Some properties of certain class of multivalent functions with negative coefficients (Q2912655)

For \(p\)-valent functions of the form \(f(z)=z^p+\sum_{j=1}^\infty a_{p+j}z^{p+j}\), the operator \(D_\lambda^n\) is defined by \(D_\lambda^n f(z)=(1-\lambda+\lambda p)^nz^p+\sum_{j=1}^\infty (1-\lambda+\lambda(p+j))^n a_{p+j}z^{p+j}\). The authors consider the functions with negative \(a_{p+j}\) for which \(D_\lambda^{n+1} f/D_\lambda^n f\) belongs to a certain region in the complex plane. They obtain the usual sufficient coefficient conditions, coefficient estimates, growth and distortion estimates. They also investigate closure under convex linear combinations, modified convolution and an integral operator.