[Paper Review] A note on the value distribution of Differential Polynomials

Subhas S. Bhoosnurmath, Bikash Chakraborty+1 more|arXiv (Cornell University)|Jun 4, 2018

Meromorphic and Entire Functions8 references3 citations

TL;DR

This paper establishes quantitative estimates for the characteristic function $ T(r,f) $ of a transcendental meromorphic function $ f $ in terms of the counting function of a homogeneous differential polynomial $ P[f] $ generated by $ f $. By leveraging value distribution theory and Nevanlinna theory, the authors derive sharp upper bounds involving $ N(r, 1/(P[f]-1)) $ and $ \overline{N}(r, 1/(P[f]-1)) $, improving upon earlier results for differential monomials and polynomials with small function coefficients.

ABSTRACT

Let $f$ be a transcendental meromorphic function, defined in the complex plane $\mathbb{C}$. In this paper, we give a quantitative estimations of the characteristic function $T(r,f)$ in terms of the counting function of a homogeneous differential polynomial generated by $f$. Our result improves and generalizes some recent results.

Motivation & Objective

• To provide quantitative estimates of the Nevanlinna characteristic function $ T(r,f) $ in terms of the counting function of a differential polynomial $ P[f] $ generated by a transcendental meromorphic function $ f $.
• To generalize and improve upon existing results such as Theorem A (Mues), Theorem B (Huang & Gu), and Theorem D (Lahiri & Dewan) for differential monomials and polynomials.
• To investigate the value distribution of homogeneous differential polynomials $ P[f] $, particularly focusing on the zeros of $ P[f] - 1 $, under conditions on the degrees and weights of the monomials.
• To refine the error terms using $ S^*(r,f) $, which is $ o(T(r,f)) $ outside a set of logarithmic density zero, enhancing precision in asymptotic estimates.
• To establish sharp inequalities linking $ T(r,f) $ with the reduced and full counting functions of $ P[f] - 1 $, especially for differential polynomials of higher order derivatives.

Proposed method

• Utilizes Nevanlinna theory, particularly the First and Second Main Theorems, to relate the growth of $ f $ to the distribution of zeros of $ P[f] - 1 $.
• Applies key lemmas (e.g., Lemmas 3.2, 3.5, 3.6) to bound the counting functions of $ f $, $ f^{(k)} $, and $ (P[f])' $, using the structure of differential monomials and their degrees and weights.
• Defines the degree $ d(P) $, weight $ \Gamma_P $, and excess $ \nu = \max\{ q_{1j} + 2q_{2j} + \cdots + kq_{kj} \} $ to characterize the differential polynomial $ P[f] $, enabling precise estimation.
• Introduces the notation $ S^*(r,f) = o(T(r,f)) $ as $ r \to \infty $, $ r \notin E^* $, where $ E^* $ has logarithmic density zero, to refine error terms beyond standard $ S(r,f) $.
• Derives inequalities by combining the First Main Theorem with estimates on the logarithmic derivative and the derivative of $ P[f] $, leading to bounds on $ T(r,f) $ in terms of $ N(r, 1/(P[f]-1)) $ and $ \overline{N}(r, 1/(P[f]-1)) $.
• Employs the concept of small functions $ b_j(z) $ with $ T(r,b_j) = S(r,f) $, ensuring the coefficients do not dominate the growth of $ f $, and maintains control over the counting functions of $ b_jP[f] - 1 $.

Experimental results

Research questions

• RQ1 Can the characteristic function $ T(r,f) $ be bounded above by a multiple of the counting function $ N(r, 1/(P[f]-1)) $ for a homogeneous differential polynomial $ P[f] $ generated by a transcendental meromorphic function $ f $?
• RQ2 What is the optimal constant $ B_1 $ such that $ T(r,f) \leq B_1 N(r, 1/(P[f]-1)) + S^*(r,f) $, and how does it depend on the structure of $ P[f] $?
• RQ3 Can a similar bound be established using the reduced counting function $ \overline{N}(r, 1/(P[f]-1)) $, and what is the best possible constant $ B_2 $ in this case?
• RQ4 How do the degree $ d(P) $, weight $ \Gamma_P $, and excess $ \nu $ of the differential polynomial influence the sharpness of the bounds on $ T(r,f) $?
• RQ5 To what extent do the results generalize prior theorems such as Theorem A (Mues), Theorem B (Huang & Gu), and Theorem D (Lahiri & Dewan) for specific forms of $ P[f] $?

Key findings

• For a homogeneous differential polynomial $ P[f] $ of degree $ d(P) $, with $ q_0 \geq 2 $, $ q_k \geq 2 $, and $ k \geq 2 $, the inequality $ T(r,f) \leq \frac{1}{q_0 - 1} N\left(r, \frac{1}{P[f] - 1}\right) + S^*(r,f) $ holds.
• For the reduced counting function, the bound $ T(r,f) \leq \frac{1}{d(P) - \nu - 2} \overline{N}\left(r, \frac{1}{P[f] - 1}\right) + S(r,f) $ is established, where $ \nu = \max_j \{ q_{1j} + 2q_{2j} + \cdots + kq_{kj} \} $.
• A third bound is derived: $ T(r,f) \leq \frac{k+1}{d(P) + kq_* - \nu - 2(k+1)} \overline{N}\left(r, \frac{1}{P[f] - 1}\right) + S(r,f) $, where $ q_* = \min_j q_{1j} $, under the condition $ q_* \geq 2 $.
• The results improve upon Theorem D (Lahiri & Dewan) by providing tighter constants and extending the framework to general homogeneous differential polynomials with small function coefficients.
• The use of $ S^*(r,f) $, which is $ o(T(r,f)) $ outside a set of logarithmic density zero, provides a stronger error term than the standard $ S(r,f) $, enhancing the precision of the estimates.
• The bounds are sharp in the sense that they depend explicitly on structural parameters of $ P[f] $, such as $ d(P) $, $ \nu $, and $ q_0 $, and reduce to known results when specialized to monomials like $ f^2 f' $ or $ f^l (f^{(k)})^n $.