Classes of uniformly convex and uniformly starlike functions as dual sets
Let \(A\) denote the class of functions regular in the unit disc \(E\) which are of the form \(f(z)= z+\sum_{n=2}^\infty a_nz^n\) and let UCV and USV denote respectively the class of uniformly convex and uniformly starlike functions in \(E\). In the present paper, the author first shows that the classes UCV and USV can be expressed as dual sets, in the sense of convolution theory, of certain function families. He then uses these results to obtain some sharp sufficient conditions for functions in \(A\) to belong to the above classes. For instance, it is shown that if \(f\in A\) satisfies the condition \(\sum_{n=2}^\infty n(2n-1)|a_n|\leq 1\), then \(f\in \text{UCV}\), the constant 1 on the right hand side being the best possible.
scientific article; zbMATH DE number 1115521