On convolution theorems (Q917736)

Let \(K(\alpha)\) denote the set of functions which are convex of order \(\alpha\) in the open unit disk. The author defines a family of functions denoted \(K(\alpha,\beta)\) and which are called close-to-convex of order \(\alpha\) and type \(\beta\). Let \(f*g\) denote the Hadamard product of two power series \(f\) and \(g\) each defined in the open unit disk. The author proves two theorems and makes two conjectures about \(f*g\) where \(f\) and \(g\) are assumed to belong to various families such as \(K(\alpha)\) and \(K(\alpha,\beta)\). For example, the first theorem asserts that if \(f\in K(\alpha,\beta)\) and \(g\in K(\gamma)\) then \(f*g\in K(\alpha,\delta)\) where \(\delta =\max (\beta,\gamma)\).