Let \(S_ H\) denote the class of univalent harmonic functions \(f(z)=\sum^{\infty}_{k=-\infty}a_ kz^ k\) in the unit disk \({\mathbb{D}}\), with \(a_ 0=0\), \(a_ 1=1\). If, in addition, \(a_{-1}=0\) then \(f\in S^ 0_ H\). The present article describes the present knowledge of some extremal problems for \(S_ H\) and subclasses, and achieves considerable progress in some of them. Let L be a both affine and linear invariant subclass of \(S_ H\), \(L^ 0=L\cap S^ 0_ H\), and \(m(r,f)=\min \{| f(z)|:\;| z| =r\},\) \(M(r,f)=\max \{| f(z)|:\;| z| =r\},\alpha =\alpha (L)=\sup \{| a_ 2(f)|:\) \(f\in L\}\), \(d(L^ 0)=\liminf_{r\to 1}\{m(r,f):\;f\in L^ 0\}.\) Then it is shown that \[ \frac{1}{2\alpha}(1-(\frac{1- r}{1+r})^{\alpha})\leq m(r,f)\leq M(r,f)\leq \frac{1}{2\alpha}((\frac{1+r}{1-r})^{\alpha}-1) \] and \(d(L^ 0)\geq 1/2\alpha (L).\)
An interesting new approach is to replace linear invariance by composition invariance: for L as above one forms \(\tilde L\) as the closure of the functions \(\frac{f_ n(\sigma (z))-f(\sigma (0))}{\sigma '(0)h'(0))}\), where \(f=g+\bar h\in L\), \(\sigma: {\mathbb{D}}\to {\mathbb{D}}\) analytic univalent. It is then shown that \(\tilde L\) is affine and linear invariant in \(\overline{S_ H}\) and satisfies \(\alpha (\tilde L)=\max (2,\alpha (L)),\) \(d(\tilde L^ 0\geq \min ((1/4),(1/2\alpha (L))).\) This result is then applied to the cases where L is one of the classes of convex or close-to-convex functions in \(S_ H\), thus widely generalising the previously known results for those classes. The author does not exclude the possibility that the latter (\(\tilde{\;}\)-extended) class may coincide with \(\overline{S_ H}\), which would solve many of the open extremal problems for \(S_ H.\)
The next result concerns the envisaged extension of the Bieberbach conjecture for \(S_ H\). Here it is shown that, if \(f\in S^ 0_ H\) maps \({\mathbb{D}}\) onto a domain which is either starlike w.r.t. the origin or convex in one direction, the inequalities \(| | a_ n| - | a_{-n}| | \leq n\), \(n=2,3,...\), and \(| a_ n| \leq (1/6)| (n+1)(2n+1)|\), \(n\in {\mathbb{Z}}\), hold. For the general case \(f\in S_ H\) it is shown that \(| a_ 2| \leq (96\pi /\sqrt{27})- 1\), which, however, is still far away from the conjectured \(| a_ 2| \leq 3\). The paper also discusses the radius of convexity and inner mapping radius problems in \(S_ H\) and some subclasses.