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Subclasses of univalent functions. (English) Zbl 0531.30009
Complex analysis - Proc. 5th Rom.-Finn. Semin., Bucharest 1981, Part 1, Lect. Notes Math. 1013, 362-372 (1983).

[For the entire collection see Zbl 0516.00016.]

This paper is concerned with the classes S n (α)={f: f is holomorphic in the unit disk U, f(0)=f ' (0)-1=0 and Re[D n+1 f(z)/D n f(z)]>α for zU}, 0α<1, where D 0 f(z)=f(z),D 1 f(z)=Df(z)=zf ' (z) and D n f(z)=D(D n-1 f(z)), n2. Using subordination techniques the sharp result is obtaind that S n+1 (α)S n (δ(α)),0α<1, where δ(α)=(2α-1)/[2(1-2 1-2α )],α1 2, and δ(α)=1/(2ln2),α=1 2· From a corollary it is noted that for 0α<1, all functions in S n (α) are starlike for n a nonnegative integer and convex for n a positive integer. The author also obtains coefficients bounds that generalize a result of H. Silverman and E. M. Silvia [Rocky Mt. J. Math. 10, 469-474 (1980; Zbl 0455.30011)].

Reviewer: D.V.V.Wend

30C45Special classes of univalent and multivalent functions
30C50Coefficient problems for univalent and multivalent functions
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)