Starlike functions with a fixed coefficient. (English) Zbl 0653.30004

The author investigates the radius of convexity for a certain class of functions: \[ S^*_{k,b}(A,B)=\{f(z)=z+a_{k+1}z^{k+1}+a_{2k+1}z^{2k+1}+...,\quad zf'(z)/f(z)\in P_{k,b}(A,B)\}, \] where \[ P_{k,b}(A,B)=\{p(z)=1+b(A-B)z^ k+p_{2k}z^{2k}+...,\quad p(z)\prec (1+Az^ k)/(1+Bz^ k)\}, \] k\(=1,2,3,...\), \(-1\leq B<A\leq 1\), \(0\leq b\leq 1\). One of the tools is a theorem of Pfaltzgraff and Pinchuk (Lemma 1 in the paper).
Reviewer: D.Aharonov


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
Full Text: DOI


[1] DOI: 10.2307/2039693 · Zbl 0286.30005 · doi:10.2307/2039693
[2] DOI: 10.2307/2033832 · doi:10.2307/2033832
[3] Janowski, Ann. Polon. Math. 23 pp 159– (1970)
[4] Anh, Bull. Austral. Math. Soc. 32 pp 419– (1985)
[5] Padmanabhan, J. Indian Math. Soc. 32 pp 89– (1968)
[6] Tepper, Trans. Amer. Math. Soc. 150 pp 1970–
[7] DOI: 10.2307/1968451 · Zbl 0014.16505 · doi:10.2307/1968451
[8] DOI: 10.1007/BF02790372 · Zbl 0247.30012 · doi:10.1007/BF02790372
[9] Tuan, Czechoslovak Math. J. 30 pp 302– (1980)
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