A note on the $2/3$ conjecture for starlike functions
نویسندگان
چکیده
منابع مشابه
A Note on Multivalent Starlike Functions
In a recent paper [2], G. R. Blakley raised a question about the decomposition of a ^-valent starlike function into the product of two ^-valent starlike functions. It is the purpose of this note to prove the conjecture of Blakley. Definition 1. The function/(z), regular in |z| <1, is said to be in the class S(p, m), where p and m are positive integers with p^m, if and only if (1) there exists a...
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We prove the case ρ = 4 of the following conjecture of Koumandos and Ruscheweyh: let s μ n (z) := ∑n k=0 (μ)k k! z k , and for ρ ∈ (0, 1] let μ(ρ) be the unique solution of ∫ (ρ+1)π 0 sin(t − ρπ)tμ−1dt = 0 in (0, 1]. Then we have | arg[(1− z)ρs n (z)]| ≤ ρπ/2 for 0 < μ ≤ μ(ρ), n ∈ N and z in the unit disk of C and μ(ρ) is the largest number with this property. For the proof of this other new re...
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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 orde...
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In 1985, Chor et al 2] conjectured that the only 1-resilient symmetric functions are the exclusive-or of all n variables and its negation. In this note the existence of symmetric resilient functions is shown to be equivalent to the existence of a solution to a simultaneous subset sum problem. Then, using arithmetic properties of certain binomial coeecients, an innnite class of counterexamples t...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1972
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1972-0304632-3