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A local mean-value theorem for analytic functions with smooth boundary values
| Content Provider | Scilit |
|---|---|
| Author | Novinger, W. P. |
| Copyright Year | 1974 |
| Description | Let f be an analytic function on a connected open set Ώ in the complex plane. Then, for given a, b ∈ Ώ, the equation need not have a solution z ∈ Ώ. As a matter of fact, this would happen with each locally one-to-one analytic function which is not one-to-one on Ώ. But if we fix a a ∈ Ώ, then, for all b sufficiently close to a, (1) is solvable for z. This is an easy consequence of the Open Mapping Theorem applied to f'. For, assuming that f' is non-constant (otherwise, (1) holds for all a, b, z ∈Ώ), the Open Mapping Theorem tells us that f'(Ώ), the image under f' of Ώ, is an open neighbourhood of f'(a); so it is a direct consequence of the definition of f'(a) that there exists δ > 0 such that 0 < |b – a| < δ implies (f(b) – f(a))/(b – a) ∈f'(Ώ). A stronger statement has been obtained by J. M. Robertson [1, p. 329], who has shown that and, if f''(a) ≠ 0, then . |
| Related Links | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/D6BB6F7615D1F108CD82D95CE379831B/S0017089500002068a.pdf/div-class-title-a-local-mean-value-theorem-for-analytic-functions-with-smooth-boundary-values-div.pdf |
| Ending Page | 29 |
| Page Count | 3 |
| Starting Page | 27 |
| ISSN | 00170895 |
| e-ISSN | 1469509X |
| DOI | 10.1017/s0017089500002068 |
| Journal | Glasgow Mathematical Journal |
| Issue Number | 1 |
| Volume Number | 15 |
| Language | English |
| Publisher | Cambridge University Press (CUP) |
| Publisher Date | 1974-03-01 |
| Access Restriction | Open |
| Subject Keyword | Glasgow Mathematical Journal Applied Mathematics Mean Value Theorem Analytic Function |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
