The pre-Schwarzian norm estimate for analytic concave functions
Summary: Let \(\mathbb{D}\) denote the open unit disk and let \(\mathcal{S}\) denote the class of normalized univalent functions which are analytic in \(\mathbb{D}\). Let \(\mathrm{Co}(\alpha)\) be the class of concave functions \(f \in \mathcal{S}\), which have the condition that the opening angle of \(f(\mathbb{D})\) at infinity is less than or equal to \(\pi \alpha\), \(\alpha \in(1,2]\). In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class \(\mathrm{Co}(\alpha)\). And we define a class \(\mathrm{Co}(\alpha, A, B)\), \((- 1 \leq B < A \leq 1)\), which is a subclass of \(\mathrm{Co}(\alpha)\) and we find the set of variabilities for the functional \((1 - | z |^2)(f''(z) / f'(z))\) for \(f \in \mathrm{Co}(\alpha, A, B)\). This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in \(\mathrm{Co}(\alpha, A, B)\). We also give a characterization for functions in \(\mathrm{Co}(\alpha, A, B)\) in terms of Hadamard product.
scientific article; zbMATH DE number 6873265
scientific article