Certain integral operators on the classes \(\mathcal M(\beta_i)\) and \(\mathcal N(\beta_i)\) (Q938399)

The author considers the classes \(M(\beta)\) and \(N(\beta)\) of analytic functions and two general integral operators. He proves some properties for these operators on these classes. The class \(M(\beta)\) is the subclass of \(A\), consisting of the functions \(f(z)\), which satisfy the inequality \[ \text{Re}\left \{ \frac{zf'(z)}{f(z)}\right\}< \beta,\quad z\in U,\;\beta >1, \] and \(N(\beta)\) is the subclass of \(A\), consisting of the functions \(f(z)\), which satisfy the inequality \[ \text{Re}\left \{ \frac{zf''(z)}{f'(z)}+1\right \}< \beta,\quad z\in U. \] The integral operators studied in this paper are \[ F_{n}(z)=\int_{0}^{z} \left( \frac{f_{1}(t)}{t}\right)^{\alpha_{1}}\cdots \left( \frac{f_{n}(t)}{t}\right)^{\alpha_{n}}\,dt, \] where \(f_{i}(z)\in A\) and \(\alpha_{i} >0\), for all \(i\in \{1,2,\dots ,n\}\), and \[ F_{\alpha_{1},\dots ,\alpha_{n}}(z)= \int_{0}^{z} [f_{1}'(t)]^{\alpha_{1}}\cdots [f_{n}^{\prime}(t)]^{\alpha_{n}} \,dt, \] where \(f_{i}(z)\in A\) and \(\alpha_{i} >0\), for all \(i\in \{1,2,\dots,n\}\).