## Argument inequalities for certain analytic functions.(English)Zbl 1205.30014

Summary: Let $$q$$ be analytic in the open unit disk $$U$$ with $$q(0)=1$$ and $$q(z)\neq 0$$ $$z\in U$$. By using the method of differential subordinations, we derive certain conditions involving $$q$$ and $$zq'$$ under which the functions $$q$$ satisfy the following two-sided inequality: $-\frac{\alpha_2\pi}{2}<\mathrm{arg} q(z)<-\frac{\alpha_1\pi}{2}~(z\in U)$ for some $$\alpha_1$$ and $$\alpha_2$$ $$(0<\alpha_1, \alpha_2\leq 1$$. Several interesting consequences of the main results are also given. All these results presented here are sharp.

### MSC:

 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
Full Text:

### References:

 [1] Brannan, D.A.; Kirwan, W.E., On some classes of bounded univalent functions, J. London math. soc., 1, 2, 431-443, (1969) · Zbl 0177.33403 [2] Liu, J.-L., The Noor integral operator and strongly starlike functions, J. math. anal. appl., 261, 441-447, (2001) · Zbl 1040.30005 [3] Nunokawa, M., On the order of strongly starlikeness of strongly convex functions, Proc. Japan acad. ser. A, 69, 234-237, (1993) · Zbl 0793.30007 [4] Nunokawa, M.; Owa, S.; Saitoh, H.; Ikeda, A.; Koike, N., Some results for strongly starlike functions, J. math. anal. appl., 212, 98-106, (1997) · Zbl 0880.30012 [5] Nunokawa, M.; Thomas, D.K., On convex and starlike functions in a sector, J. aust. math. soc. ser. A, 60, 363-368, (1996) · Zbl 0864.30008 [6] Obradović, M.; Owa, S., Some sufficient conditions for strongly starlikeness, Int. J. math. math. sci., 24, 643-647, (2000) · Zbl 0980.30012 [7] Padmanabhan, K.S., On sufficient conditions for starlikeness, Indian J. pure appl. math., 32, 543-550, (2001) · Zbl 0979.30007 [8] Ponnusamy, S.; Singh, V., Criteria for strongly starlike functions, Complex variables theory appl., 34, 267-291, (1997) · Zbl 0892.30005 [9] Srivastava, H.M.; Yang, Ding-Gong; Xu, N-Eng, Subordinations for multivalent analytic functions associated with the dziok – srivastava operator, Integral transforms spec. funct., 20, 581-606, (2009) · Zbl 1170.30006 [10] Xu, N-Eng; Yang, Ding-Gong, An application of differential subordinations and some criteria for starlikeness, Indian J. pure appl. math., 36, 541-556, (2005) · Zbl 1147.30303 [11] Miller, S.S.; Mocanu, P.T., () [12] Takahashi, N.; Nunokawa, M., A certain connection between starlike and convex functions, Appl. math. lett., 16, 653-655, (2003) · Zbl 1064.30008
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.