Successive coefficients of starlike functions

DefinitionNormalized univalent functions (the class \(S\))#✓fully proved

A function \(f : \mathbb{C} \to \mathbb{C}\) is in the class \(S\) when it is analytic on the open unit disc \(\mathbb{D} = \{z : |z| < 1\}\), satisfies \(f(0) = 0\) and \(f'(0) = 1\), and is injective on \(\mathbb{D}\).

def IsInClassS (f : ℂ → ℂ) : Prop :=
  AnalyticOn ℂ f (Metric.ball (0 : ℂ) 1) ∧
    f 0 = 0 ∧ HasDerivAt f 1 0 ∧ Set.InjOn f (Metric.ball (0 : ℂ) 1)

Sources

• Hayman and Lingham, Research Problems in Function Theory (2018), Problem 6.46, https://arxiv.org/abs/1809.07200.

Used by 1

Used byHayman's successive coefficient problem for starlike functions

DefinitionStarlike functions on the unit disc#✓fully proved

A function \(f : \mathbb{C} \to \mathbb{C}\) is starlike when it is analytic on the open unit disc \(\mathbb{D} = \{z : |z| < 1\}\) and, for every \(z \ne 0\) in \(\mathbb{D}\), \(f(z) \ne 0\) and \(\operatorname{Re}\bigl(z f'(z) / f(z)\bigr) > 0\).

def IsStarlike (f : ℂ → ℂ) : Prop :=
  AnalyticOn ℂ f (Metric.ball (0 : ℂ) 1) ∧
    ∀ z ∈ Metric.ball (0 : ℂ) 1, z ≠ 0 → 0 < (z * deriv f z / f z).re

Sources

• Hayman, Research Problems in Function Theory (2018), Problem 6.46, https://arxiv.org/abs/1809.07200.

Used by 1

Used byHayman's successive coefficient problem for starlike functions

DefinitionHayman's successive coefficient problem for starlike functions#✓fully proved

The statement that for every function \(f : \mathbb{C} \to \mathbb{C}\) in the class \(S\) (analytic on the open unit disc \(\mathbb{D} = \{z : |z| < 1\}\), with \(f(0) = 0\) and \(f'(0) = 1\), and injective on \(\mathbb{D}\)) that is moreover starlike (\(\operatorname{Re}\bigl(z f'(z) / f(z)\bigr) > 0\) for every \(z \ne 0\) in \(\mathbb{D}\)), the Taylor coefficients \(a_n = f^{(n)}(0) / n!\) of \(f\) at \(0\) satisfy \(\bigl| |a_{n+1}| - |a_n| \bigr| \le 1\) for every natural number \(n \ge 1\).

def conjecture : Prop :=
  ∀ f : ℂ → ℂ, IsInClassS f → IsStarlike f →
    ∀ n : ℕ, 1 ≤ n → |‖wimanTaylorCoefficient f (n + 1)‖ - ‖wimanTaylorCoefficient f n‖| ≤ 1

Sources

• Hayman and Lingham, Research Problems in Function Theory (2018), Problem 6.46, https://arxiv.org/abs/1809.07200.

Uses 3 · Used by 1

Statement usesWiman Taylor coefficient · Normalized univalent functions (the class \(S\)) · Starlike functions on the unit disc

Used byLeung's theorem on successive coefficients of starlike functions

TheoremLeung's theorem on successive coefficients of starlike functions#●ready to prove

Let \(f : \mathbb{C} \to \mathbb{C}\) be analytic on the open unit disc \(\mathbb{D} = \{z : |z| < 1\}\), with \(f(0) = 0\) and \(f'(0) = 1\), injective on \(\mathbb{D}\), and starlike, meaning that \(\operatorname{Re}\bigl(z f'(z) / f(z)\bigr) > 0\) for every \(z \ne 0\) in \(\mathbb{D}\). Let \(a_n = f^{(n)}(0) / n!\) be the Taylor coefficients of \(f\) at \(0\). Then \(\bigl| |a_{n+1}| - |a_n| \bigr| \le 1\) for every natural number \(n \ge 1\).

This answers Problem 6.46 of Hayman and Lingham, Research Problems in Function Theory (2018), affirmatively. It was proved by Y. Leung (Bull. London Math. Soc. 10 (1978), 193-196) and independently by D. H. Hamilton.

public theorem_wanted conjecture_holds : conjecture

Sources

• Moduli difference of initial inverse logarithmic coefficients for starlike and convex functions (2026), arXiv:2603.17600, https://arxiv.org/abs/2603.17600.

Uses 1

Statement usesHayman's successive coefficient problem for starlike functions

3 fully proved1 ready to prove

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