Borel–Carathéodory theorem

In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory.

Statement of the theorem

Let a function $f$ be analytic on a closed disc of radius "R" centered at the origin. Suppose that "r" < "R". Then, we have the following inequality:

:$|f|\_r\; le\; frac\{2r\}\{R-r\}\; sup\_\; le\; frac\{1\}\{R\}.$

So

:$|f(omega)|\; le\; frac\{r\}\{R\}\; |2A-f(omega)|\; le\; frac\{r\}\{R\}\; (2A+|f(omega)|),$

whence

:$(R-r)\; |f(omega)|\; le\; 2Ar.$

Thus,

:$|f|\_r\; le\; frac\{2r\}\{R-r\}\; A.$

In the general case, where "f"(0) does not necessarily vanish, let $h(z)\; =\; f(z)\; -\; f(0)$. Then, by the triangle inequality,

:$sup\_\{|z|\; le\; R\}\; operatorname\{Re\}\; ,\; h(z)\; le\; sup\_\{|z|\; le\; R\}\; operatorname\{Re\}\; ,\; f(z)\; +\; |f(0)|.$

Because $h(0)\; =\; 0$, we can say that

:$|f(z)-f(0)|\; le\; frac\{2r\}\{R-r\}\; (A+|f(0)|),$

if |"z"| &le; "r". Furthermore,

:$|f(z)-f(0)|\; ge\; |f(z)|\; -\; |f(0)|,$

so

:$|f(z)|\; le\; |f(0)|\; +\; frac\{2r\}\{R-r\}\; (A+|f(0)|).$

Therefore,

:$|f(z)|\; le\; frac\{2r\}\{R-r\}\; A\; +\; frac\{R+r\}\{R-r\}\; |f(0)|.$

This completes the proof.

References

* Lang, Serge (1999). "Complex Analysis" (4th ed.). New York: Springer-Verlag, Inc. ISBN 0-387-98592-1.
* Titchmarsh, E. C. (1938). "The theory of functions." Oxford University Press.