On coefficient inequalities in the class \(\Sigma\) (Q1093764)

Let \(\Sigma_ 0\) denote the class of one-to-one analytic functions \(f(z)=z+\sum^{\infty}_{n=1}b_ nz^{-n}\) in \(| z| >1\). For the coefficient functional \[ Re\{b_ 5+b_ 1b_ 3+b^ 2_ 2+\lambda (2b_ 3+b^ 2_ 1)\} \] the authors obtain the sharp bounds \(1/3+\lambda^ 2-(1/3)\lambda^ 3\) if \(-1/3\leq \lambda <1\) and \(\lambda\) if \(\lambda\geq 1\). The extremal functions are all explicit algebraic functions. In the case -1/3\(\leq \lambda \leq 1\), a profound analysis of Schiffer's differential equation is used to show that extremal functions must be odd. Then the result follows by applying Jenkins' inequality \[ Re\{(1/2)b_ 2+\lambda b_ 1\}\leq 1/3+\lambda^ 2-(1/3)\lambda^ 3\quad to\quad f(\sqrt{z})^ 2. \] Next, the case \(\lambda >1\) is a consequence of what has been proved for \(\lambda =1\) and Grunsky's inequality \(Re\{2b_ 3+b^ 2_ 1\}\leq 1.\) Finally, the authors show that \(Re\{b_ 4+b_ 1b_ 2\}\leq 2/5\). Again, the extremal functions are algebraic. The heart of the proof is showing that extremal functions never vanish. Then the result follows by applying another of Grunsky's inequalities to \(\sqrt{f(z^ 2)}\).