Neighborhoods of certain analytic functions with negative coefficients
\(A(n)\) denotes the family of regular functions in the unit circle \(| z|>1\) such that \(f(z)=z-\sum^\infty_{k=n+1}a_kz^k(a_k\geq 0)\). If \(f(z)\in A(n)\) satisfies the following condition \(\text{Re}\left({zf'(z)\over f(z)}\right)>\alpha\) \((0\leq\alpha<1)\), then \(f(z)\) is called to be in the family \(S^*_n(\alpha)\). If \(f(z)\in A_n\) satisfies the following condition \(\text{Re} (1+{zf''(z) \over f'(z)})>\alpha\) \((0\leq \alpha<1)\), then \(f(z)\) is called to be in the family \(C_n(\alpha)\). In this paper neighborhoods are considered with respect to \(S_n^*(\alpha)\) or \(C_n(\alpha)\) by means of \textit{S. K. Chatterjea}'s results [J. Pure Math. 1, 23-26 (1981; Zbl 0514.30010)]. Neighborhoods are defined by \textit{S. Ruscheweyh} [Proc. Am. Math. Soc. 81, 521-524 (1981; Zbl 0458.30008)].
scientific article; zbMATH DE number 930059