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On functions realizing the maxima of two functionals at a time. (English) Zbl 0576.30013

The main results are described in the authors’ abstract: Let \(S_ R(M)\), \(M>1\), denote the family of functions \(F(z)=z+\sum^{\infty}_{n=2}A_ nz^ n\) holomorphic and univalent in the disc \(\Delta\) \(=\{z: | z| <1\}\), satisfying the conditions: \(| F(z)| <M\) for \(z\in \Delta\), \(A_ n=\overline{A_ n}\) for \(n=2,3,..\). In the paper it has been proved that if there exists a function \(w=F(z)\) for which in the family \(S_ R(M)\) the maxima of the coefficients \(A_ N\) and \(A_{N+1}(N=4,5,..)\) are attained simultaneously, then it satisfies in the disc \(\Delta\) the equation \[ w/(\epsilon -w/M)({\bar \epsilon}- w/M)=z/(\epsilon -z)({\bar \epsilon}-z),\quad | \epsilon | =1. \] There has also been given an analogous theorem concerning the coefficients \(A_ K\), \(A_ N\), \(N=p+1\), \(2\leq K\leq N-1\), where p is an arbitrary prime number.
Reviewer: D.W.De Temple

MSC:

30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C55 General theory of univalent and multivalent functions of one complex variable
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