[Paper Review] Sufficient conditions for univalence and study of a class of meromorphic univalent functions

Bappaditya Bhowmik, Firdoshi Parveen|arXiv (Cornell University)|May 17, 2017

Meromorphic and Entire Functions3 references6 citations

TL;DR

This paper establishes a new sufficient condition for univalence of meromorphic functions with a simple pole in the unit disk, proving that |U_f(z)| < 1 implies univalence. It introduces the class 𝒱_p(λ) as a strictly larger family than the previously studied 𝒰_p(λ), and derives sharp coefficient bounds, including the exact region of variability for the second Taylor coefficient and a conjecture on sharp bounds for higher-order coefficients.

ABSTRACT

In this article we consider the class $\mathcal{A}(p)$ which consists of functions that are meromorphic in the unit disc $\ID$ having a simple pole at $z=p\in (0,1)$ with the normalization $f(0)=0=f'(0)-1 $. First we prove some sufficient conditions for univalence of such functions in $\ID$. One of these conditions enable us to consider the class $\mathcal{V}_{p}(λ)$ that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that $\mathcal{U}_{p}(λ)\subsetneq \mathcal{V}_{p}(λ)$, where $\mathcal{U}_{p}(λ)$ was introduced and studied in \cite{BF-1}. Finally, we discuss some coefficient problems for $\mathcal{V}_{p}(λ)$ and end the article with a coefficient conjecture.

Motivation & Objective

• To improve the sufficient condition for univalence of meromorphic functions with a simple pole in the unit disk.
• To define and study a new class 𝒱_p(λ) of meromorphic univalent functions satisfying |U_f(z)| < λ.
• To establish the proper inclusion 𝒰_p(λ) ⊊ 𝒱_p(λ) ⊊ Σ(p), where Σ(p) is the class of univalent functions in 𝒜(p).
• To investigate coefficient problems for functions in 𝒱_p(λ), particularly the region of variability of Taylor coefficients.
• To propose a conjecture on sharp bounds for higher-order Taylor coefficients in 𝒱_p(λ).

Proposed method

• Transforms functions f ∈ 𝒜(p) into F ∈ 𝒫_p via F(ζ) = 1/f(1/ζ), linking them to the class 𝒫_p ⊂ 𝒫.
• Applies Theorem A (from [1]) to F ∈ 𝒫_p, showing that |F’(ζ) - 1| ≤ 1 implies univalence of F, hence univalence of f.
• Uses the identity F’(ζ) - 1 = U_f(z) with z = 1/ζ to translate the condition |U_f(z)| < 1 into a sufficient univalence criterion.
• Derives the representation f(z) = z / (1 - (f''(0)/2)z + λz ∫₀^z w(t)dt) for f ∈ 𝒱_p(λ), where w ∈ ℬ (bounded analytic functions in 𝔻).
• Applies this representation to compute the exact region of variability for the second Taylor coefficient a₂(f), showing |a₂(f) - 1/p| ≤ λp.
• Proposes a conjecture on sharp bounds for |a_n(f)| in terms of λ, p, and n, based on extremal function k_p^λ.

Experimental results

Research questions

• RQ1 What is the optimal sufficient condition for univalence of meromorphic functions in 𝒜(p) with a simple pole at z = p ∈ (0,1)?
• RQ2 How does the new class 𝒱_p(λ) defined by |U_f(z)| < λ relate to the previously studied class 𝒰_p(λ)?
• RQ3 What is the exact region of variability for the second Taylor coefficient a₂(f) of functions in 𝒱_p(λ)?
• RQ4 Are the sharp coefficient bounds for |a_n(f)| in 𝒰_p(λ) also valid and sharp in the larger class 𝒱_p(λ)?
• RQ5 Can the extremal function k_p^λ be used to conjecture sharp bounds for higher-order Taylor coefficients in 𝒱_p(λ)?

Key findings

• The condition |U_f(z)| < 1 is a sufficient condition for univalence of f ∈ 𝒜(p), improving upon the earlier bound ((1−p)/(1+p))².
• The class 𝒱_p(λ) is strictly larger than 𝒰_p(λ), with 𝒰_p(λ) ⊊ 𝒱_p(λ) ⊊ Σ(p), as shown by constructing functions in 𝒱_p(λ) not in 𝒰_p(λ).
• The exact region of variability for the second Taylor coefficient a₂(f) is the closed disk |a₂(f) - 1/p| ≤ λp, with equality attained for f_θ(z) = z / (1 - (z/p)(1 + λp²e^{iθ}) + λe^{iθ}z²).
• The bound |a₂(f)| ≤ 1/p + λp is sharp, with equality achieved by the function k_p^λ(z) = -pz / ((z−p)(1−λpz)).
• The function k_p^λ is extremal for the class 𝒱_p(λ), and a conjecture is proposed that |a_n(f)| ≤ (1−λ^n p^{2n}) / (p^{n−1}(1−λp²)) for all n ≥ 3, with equality for k_p^λ.