Complex linear combinations of n functions in S*(A,B)

Bhargava, S. and Nanjunda Rao, S. (1987) Complex linear combinations of n functions in S*(A,B). Indian J. Pure Appl. Math., 18 (9). pp. 835-839.

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Let A and B be two real numbers with −1≤B<A≤1. Let P(A,B) be the class of functions p(z)=1+p1z+⋯ holomorphic in the unit disc K such that p(z)=(1+Aw(z))/(1+Bw(z)), z∈K, where w(z) is holomorphic in K with w(0)=0 and |w(z)|<1. Let S∗(A,B) consist of functions f(z)=z+∑∞n=2anzn holomorphic in K such that zf′(z)/f(z)∈P(A,B). The authors use results of V. V. Anh and P. D. Tuan [Rev. Roumaine Math. Pures Appl. 26 (1979), no. 10, 1413–1424; MR0554559 (81b:30019)] and B. Pinchuk [Duke Math. J. 35 (1968), 721–734; MR0230896 (37 #6454)] to obtain the discs of starlikeness and convexity of linear combinations γ1f1(z)+⋯+γnfn(z) (∑ni=1γi=1) of n functions f1,f2,⋯,fn in S∗(A,B) under the condition on the "joint parameters'', 0≤max1≤i,j≤n(argγi/γj)<π. A generalization of the result obtained by R. K. Stump [Canad. J. Math. 23 (1971), 712–717; MR0285701 (44 #2919)] is used in obtaining the results.

Item Type: Article
Subjects: Physical Sciences > Mathematics
Divisions: PG Campuses > Manasagangotri, Mysore > Mathematics
Depositing User: Kodandarama
Date Deposited: 21 May 2013 05:47
Last Modified: 21 May 2013 05:47

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