## 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.)

Zbl 1012.30012
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### 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
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