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SUMMARY:Yueyang Zhang (University of Science and Technology Beijing)
DTSTART:20210420T130000Z
DTEND:20210420T140000Z
DTSTAMP:20260422T201404Z
UID:CAvid/37
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/37/">O
 n entire function $e^{p(z)}\\int_0^{z}\\beta(t)e^{-p(t)}dt$ with applicati
 ons to Tumura--Clunie equations and complex dynamics</a>\nby Yueyang Zhang
  (University of Science and Technology Beijing) as part of CAvid: Complex 
 Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nLet $p(z)$ be 
 a non-constant polynomial and $\\beta(z)$ be a small entire function of $e
 ^{p(z)}$ in the sense of Nevanlinna. By using the classical Phragm\\'{e}n-
 -Lindel\\"{o}f theorem\, we analyze the growth behavior of the entire func
 tion $H(z):=e^{p(z)}\\int_0^{z}\\beta(t)e^{-p(t)}dt$ on the complex plane 
 $\\mathbb{C}$. We then apply these results to Tumura--Clunie type differen
 tial equation $f(z)^n+P(z\,f)=b_1(z)e^{p_1(z)}+b_2(z)e^{p_2(z)}$\, where $
 b_1(z)$ and $b_2(z)$ are non-zero polynomials\, $p_1(z)$ and $p_2(z)$ are 
 two polynomials of the same degree~$k\\geq 1$ and $P(z\,f)$ is a different
 ial polynomial in $f$ of degree $\\leq n-1$ with meromorphic functions of 
 order less than~$k$ as coefficients\, and precisely characterize entire so
 lutions of this equation. This gives an answer to a problem in the literat
 ure and allows to find all zero-free solutions of the second-order differe
 ntial equation $f''-(b_1e^{p_1}+b_2e^{p_2}+b_3)f=0$\, where $b_3$ is a pol
 ynomial. We also use the Phragm\\'{e}n--Lindel\\"{o}f theorem to prove a t
 heorem on certain first-order non-homogeneous linear differential equation
  related to complex dynamics.\n
LOCATION:https://researchseminars.org/talk/CAvid/37/
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