Mutual variation of the initial coefficients of univalent functions (Q1073948)

Let \(S^ M_ k\) denote the class of functions \[ f(z)=z+a_{k+1}z^{k+1}+a_{2k+1}z^{2k+1}+... \] that are analytic and satisfy the condition \(| f(z)| <M\) in the unit disc. The classes \(S^ M_ k(\alpha,\gamma)\) (0\(\leq \alpha \leq 2/3\) and \(0\leq \gamma \leq 1/(1-\alpha))\) are more general, and \(S^ M_ k(0,0)=S^ M_ k.\) The author finds the set of values (in parametric form) of the functionals \({\mathcal I}_ 1(f)=(Re a_{k+1},Im a_{k+1},Re a_{2k+1})\) and \({\mathcal I}_ 2(f)=(Re a_{k+1},Re a_{2k+1})\) for \(f\in S^ M_ k(\alpha,\gamma)\). The proofs use methods of optimal control theory.