Criteria for Holomorphic

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 $\Re(z)>0$?

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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.
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