On certain multivalent functions (Q5905467)

Let \(A(p)\) denote the class of functions analytic in the unit disc \(U\) which are of the form \(f(z)=z^ p+\sum^ \infty_{k=1}a_{p+k}z^ k\). The author obtains sharp results of the form: If \(f(z)\in A(p)\) and \(\text{Re}\{f^{(j)}(z)/z^{p-j}\}>\alpha\) for all \(z\in U\) where \(0\leq\alpha<q=p!/(p-j)!\) and \(1\leq j\leq p\), then \[ \text{Re}\{f^{(j- 1)}(z)/z^{p-j+1}\}>(2\alpha-q)/(p-j+1)+2(q-\alpha)\sum^ \infty_{k=1}(-1)^{k-1}/(p-j+k). \] Other similar theorems involve conditions on \(F(z)=(1-\lambda)f(z)+\lambda zf'(z)\). For example, a special case gives: If \(f(z)\in A(1)\) and \(\text{Re}\{f'(z)+\lambda zf''(z)\}>\alpha\) for all \(z\in U\), where \(0\leq\alpha<1\), then \[ \text{Re}\{f'(z)\}>1+2(1-\alpha)\sum^ \infty_{k=1}(-1)^ k/(1+\lambda k). \]