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Zbl 1203.30025
Kuroki, K.; Owa, S.
Differential subordinations for certain analytic functions missing some coefficients. (English)
[J] Ann. Funct. Anal. AFA 1, No. 1, 13-25, electronic only (2010). ISSN 2008-8752/e

Summary: For a positive integer $n$, applying the Schwarz lemma for analytic functions $w(z) = c_nz^n + \ldots$ in the open unit disk $\mathbb U$, an assertion on a lemma well-known as Jack's lemma, proved by {\it S. S. Miller} and {\it P. T. Mocanu} [J. Math. Anal. Appl. 65, 289--305 (1978; Zbl 0367.34005)], is given. Further, using a method from the proof of a subordination relation discussed by {\it T. J. Suffridge} [Duke Math. J. 37, 775--777 (1970; Zbl 0206.36202)] and {\it T. H. MacGregor} [J. Lond. Math. Soc., II. Ser. 9, 530--536 (1975; Zbl 0331.30011)], some differential subordination property concerning the subordination $$p(z) \prec q(z^n),\qquad z \in\mathbb U,$$ for functions $p(z) = a + a_nz^n + \ldots $ and $q(z) = a + b_1z + \ldots $ which are analytic in $\mathbb U$, is deduced, and an extension of some subordination relation is given.

MSC 2000:: *30C80 Maximum principle, etc. (one complex variable)
30C45 Special classes of univalent and multivalent functions

Keywords: differential subordination; Schwarz's Lemma; Jack's Lemma; convex function; Briot-Bouquet differential equation

Citations: Zbl 0367.34005; Zbl 0206.36202; Zbl 0331.30011

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