The Pre-Schwarzian Norm Estimate for Analytic Concave Functions

Abstract EN

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

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.