×

Argument estimates for certain analytic functions. (English) Zbl 1137.30332

Summary: Let \(p(z)\) be analytic in the open unit disk \(\mathbf U\) with \(p(0)=1\) and \(p'(0)=0\). S. S. Miller and P. T. Mocanu [J. Math. Anal. Appl. 276, No. 1, 90–97 (2002; Zbl 1012.30012)] have obtained some interesting subordination theorems for such functions \(p(z)\). The object of the present paper is to discuss some sufficient conditions for the argument of \(p(z)\) to satisfy \(|\text{arg}\,p(z)|<(\pi/2)\rho\) for \(z\in\mathbf U\).

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.)

Citations:

Zbl 1012.30012
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] Fukui, S., and Sakaguchi, K.: An extension of a theorem of S. Ruscheweyh. Bull. Fac. Ed. Wakayama Univ. Natur. Sci., 29 , 1-3 (1980). · Zbl 1255.30012
[2] Hallenbeck, D. J., and Ruscheweyh, S.: Subordinations by convex functions. Proc. Amer. Math. Soc., 52 , 191-195 (1975). · Zbl 0311.30010
[3] Miller, S. S., and Mocanu, P. T.: Libera transform of functions with bounded turning. J. Math. Anal. Appl., 276 , 90-97 (2002). · Zbl 1012.30012
[4] Nunokawa, M.: On the order of strongly starlikeness of strongly convex functions. Proc. Japan Acad., 69A , 234-237 (1993). · Zbl 0793.30007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.