Neighborhoods of analytic functions (Q1114806)

As a generalization of the definition of a \(\delta\)-neighbourhood of a function \[ f(z)=z+\sum^{\infty}_{k=2}a_ kz^ k \] holomorphic in the unit disc D given by \textit{St. Ruscheweyh} [Proc. Am. Math. Soc. 81, 521-527 (1981; Zbl 0458.30008)] the authors define T-\(\delta\)- neighbourhoods \(TN_{\delta}(f)\) of f as follows: Let \(T=\{T_ k\}^{\infty}_{k=2}\) be a sequence of nonnegative reals and A the set of functions \[ g(z)=z+\sum^{\infty}_{k=2}b_ kz^ k \] holomorphic in D. Then \[ TN_{\delta}(f)=\{g| \quad g\in A\quad and\quad \sum^{\infty}_{k=2}T_ k| a_ k-b_ k| \leq \delta \}. \] Sections 2-4 of this paper contain such a lot of interesting results on such neighborhoods concerning special subclasses of A that it is impossible to give an impression in few words. Anyone who is interested in Hadamard convolution should study this paper. The paper concludes with the discussion of two different neighbourhood systems and some open problems.