A note on the avoidance criterion for normal functions
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A note on the avoidance criterion for normal functions Liu Yang1 Received: 28 February 2018 / Revised: 27 July 2020 / Accepted: 27 July 2020 © Springer Nature Switzerland AG 2020
Abstract Let ϕ1 , ϕ2 , ϕ3 be three functions meromorphic in the unit disc and continuous on the closure of such that ϕi (z) = ϕ j (z) on the unit circle |z| = 1. Let F be a family of meromorphic functions such that f = ϕ j on for j = 1, 2, 3 and f ∈ F. Then there exists a constant M such that (1 − |z|2 ) f # (z) ≤ M for each z ∈ and f ∈ F. In addition, this constant M depends only on the three functions ϕ1 , ϕ2 and ϕ3 . This generalizes the related theorems due to Lappan (In: Progress in analysis: proceedings of the 3rd international ISAAC congress in progress in analysis, vol 1, pp 221–228. World Scientific, 2003), and Xu and Qiu (C R Math 349:1159–1160, 2011), respectively. Keywords Meromorphic functions · Normal functions · Normal family · Spherical derivative Mathematics Subject Classification 30D45 · 30D35
1 Introduction Let F be a family of meromorphic functions on a domain D ⊂ C. Then F is said to be normal on D in the sense of Montel, if each sequence of F contains a subsequence which converges spherically uniformly on each compact subset of D to a meromorphic function which may be ∞ identically. See [3,6]. Suppose f is meromorphic on the unit disc . Then the functions f ◦ S, where z = S(z) is an arbitrary one-to-one conformal mapping of onto itself, form the meromorphic family F f . The function f is called normal if the family F f is normal.
This research was supported by NNSF of China (No. 11701006 ), and by Natural Science Foundation of Anhui Province (No. 1808085QA02), China.
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Liu Yang [email protected] Department of Mathematics and Physics, Anhui University of Technology, Maanshan 243032, People’s Republic of China 0123456789().: V,-vol
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L. Yang
It is a classical result that a meromorphic function f in is normal if the function f omits three distinct values. See [3,4]. For two functions f and g defined in , we write f = g on if f (z) = g(z) for every z ∈ ; write f ≡ g if f (z 0 ) = g(z 0 ) for some z 0 ∈ . In 2003, P. Lappan [1] proved a criterion for a normal function avoiding three continuous functions. Theorem A Let g1 , g2 , g3 be three continuous functions that avoid each other uniformly in the unit disc . Further, for each j = 1, 2, 3, let the family { g j ◦ φ : φ is conformal mapping of onto itself } be normal in . Let f be a function meromorphic in such that f = g j for j = 1, 2, 3. Then f is a normal function. In [5], the authors pointed out that when the functions avoided are all meromorphic in and continuous on the closure of , then they only need avoid each other on the unit circle instead of the unit disc. Theorem B Let ϕ1 , ϕ2 , ϕ3 be three functions meromorphic in and continuous on the closure of . such that ϕi (z) = ϕ j (z) on the unit circle |z| = 1. Let f be a function meromorphic such that f = ϕ j on for j = 1, 2, 3. Then f is a nor
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