We present an overview of the known results describing the isometric and closed-range composition operators on different types of holomorphic function spaces. We add new results and give a complete characterization of the isometric univalently induced composition operators acting between Bloch-type spaces. We also add few results on the closed-range determination of composition operators on Blo...

and Applied Analysis 3 Here H∞ ω denotes the weighted-type space consisting of all f ∈ H B with ∥ ∥f ∥ ∥ H∞ ω sup z∈B ω z ∣ ∣f z ∣ ∣ < ∞ 1.9 see, e.g., 23, 24 . Associated weights assist us in studying of weighted-type spaces of holomorphic functions. It is known that associated weights are also continuous, 0 < ω ≤ ω̃, and for each z ∈ B, we can find an fz ∈ H∞ ω , ‖fz‖H∞ ω ≤ 1 such that fz z 1/...