Coefficient estimates for a subclass of meromorphic functions (Q1885182)

For any complex number \(b\neq 0\) and \(-1\leq B < A\leq 1\) let \(F^*(b, A, B)\) denote the class of functions \(f(z) =\frac12+\sum^\infty_{k=1} a_kz^k\) analytic in \(0 < |z| < 1\) and satisfying \[ 1-\tfrac1b\,\left(\frac{zf'(z)}{f(z)}+1\right)\prec \frac{1+Az}{1+Bz} \] (\(\prec\) means subordination in \(|z| < 1\)). Theorem: If \(f\in F^*(b, A, B)\) then \[ \sum^\infty_{k=1}[(k+1)^2(1-B^2)-2(k+1)B(A - B)re\{b\}-(A-B)^2|b|^2]|a_k|^2\leq (A - B)^2|b|^2. \] If moreover \((k + 1)^2(1 - B^2)-2(k+ 1)B(A - B)re\{b\} - (A - B)^2|b|^2 > 0\) holds for \(k = 1,2,\dots,n - 2\), then \[ |a_n|\leq\frac{(A B)|b|}{n+1}\,,\quad n=1,\dots\,. \] The equality is attained for the \(n\)-th coefficient of the function \(\frac1z(1 + Bz^{n+1})^{\frac{-(A-B)b}{B(n+1)}}\), \(B\neq 0\) or \(\frac1z\,\text{exp}\,\frac{-Ab}{n+1}\,z^{n+1}\), \(B\neq 0\).