On a certain condition for p-valently starlikeness
Let A(p) be the class of functions of the form \(f(z)=z^ p+\sum^{\infty}_{n=p+1}a_ nz^ n\) \((p=1,2,...)\) which are analytic in \(\Delta =(z: | z| <1)\). A function \(f\in A(p)\) is said to be p-valently starlike if and only if it satisfies the condition \(Re zf'(z)/f(z)>0(z\in \Delta).\) The author's main theorem gives conditions which imply \(| \arg zf'(z)/f(z)| <(\pi /2)\alpha (0<\alpha \leq 1)\) for \(z\in \Delta\). This implies that such a function is p-valently starlike. One of the two assumptions made by the authors is that \(| \arg f'(z)/z^{p-1}| <(\pi /2)\alpha\) for z in \(\Delta\). The other assumption deals with the imaginary part of \(f'(z)/z^{p-1}\) and we do not give it here.
scientific article; zbMATH DE number 4093840