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 $\Re(z)>0$?