Abstract: Various biholomorphic mappings'distortion estimation have been one of the hot issues studied for the theory of several complex variables. Since the definition of strongly spirallike mappings of type α was presented, the researchers only got growth and covering results, but its distortion estimation have been poorly understood. In response to this situation, we use growth and covering theorem of strongly spirallike mappings of type α, for the first time we obtained the distortion estimation of strongly spirallike mappings of type α in the open unit ball B of a complex Banach spaces and unit polydisc Dⁿ of the European space along unit direction. The results made us have a new understanding for strongly spirallike mappings of type α.

Key words: Strongly spirallike mappings of type α, Growth and covering theorem, Unit direction, Distortion estimation