Removable singularity and Liouville's Theorem

Technically, there is a problem with the question. More precisely, one should write $$\sup_{z\in{\mathbb C}\setminus\{0\}}\left|\frac{f(z)}z\right|<\infty,\quad (1)$$ as $g(z)=\frac{f(z)}{z}$ may be undefined at $z=0$.

1(a). Clearly (1) implies that $f(0)=0$, otherwise by continuity there exists $r>0$ such that $$|f(z)|\geq\frac 12|f(0)|\neq 0,\forall z~{\rm with~}|z|\leq r.$$ It then follows that $$\sup_{z\in{\mathbb C}\setminus\{0\}}\left|\frac{f(z)}z\right|\geq\frac 12|f(0)|\sup_{0<|z|\leq r}\frac 1{|z|}=\infty,$$ a contradiction.

Now since $f(z)$ is entire and $f(0)=0,$ one has $$\lim_{z\rightarrow 0}\frac{f(z)}z=\lim_{z\rightarrow 0}\frac{f(z)-f(0)}{z-0}=f’(0).$$ By Riemann’s Removable Singularity Theorem, $z=0$ is a removable singularity of $g(z)$, and $g(z)$ is analytic if $g(0)$ is defined as $f’(0).$

1(b). Define $h(z)=\frac{f(z)}{g(z)}$ for all $z\notin S:=\{z~|~g(z)=0\}.$ By a similar argument as in 1(a), any $z_0\in S$ is a removable singularity of $h(z),$ where one removes singularity by defining $h(z_0)=f^{(k)}(z_0)/g^{(k)}(z_0)$ (with $k$ the multiplicity of the zero $z_0$ for $g(z)$). Now $h(z)$ is entire and bounded, so by Liouville, one has $h(z)=c$ is a constant, hence $f(x)=cg(z),$ as required.

Remark. In 1(b), if $\sup_{{\mathbb C}\setminus S}|h(z)|$ is bounded, one can work out Riemann’s removable singularity theorem by power series expansion. If $z_0\in S$, then analogous to the proof in 1(a), one has $m\geq n$, where $m$ (resp. $n$) is the multiplicity of zero $z_0$ of $f(z)$ (resp. $g(z)$). Expanding in power series at $z_0$, one has $$f(z)=(z-z_0)^mf_1(z)=a_n(z-z_0)^n+\cdots+a_m(z-z_0)^m+\cdots$$ and $$g(z)=(z-z_0)^ng_1(z)=b_n(z-z_0)^n+\cdots,$$ where $$f_1(z_0)\neq 0,g_1(z_0)\neq 0,a_n=\frac{f^{(n)}(z_0)}{n!},b_n=\frac{g^{(n)}(z_0)}{n!}\neq 0.$$ (Note that $a_n=\cdots=a_{m-1}=0$ if $m>n$.)

After canceling common zeros at $z_0$, one sees that $\frac{f(z)}{g(z)}=(z-z_0)^{m-n}\frac {f_1(z)}{g_1(z)}$ has power series expansion at $z_0$ with constant term $$\frac{a_n}{b_n}=\frac{f^{(n)}(z_0)}{g^{(n)}(z_0)},$$ which is the value to be redefined for $h(z_0)$. (Note that $h(z_0)=0$ if $m>n$.)