On a conjecture for trigonometric sums and starlike functions
نویسندگان
چکیده
We pose and discuss the following conjecture: let s n(z) := ∑n k=0 ( )k k! zk , and for ∈ (0, 1] let ∗( ) be the unique solution ∈ (0, 1] of ∫ ( +1) 0 sin (t − ) t −1 dt = 0. Then for 0< ∗( ) and n ∈ N we have | arg[(1− z) s n(z)]| /2, |z|< 1. We prove this for = 1 2 , and in a somewhat weaker form, for = 3 4 . Far reaching extensions of our conjectures and results to starlike functions of order 1− /2 are also discussed. Our work is closely related to recent investigations concerning the understanding and generalization of the celebrated Vietoris’ inequalities. © 2007 Elsevier Inc. All rights reserved. MSC: 42A05; 42A32; 26D05; 26D15; 30C45; 33C45
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عنوان ژورنال:
- Journal of Approximation Theory
دوره 149 شماره
صفحات -
تاریخ انتشار 2007