Properties of certain analytic functions
If \(p\in {\mathbb{N}}\), let A(p) denote the class of functions f analytic in the unit disc U with Taylor expansion \(f(z)=z^ p+\sum^{\infty}_{k=0}a_{p+k}z^{p+k}\). It is shown that for f in A(p), and \(0<\alpha <p!/(p-j)!\), the inequality \(Re(f^{(j)}(z)z^{j- p})>\alpha,\) (z\(\in U)\) implies \[ Re(f^{(j-1)}(z)z^{j-p- 1})>\frac{(p-j+1)!2\alpha +p!}{(p-j+1)!(2p-2j+3)}\quad (z\in U) \] for \(1\leq j\leq p\). Similar inequalities are deduced for \((1- \lambda)f(z)+\lambda zf'(z)\) when \(\lambda\geq 0\), and a number of corollaries are obtained.
scientific article; zbMATH DE number 4132501