Bounded p-valent Robertson functions of order \(\alpha\)
Let \(F_ M(\lambda,\alpha,p)\) \((M>\), \(| \lambda | <\pi /2\), \(0\leq \alpha <p)\), denote the class of functions \(f(z)=z^ p+\sum^{\infty}_{n=p+1}a_ nz^ n\) analytic in \(U=\{z: | z| <1\}\), which satisfy for fixed M, \(z=re^{i\theta}\in U\) and \[ | (e^{i\lambda}(zf'(z)/f(z))-\alpha \cos \lambda -ip \sin \lambda)(p- \alpha)^{-1}(\cos x)^{-1}-M| <M \] and let \(G_ M(\lambda,\alpha,p)\) \((M>\), \(| z| <\pi /2\), \(0\leq \alpha <p)\), denote the class of functions \(g(z)=z^ p+\sum^{\infty}_{n=p+1}a_ nz^ n\) analytic in U, which satisfy for fixed M, \(z=re^{i\theta}\in U\) and \[ | (e^{i\lambda}(1+(zg''(z))/g'(z))-\alpha \cos \lambda - ip \sin \lambda)(p-\alpha)^{-1}(\cos \alpha)^{-1}-M| <M. \] The author obtains representation formulas for these classes. He uses a technique of \textit{J. Clunie} [J. Lond. Math. Soc. 34, 215-216 (1959; Zbl 0087.077)] to obtain sharp coefficient bounds.
scientific article; zbMATH DE number 3935385