Behavior of holomorphic functions in the unit disk on arcs of positive hyperbolic diameter

Rung, D. C.

doi:10.1215/kjm/1250524061

Cited by 11 publications

(6 citation statements)

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“…Then [3,Theorem 1] the family is normal in some neighbourhood of z = 0, and hence [8,Theorem 15.2.2] equicontinuous. It follows from this and from (10), that for all sufficiently large n > N(a), a meets (\z\ = r n ), and |/(2)| < 1, z e A(a,é/n)r\ (\z\ ^ r n ).…”

Section: Theorem 5' If F(z) Is a Normal Meromorphic Function In The mentioning

confidence: 76%

“…THEOREM 6. There exists a continuous positive junction X(r), 0 ^ r < 1, such that if j(z) is any meromorphic junction in the unit disc D, satisfying on a boundary path a the inequality (10) \j{z)\ g A(|s|), z G a, then either j(z) = 0 or a is a p-path.…”

Section: Theorem 5' If F(z) Is a Normal Meromorphic Function In The mentioning

confidence: 99%

“…Suppose now, to prove the theorem, that a is a boundary path on which f(z) satisfies (10), and suppose that a is not a p-path. Then [3,Theorem 1] the family is normal in some neighbourhood of z = 0, and hence [8,Theorem 15.2.2] equicontinuous.…”

Section: Theorem 5' If F(z) Is a Normal Meromorphic Function In The mentioning

confidence: 99%

“…Under the hypotheses of Corollary 2, if a approaches z = 1 from within a boundary disc i7 0 , then in each boundary disc Hi (strictly) containing H 0 , there is a sequence of p-points.The proof is the same as the proof of Theorem 3.COROLLARY 5. If' f(z) satisfies the hypotheses of Corollary 2 on a non-tangential boundary path a ending at z = 1, then there is a sequence of p-points which is eventually in each boundary disc at z = 1.COROLLARY 6[10, Theorem 7]. Let f(z) be a function mer omorphic in D, and suppose that for some boundary path a tending to z = 1 non-tangentially and some a > 1,…”

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The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution

Gauthier

1970

Can. j. math.

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Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulusand the minimum modulusWhen no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then(1)This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.

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Section: Theorem 5' If F(z) Is a Normal Meromorphic Function In The mentioning

confidence: 76%

Section: Theorem 5' If F(z) Is a Normal Meromorphic Function In The mentioning

confidence: 99%

Section: Theorem 5' If F(z) Is a Normal Meromorphic Function In The mentioning

confidence: 99%

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confidence: 99%

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The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution

Gauthier

1970

Can. j. math.

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“…For results concerning sequential limits, see also [2]. This topic has been studied by several authors, in particular by Rung, who studied the connection between the boundary behavior of analytic functions and the hyperbolic metric in [15]. In Rung's results, the values attained by the function are assumed to approach a limit at a certain rate on a sequence of continua of given hyperbolic diameter.…”

Section: Introductionmentioning

confidence: 99%