and LipA(D,K, α) = A(D,K) Lip(D,K, α). It is known that A(D,K) and LipA(D,K, α) are natural uniform algebra and natural Banach function algebra on D under the norms ||f|| = sup{|f(z)| : z ∈ D} (f ∈ A(D,K)) and ||f||α,K = ||f|| + pα,K(f) (f ∈ LipA(D,K, α)) , respectively.
