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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(z)=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(z)q(z n ),z𝕌,

for functions p(z)=a+a n z n +... and q(z)=a+b 1 z+... which are analytic in 𝕌, is deduced, and an extension of some subordination relation is given.

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