Inner composition of analytic mappings on the unit disk (Q810191)

Let \(\{f_ n\}\) be a sequence of analytic self mappings on the unit disk D, and form the inner compositions \[ (1)\quad F_ n(z)=f_ 1\circ f_ 2\circ...\circ f_ n(z)\text{ for } n=1,2,3,.... \] The paper gives sufficient conditions for the interesting property \[ (2)\quad \lim_{n\to \infty}F_ n(z)=\lambda \text{ for } z\in D, \] where \(\lambda\) is a constant (independent of z). The conditions given are too restrictive.(See papers by I. N. Baker and P. J. Rippon and the reviewer).