The Pre-Schwarzian Norm Estimate for Analytic Concave Functions - Young Jae Sim; Oh Sang Kwon (International Journal of Mathematics and Mathematical Sciences)
Let D denote the open unit disk and let S denote the class of normalized univalent functions which are analytic in D. Let Co(α) be the class of concave functions f∈S, which have the condition that the opening angle of f(D) at infinity is less than or equal to πα, α∈(1,2]. In this paper, we find a sufficient condition for the Gaussian hypergeometric functions to be in the class Co(α). And we define a class Co(α,A,B), (-1≤B<A≤1), which is a subclass of Co(α) and we find the set of variabilities for the functional (1-|z|2)(f″(z)/f′(z)) for f∈Co(α,A,B). This gives sharp upper and lower estimates for the pre-Schwarzian norm of functions in Co(α,A,B). We also give a characterization for functions in Co(α,A,B) in terms of Hadamard product.