Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-16T18:39:53.609Z Has data issue: false hasContentIssue false
  • English
  • Français

The mean value theorem and analytic functions

Published online by Cambridge University Press:  20 January 2009

Elgin H. Johnston
Elgin H. Johnston
Affiliation:
Iowa State UniversityAmes, Iowa 50010
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that the mean value theorem (MVT) does not, in general, hold for analytic functions. The most familiar example to this effect is f(z) = ez since e2πi−e0≠2πiez0 for any z0∈ℂ. On the other hand, it is easy to show that the MVT holds in ℂ if f(z) is a polynomial of degree at most 2. Thus it is natural to ask what conditions on a function f(z) analytic in a domain D are necessary and sufficient for f(z) to satisfy the MVT in D. This is one of the questions answered in this paper.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 26 , Issue 3 , October 1983 , pp. 289 - 295
Copyright
Copyright © Edinburgh Mathematical Society 1983

References

REFERENCES

1.Mcleod, R. M., Mean value theorems for vector valued functions, Proc. Edinburgh Math. Soc. (2) 14 (1965), 197–209.CrossRefGoogle Scholar
2.Novinger, W. P., A local mean value theorem for analytic functions with smooth boundary values, Glasgow Math. J. 15 (1974), 27–29.CrossRefGoogle Scholar
3.Robertson, J. M., A local mean value theorem for the complex plane, Proc. Edinburgh Math. Soc. (2) 16 (1968/1969), 329–331.CrossRefGoogle Scholar
4.Rudin, Walter, Real and Complex Analysis, (McGraw-Hill, New York, 1966).Google Scholar
5.Samuelson, Åke, A local mean value theorem for analytic functions, American Math. Monthly 80 (1973), 45–46.CrossRefGoogle Scholar