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Some applications of differential subordination. (English) Zbl 0575.30018

Using the techniques of differential subordination developed by P. Eenigenburg, S. S. Miller, P. T. Mocanu and M. O. Reade [General inequalities 3, 3rd int. Conf., Oberwolfach 1981, ISNM 64, 339-348 (1983; Zbl 0527.30008)] the authors show that if f(z) is in a certain class of regular functions, then so is \[ F(z)=((c+1)/c)\int^{z}_{0}t^{c-1}f(t)dt. \] The function f(z) must satisfy a subordination condition. The authors’ work generalizes and combines prior similar results of, for example, the reviewer [Proc. Am. Math. Soc. 16, 755-758 (1965; Zbl 0158.077)], St. Ruscheweyh [Proc. Am. Math. Soc. 49, 109-115 (1975; Zbl 0303.30006)], etc.
Reviewer: R.Libera

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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References:

[1] Goel, Indian J. Pure. Appl. Math. 12 pp 1240– (1981)
[2] Eenigenburg, On a Briot. Bouquet differential subordination 3 pp 339– (1983)
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[4] Hassoon, Rev. Roum. Math. Pures et Appl. 23 pp 1449– (1978)
[5] DOI: 10.1307/mmj/1028988895 · Zbl 0048.31101
[6] DOI: 10.2307/2039801 · Zbl 0303.30006
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[8] DOI: 10.2307/2033917 · Zbl 0158.07702
[9] Zmorovich, Ukrain. Mat. 33 pp 670– (1981)
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