Properties of functions concerned with Caratheodory functions

Mamoru Nunokawa, Emel Yavuz Duman, Shigeyoshi Owa

Abstract


Let \(\mathcal{P}_n\) denote the class of analytic functions \(p(z)\) of the form \(p(z)=1+c_nz^n+c_{n+1}z^{n+1}+\dots\) in the open unit disc \(\mathbb{U}\). Applying the result by S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 65 (1978), 289-305), some interesting properties for \(p(z)\) concerned with Caratheodory functions are discussed. Further, some corollaries of the results concerned with the result due to M. Obradovic and S. Owa (Math. Nachr. 140 (1989), 97-102) are shown.

Keywords


Analytic; starlike; Caratheodory function.

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References


Caratheodory, C., Uber den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64(1907), 95–115.

Jack, I. S., Functions starlike and convex of order (alpha), J. London Math. Soc. 3 (1971), 469–474.

Miller, S. S., Mocanu, P. T., Second order differential inequalities in the complex plane, J. Math. Anal. Appl. 65 (1978), 289–305.

Nunokawa, M., On the Bazilevic analytic functions, Sci. Rep. Fac. Edu. Gunma Univ. 21 (1972), 9–13.

Obradovic, M., Owa, S., A criterion for starlikeness, Math. Nachr. 140 (1989), 97–102.

Umezawa, T., Analytic functions convex in one direction, J. Math. Soc. Japan 4 (1952), 194–202.




DOI: http://dx.doi.org/10.2478/v10062-012-0028-7
Date of publication: 2015-07-15 00:00:00
Date of submission: 2016-01-12 09:25:02


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