Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose that $f,g_k:\mathbb{C}\to\mathbb{C}$ satisfy for all $\Re(z)>0$ the following conditions:

(i) $f(z)=\sum_{k=0}^\infty g_k(z)$

(ii) $|g_k(z)|$ is bounded for $k=0,1,2,\ldots$

(iii) $\lim_{k \rightarrow \infty} |g_k(z)|=0$

(iv) $\sum_{k=0}^\infty g_k(x)$ converges

Is it enough to deduce from (i),(ii),(iii),(iv) that $f(z)$ is holomorphic for all $\Re(z)>0$? If not, what other conditions that $g_k(z)$ needs to satisfy so that $f(z)$ is holomorphic for all $\Re(z)>0$?

share|cite|improve this question

You at least need to ask that all the $g_k$ are holomorphic. Then I think (ii)+(iii)+(iv) imply that $\sum_k g_k$ converges uniformly on compact subsets.

Now apply a well-known theorem by Weierstrass (cf. Rudin, Real and Complex analysis, Th. 10.27) and get that $f$ is holomorphic.

share|cite|improve this answer
But if we suppose $g_k$ holomorphic, we get an issue that $|g_k(z)|$ bounded implies $g_k$ constant. – Matt Aug 25 '12 at 14:11
@Matt Not in a half-plane. Though the functions are assumed to be defined in $\mathbb C$, all the properties and conclusion are stated for the right half-plane. – user31373 Aug 26 '12 at 1:16
Oops. Thanks. I completely overlooked that sentence. – Matt Aug 26 '12 at 2:19

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.