Behavior of meromorphic functions at the boundary of the unit disc

Nafi, Bülent Örnek

doi:10.2298/fil1711443o

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2019

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“…In addition, the concerning results in more general aspects is discussed by M. Mateljević in [12] where was announced on ResearchGate. In recent years, the Schwarz lemma at the boundary for holomorphic mappings (see, [1], [3], [4], [6], [7], [16] [17], [18], [19], [20], [21], and references therein). Some of them are about the estimates from below for the modulus of the derivative of the function at the boundary points which satisfy the condition f (b) = 1.…”

Section: Introductionmentioning

confidence: 99%

On the Schwarz lemma at the upper half plane

Nafi

2019

Filomat

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In this paper, we give a simple proof for the boundary Schwarz lemma at the upper half plane. Considering that f (z) is a holomorphic function defined on the upper half plane, we derive inequalities for the modulus of derivative of f (z), f (0) , by assuming that the f (z) function is also holomorphic at the boundary point z = 0 on the real axis with f (0) = f (i). Let E be the unit disc and S = {z ∈ C : z > 0} the upper half plane in C. For i ∈ E, w−i 1+iw defines a conformal self-map of E carrying i to 0. Similary, for any i ∈ S, w → z − i z + i is conformal map of S onto E, i to 0. It follows in particular the S and E are conformal equivalent. Consider the function f (w) = f (z) − f (i)

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