Radius of convexity of starlike functions

DefinitionConjecture: a sharp radius of convexity for starlike functions of order 1/2#✓fully proved

Let \(\mathbb{D}\) be the open unit disc in \(\mathbb{C}\). Call \(f : \mathbb{C} \to \mathbb{C}\) a normalized starlike function of order \(1/2\) if \(f\) is holomorphic and injective on \(\mathbb{D}\), \(f(0) = 0\), \(f'(0) = 1\), and \(\operatorname{Re}(z f'(z)/f(z)) \ge 1/2\) for every \(z \in \mathbb{D}\) with \(z \ne 0\) and \(f(z) \ne 0\). Call \(P : \mathbb{C} \to \mathbb{C}\) a Carathéodory function if \(P\) is holomorphic on \(\mathbb{D}\), \(P(0) = 1\), and \(\operatorname{Re} P(z) > 0\) for every \(z \in \mathbb{D}\). Call a real number \(\rho\) a good radius if \(\operatorname{Re}(1 + z f''(z)/f'(z)) > 0\) for every normalized starlike function \(f\) of order \(1/2\) and every \(z\) with \(0 < |z| < \rho\) and \(f'(z) \ne 0\), and \(\operatorname{Re}\big((P(z) + 1)/2 + z P'(z)/(P(z) + 1)\big) > 0\) for every Carathéodory function \(P\) and every \(z\) with \(|z| < \rho\). The conjecture is the proposition that there is a good radius \(r\) with \(0 < r \le 1\) such that no real number \(r'\) with \(r < r' \le 1\) is a good radius.

This is Problem 6.22 of Hayman, Research Problems in Function Theory (2018), source record AMR-022-6022.

MathlibExt.Analysis.HaymanProblem622Wanted.conjecture · 33 lines

def conjecture : Prop :=
  ∃ r : ℝ, 0 < r ∧ r ≤ 1 ∧
    ((∀ f : ℂ → ℂ,
        DifferentiableOn ℂ f (Metric.ball (0 : ℂ) (1 : ℝ)) →
        f 0 = 0 →
        deriv f 0 = 1 →
        Set.InjOn f (Metric.ball (0 : ℂ) (1 : ℝ)) →
        (∀ z ∈ Metric.ball (0 : ℂ) (1 : ℝ), z ≠ 0 → f z ≠ 0 →
          (1 / 2 : ℝ) ≤ (z * deriv f z / f z).re) →
        ∀ z ∈ Metric.ball (0 : ℂ) r, z ≠ 0 → deriv f z ≠ 0 →
          0 < (1 + z * deriv (deriv f) z / deriv f z).re) ∧
      (∀ P : ℂ → ℂ,
        DifferentiableOn ℂ P (Metric.ball (0 : ℂ) (1 : ℝ)) →
        P 0 = 1 →
        (∀ z ∈ Metric.ball (0 : ℂ) (1 : ℝ), 0 < (P z).re) →
        ∀ s : ℝ, 0 ≤ s → s < r → ∀ z ∈ Metric.sphere (0 : ℂ) s,
          0 < ((P z + 1) / (2 : ℂ) + z * deriv P z / (P z + 1)).re)) ∧
    ∀ r' : ℝ, r < r' → r' ≤ 1 →
      ¬ ((∀ f : ℂ → ℂ,
        DifferentiableOn ℂ f (Metric.ball (0 : ℂ) (1 : ℝ)) →
        f 0 = 0 →
        deriv f 0 = 1 →
        Set.InjOn f (Metric.ball (0 : ℂ) (1 : ℝ)) →
        (∀ z ∈ Metric.ball (0 : ℂ) (1 : ℝ), z ≠ 0 → f z ≠ 0 →
          (1 / 2 : ℝ) ≤ (z * deriv f z / f z).re) →
        ∀ z ∈ Metric.ball (0 : ℂ) r', z ≠ 0 → deriv f z ≠ 0 →
          0 < (1 + z * deriv (deriv f) z / deriv f z).re) ∧
      (∀ P : ℂ → ℂ,
        DifferentiableOn ℂ P (Metric.ball (0 : ℂ) (1 : ℝ)) →
        P 0 = 1 →
        (∀ z ∈ Metric.ball (0 : ℂ) (1 : ℝ), 0 < (P z).re) →
        ∀ s : ℝ, 0 ≤ s → s < r' → ∀ z ∈ Metric.sphere (0 : ℂ) s,
          0 < ((P z + 1) / (2 : ℂ) + z * deriv P z / (P z + 1)).re))

Sources

• https://arxiv.org/abs/1809.07200

Used by 1

Used byA sharp radius of convexity for starlike functions of order 1/2 exists

TheoremA sharp radius of convexity for starlike functions of order 1/2 exists#●ready to prove

Let \(\mathbb{D}\) be the open unit disc in \(\mathbb{C}\). Call \(f : \mathbb{C} \to \mathbb{C}\) a normalized starlike function of order \(1/2\) if \(f\) is holomorphic and injective on \(\mathbb{D}\), \(f(0) = 0\), \(f'(0) = 1\), and \(\operatorname{Re}(z f'(z)/f(z)) \ge 1/2\) for every \(z \in \mathbb{D}\) with \(z \ne 0\) and \(f(z) \ne 0\). Call \(P : \mathbb{C} \to \mathbb{C}\) a Carathéodory function if \(P\) is holomorphic on \(\mathbb{D}\), \(P(0) = 1\), and \(\operatorname{Re} P(z) > 0\) for every \(z \in \mathbb{D}\). Call a real number \(\rho\) a good radius if \(\operatorname{Re}(1 + z f''(z)/f'(z)) > 0\) for every normalized starlike function \(f\) of order \(1/2\) and every \(z\) with \(0 < |z| < \rho\) and \(f'(z) \ne 0\), and \(\operatorname{Re}\big((P(z) + 1)/2 + z P'(z)/(P(z) + 1)\big) > 0\) for every Carathéodory function \(P\) and every \(z\) with \(|z| < \rho\). Then there is a good radius \(r\) with \(0 < r \le 1\) such that no real \(r'\) with \(r < r' \le 1\) is a good radius.

MacGregor (Proc. AMS 14 (1963) 71-76, Theorem 1) proved that the sharp radius of convexity of normalized starlike functions of order \(1/2\) is \(\sqrt{2\sqrt{3}-3} \approx 0.681\); this value is such a radius \(r\). The statement above does not assert the value of \(r\).

public theorem_wanted conjecture_holds : conjecture

Sources

• B. Bhowmik and S. Biswas, Radius of convexity of certain classes of functions defined by convolution, preprint (2026), arXiv:2606.20872, https://arxiv.org/abs/2606.20872.

Uses 1

Statement usesConjecture: a sharp radius of convexity for starlike functions of order 1/2

1 fully proved1 ready to prove

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