Abstract
The object of this paper is to study relationship between successive coefficients of some subclasses of the class of univalent functions in the unit disk. the result obtained is sharp, and is used to provide a new, short proof of the well-known conjecture of Robertson on the coefficients of close-to-convex functions.
Results & Lemmas (7)
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Lemma 2.1
Lemma 2.1([1]) Let φ(z) = P j=0 λjzj be an arbitrary power series having a positive radius of convergence and normalized by φ(0) = 0. Also…
Lemma 2.1([1]) Let φ(z) = P j=0 λjzj be an arbitrary power series having a positive radius of convergence and normalized by φ(0) = 0. Also let exp(φ(z)) = P j=0 βjzj, β0 = 1, be the exponentiated power series of φ(z). Then
Lemma 2.2
Lemma 2.2([2]) For every p ∈P and every positive integer k, there exists a complex number ν with |ν| = 1 such that: k X j=1 1 j pj −νj 2 ≤…
Lemma 2.2([2]) For every p ∈P and every positive integer k, there exists a complex number ν with |ν| = 1 such that: k X j=1 1 j pj −νj 2 ≤ k X j=1 1 j . 3.0 Main Result
Theorem 3.1
Theorem 3.1 Let f ∈Sn, then for any complex number ν such that |ν| = 1, |(k + 1)nak+1 −νknak| ≤1; k = 1, 2, 3, · · · (3.1) For any fixed n…
Theorem 3.1 Let f ∈Sn, then for any complex number ν such that |ν| = 1, |(k + 1)nak+1 −νknak| ≤1; k = 1, 2, 3, · · · (3.1) For any fixed n ∈N0, the inequality is the best possible, with equality only for the function: fn(z) = In z (1 −νz)(1 −γz) , |γ| = 1. (3.2) By triangle inequality, the first corollary below is immediate and the next two are its consequences.
Corollary 3.2
Corollary 3.2 Let f ∈Sn, then |(k + 1)n|ak+1| −kn|ak|| ≤1; k = 1, 2, 3, · · ·
Corollary 3.2 Let f ∈Sn, then |(k + 1)n|ak+1| −kn|ak|| ≤1; k = 1, 2, 3, · · ·
Corollary 3.3
Corollary 3.3 For every f ∈Sn |ak| ≤k1−n; k = 2, 3, · · ·
Corollary 3.3 For every f ∈Sn |ak| ≤k1−n; k = 2, 3, · · ·
Corollary 3.3
Corollary 3.3 For every odd function f ∈Sn |a2k+1| ≤(2k + 1)−n; k = 1, 2, 3, · · ·
Corollary 3.3 For every odd function f ∈Sn |a2k+1| ≤(2k + 1)−n; k = 1, 2, 3, · · ·
Theorem 3.1
Theorem 3.1 For every f ∈K and positive integers k, m; |k|ak| −m|am|| ≤|k2 −m2|. (3.17)
Theorem 3.1 For every f ∈K and positive integers k, m; |k|ak| −m|am|| ≤|k2 −m2|. (3.17)
Function classes studied:
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