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An inequality between coefficients of schlicht subordinate functions. (Chinese) Zbl 0609.30024

Let \(F(z)=\sum^{\infty}_{n=0}A_ nz^ n\) and \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\) be analytic in \(| z| <1\). f(z) is said to be schlicht subordinate to F(z) in \(| z| <1\), if \(f(0)=F(0)\) and if there exists a function \(\omega\) (z), analytic, univalent and \(| \omega (z)| <1\) in \(| z| <1\), such that \(f(z)=F(\omega (z))\). In this note, the author points out that Chang Kaiming’s result \(| a_ 3| \leq \max (| A_ 1|,| A_ 2|,| A_ 3|)\) is not true, if f is schlicht subordinate to F in \(| z| <1\) and the correct result should be \[ | a_ 3| \leq ((1+\sqrt{2})/2)\max (| A_ 1|,| A_ 2|,| A_ 3|). \] It is interesting that some results on bounded functions of O. Tammi, Extremum problems for bounded univalent functions. I and II (1978; Zbl 0375.30006) and (1982; Zbl 0481.30020) ae used in his proof.
Reviewer: Ren Fuyao

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
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