Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 45, 2020, 479-491

# A LEMMA ON THE DIFFERENCE QUOTIENTS

## Risto Korhonen, Kazuya Tohge, Yueyang Zhang and Jianhua Zheng

University of Eastern Finland,
Department of Physics and Mathematics

P.O. Box 111, FI-80101 Joensuu, Finland; risto.korhonen 'at' uef.fi

Kanazawa University,
College of Science and Engineering

Kakuma-machi, Kanazawa, 920-1192, Japan;
tohge 'at' se.kanazawa-u.ac.jp

University of Science and Technology Beijing,
School of Mathematics and Physics

No. 30 Xueyuan Road, Haidian, Beijing, 100083, P.R. China;
zhangyueyang 'at' ustb.edu.cn

Tsinghua University,
Department of Mathematical Sciences

Beijing 100084, P.R. China;
zheng-jh 'at' mail.tsinghua.edu.cn

**Abstract.**
First, we are concerned with a lemma on the difference quotients due to Halburd, Korhonen and Tohge. We show that for meromorphic functions whose deficiency is origin dependent the exceptional set associated with this lemma is of infinite linear measure. In particular, for such entire functions in this set there is an infinite sequence {*r*_{n}} such that
*m*(*r*_{n},*f*(*z* +
*c*)/*f*(*z*)) ≠
*o*(*T*(*r*_{n},*f*)) for all
*r*_{n}. Then we extend this lemma to the case
of meromorphic functions *f*(*z*) such that
log* T*(*r*,*f*) ≤
*ar*/(log *r*)^{2+ν},
*a*, ν > 0, for all sufficiently large *r*,
by using a new Borel type growth lemma. Second, we give a discrete version of this Borel type growth lemma and use it to provide an extension of Halburd's result on first order discrete equations of Malmquist type.

**2010 Mathematics Subject Classification:**
Primary 30D35; Secondary 30D30.

**Key words:**
Difference quotient, growth of meromorphic functions, Diophantine integrability,
algebraic entropy.

**Reference to this article:** R. Korhonen, K. Tohge, Y. Zhang and J. Zheng:
A lemma on the difference quotients.
Ann. Acad. Sci. Fenn. Math. 45 (2020), 479-491.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4521

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