Sufficient conditions for certain classes of analytic functions (Q2782470)

The author considers functions of the form NEWLINE\[NEWLINEf(z)= z/\Biggl(1+ \sum^\infty_{n=1} b_n z^n\Biggr)NEWLINE\]NEWLINE defined on the open unit disc of the complex plane and obtains conditions on the coefficients \(\{b_n\}\) of the function \((1+ \sum^\infty_{n=1} b_n z^n)\) that will ensure that a typical real function \(f\) belongs to a subclass of starlike functions or to a subclass of spirallike functions etc.NEWLINENEWLINENEWLINEThe proof of the fact that under the given conditions the function mentioned in the denominator is analytic and non-vanishing (or even a mention of this fact) is conspicuous by its absence. Further in Theorem 6 on p. 195 \(f(z)\) is assumed to be of the form \(z/1+ \sum^\infty_{n=1} b_n z^n\) (implying \(f(0)= 0\)) and it is ``proved'' to belong to a class \(P(A,B)\) (implying \(f(0)= 1\)).NEWLINENEWLINEFor the entire collection see [Zbl 0980.00021].