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Non-real zeros of derivatives in the unit disc

James Langley

Abstract

The main result establishes an estimate for the growth of a real meromorphic function $f$ on the unit disc $Δ$ such that: (i) at least one of $f$ and $1/f$ has finitely many poles and non-real zeros in $Δ$; (ii)~$f^{(k)}$ has finitely many non-real zeros in $Δ$, for some $k \geq 2$.

Non-real zeros of derivatives in the unit disc

Abstract

The main result establishes an estimate for the growth of a real meromorphic function on the unit disc such that: (i) at least one of and has finitely many poles and non-real zeros in ; (ii)~ has finitely many non-real zeros in , for some .
Paper Structure (10 sections, 16 theorems, 88 equations)

This paper contains 10 sections, 16 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The definition of the Tsuji characteristic
  3. The Levin-Ostrovskii factorisation in the unit disc
  4. A growth estimate on the unit disc
  5. Proof of Theorem \ref{['thm2']}
  6. Wiman-Valiron theory in the unit disc
  7. The Tumura-Clunie method on a half-disc
  8. Proof of Theorem \ref{['thm1']}
  9. Proof of Proposition \ref{['propfirsttsuji']}
  10. Proof of Proposition \ref{['propsecondtsuji']}

Key Result

Theorem 1.1

Let the function $f$ be real meromorphic in the unit disc $\Delta$, that is, $f$ is meromorphic on $\Delta$ and maps the real interval $(-1, 1)$ into $\mathbb R \cup \{ \infty \}$. Assume that $f^{(k)}$ has finitely many non-real zeros in $\Delta$, for some $k \geq 2$, and that at least one of the f

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Proposition 6.1
  • ...and 6 more