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Differential subordinations for certain analytic functions missing some coefficients. (English) Zbl 1203.30025

Summary: For a positive integer $n$, applying the Schwarz lemma for analytic functions $w\left(z\right)={c}_{n}{z}^{n}+...$ in the open unit disk $𝕌$, an assertion on a lemma well-known as Jack’s lemma, proved by S. S. Miller and 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 T. J. Suffridge [Duke Math. J. 37, 775–777 (1970; Zbl 0206.36202)] and T. H. MacGregor [J. Lond. Math. Soc., II. Ser. 9, 530–536 (1975; Zbl 0331.30011)], some differential subordination property concerning the subordination

$p\left(z\right)\prec q\left({z}^{n}\right),\phantom{\rule{2.em}{0ex}}z\in 𝕌,$

for functions $p\left(z\right)=a+{a}_{n}{z}^{n}+...$ and $q\left(z\right)=a+{b}_{1}z+...$ which are analytic in $𝕌$, is deduced, and an extension of some subordination relation is given.

##### MSC:
 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable) 30C45 Special classes of univalent and multivalent functions