Generalized Zalcman Conjecture for Starlike and Typically Real Functions
نویسندگان
چکیده
منابع مشابه
On a conjecture for trigonometric sums and starlike functions, II
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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The sharp bounds for the third and fourth coefficients of Ma-Minda starlike functions having fixed second coefficient are determined. These results are proved by using certain constraint coefficient problem for functions with positive real part whose coefficients are real and the first coefficient is kept fixed. Analogous results are obtained for a general class of close-to-convex functions
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is said to be typically-real of order p, if in (1.1) the coefficients bn are all real and if either (I) f(z) is regular in |a| =S1 and 3/(ei9) changes sign 2p times as z = eie traverses the boundary of the unit circle, or (II) f(z) is regular in | z\ < 1 and if there is a p < 1 such that for each r in p<r<l, $f(reie) changes sign 2p times as z = reie traverses the circle \z\ =r. This set of fun...
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ژورنال
عنوان ژورنال: Journal of Mathematical Analysis and Applications
سال: 1999
ISSN: 0022-247X
DOI: 10.1006/jmaa.1999.6378