Stability of geometrical properties of convolutions of univalent functions in certain neighborhoods (Q1842486)

Let \(\mathfrak A\) denote the class of functions \(f(z) = z + \sum^ \infty_{k = 2} a_ k z^ k\) analytic in the unit disk \(E = \{| z| < 1\}\); and let \(S\), \(S^*\), and \(S^ 0\) denote the subclasses of univalent, starlike, and convex functions, respectively. It is known [\textit{A. W. Goodman}, 1957] that a function \(f \in {\mathfrak A}\) belongs to \(S^*\) if \(\sum^ \infty_{k = 2} k| a_ k| \leq 1\), and \(f\) belongs to \(S^ 0\) if \(\sum^ \infty_{k = 2} k^ 2 | a_ k| \leq 1\). \textit{T. Sheil-Small} and \textit{E. M. Silvia} [J. Anal. Math. 52, 210-240 (1989; Zbl 0664.30009)] introduced neighborhoods \(TN_ \delta\) as follows: \(TN_ \delta(f) = \{g(z) \in {\mathfrak A} : \rho_ T (f,g) = \sum^ \infty_{k = 2} T_ k | a_ k - b_ k| \leq \delta\}\), where \(T = \{T_ k\}^ \infty_{k = 2}\) is a sequence of positive numbers. If we write \(e = e(z) \equiv z\), it follows that \(TN_ 1(e) \subset S^*\) if \(T_ k = k\), and \(TN_ 1(e) \subset S^ 0\) if \(T_ k = k^ 2\). For functions \(f, g \in {\mathfrak A}\), the author defines convolutions (1) \((f * g) (z) = z + \sum^ \infty_{k = 2} a_ k b_ k z^ k\) and (2) \((f \otimes g) (z) = z + \sum^ \infty_{k = 2} (a_ k b_ k /k)z^ k\). Suppose \({\mathfrak M}, {\mathfrak N} \subset {\mathfrak A}\) and \({\mathfrak M} * {\mathfrak N} \subset {\mathfrak P} \subset {\mathfrak A}\). Convolution (1) is called \(T - {\mathfrak P}\)-stable on \(({\mathfrak M}, {\mathfrak N})\) if there exists a \(\delta > 0\) such that \(TN_ \delta({\mathfrak M}) * TN_ \delta({\mathfrak N}) \subset {\mathfrak P}\). In this paper the author finds necessary and sufficient conditions for convolutions (1) and (2) to be \(T - {\mathfrak P}\) stable for various sets \(\mathfrak M\), \(\mathfrak N\), and \(\mathfrak P\). The conditions involve restrictions on the sequence \(T\). For example, convolution (2) is found to be \(T - S^ 0\)-stable on \((S^ 0, \{e\})\) if and only if \(T^{-1}_ k = O(k^{-1})\) as \(k \to \infty\). The author also obtains information about the ``stability constants'' \(\delta_ T\), defined as \(\delta_ T({\mathfrak M} * {\mathfrak N},{\mathfrak P}) = \sup\{ \delta : TN_ \delta ({\mathfrak M}) * TN_ \delta ({\mathfrak N}) \subset {\mathfrak P}\}\) for convolution (1), for example.