Some inequalities involving meromorphically multivalent functions
Let \(\Sigma_p\) be the class of functions of the form \(f(z) = z^{-p} + \sum_{k=-p+1}^\infty a_k z^k\) which are regular in the unit disc \(U\) except for the pole at \(z = 0\). In 1947, Robinson proved that if \(S(z)\) and \(T(z)\) are regular in \(U\) with \(\text{Re} \{zS'(z)/S(z)\} > 0\), \(|T'(z)/S'(z)|< 1\) for all all \( z \in U\), and \(T(0) = 0\), then \(|T(z)/S(z)|< 1\) for all \(z \in U\). Using variants of Jack's Lemma, the authors prove several theorems similar to Robinson's for functions in \(\Sigma_p\). The results are also motivated by earlier inequalities of Owa, Nunokawa, and Saitoh. The theorems proved are rather technical. For example, Theorem 2 of the paper proves that if \(S(z) \in \Sigma_m\), \(T(z) \in \Sigma_n\), \(p = n - m \geq 1\), \(S(z)\) satisfies \(\text{Re} \{S(z)/(zS'(z))\} < \;-\alpha\), \((0 \leq \alpha < 1/m)\), \(-\alpha / p \leq \roman {Im}\{-S(z)/(zS'(z))\} \leq \alpha/p\), and \(\text{Re} \{z^p T'(z)/S'(z)\} < 0\) for all \(z \in U\), then \(\text{Re} \{z^p T(z)/S(z)\} > 0\) for all \(z \in U\).
scientific article; zbMATH DE number 1053990