By the Phragmen-Lindelof principle I mean the version for the angular region. Now we compare two versions of Phragmen-Lindelof principle for angular regions, for holomorphic and subharmonic functions respectively.
The holomorphic version is from Corrolary 4.4 on p.140 in Conway's book I.
Theorem 1： Let $$ \Omega_{\alpha}=\{z \in \mathbb{C} \ |\ |\operatorname{arg} z| < \frac{\pi}{2\alpha}\} $$ and $f$ be holomorphic in $\Omega_{\alpha}$ and continuous on its closure in $\mathbb{C}$. Assume $|f(z)| \leq M$ on $\partial \Omega_{\alpha}$ and \begin{align} \lim_{r \rightarrow \infty} \frac{\ln M(r)}{r^{\alpha}}=0, \ \ \text{where}\ M(r)=\sup_{|z|=r, z \in \Omega_{\alpha}} |f(z)|. \label{PLPgrowth} \end{align} Then $|f(z)| \leq M$ on $\Omega_{\alpha}$.
The subharmonic version is from p.94 in Protter-Weinberger's book ``Maximum Principles in differential equations", for a $C^2$ subharmonic function which is continuous on the closure.
Theorem 2： Let $$ \Omega_{\alpha}=\{z \in \mathbb{C} \ |\ |\operatorname{arg} z| < \frac{\pi}{2\alpha}\} $$ and $u \in C^2(\Omega_{\alpha})$ be subharmonic and continuous on its closure in $\mathbb{C}$. Assume $u(z) \leq N$ on $\partial \Omega_{\alpha}$ for some real number $N$ and \begin{align} \liminf_{r \rightarrow \infty} \frac{\sup_{|z|=r, z \in \Omega_{\alpha}} u(z)}{r^{\alpha}} \leq 0. \end{align} Then $u(z) \leq N$ on $\Omega_{\alpha}$.
Since $\ln |f|$ of a holomorphic function is subharmonic, we see that Protter-Weinberger makes a weaker assumption. In the proof given in their book, they used a harmonic barrier function $$ w_R(r, \theta)=\operatorname{Im}(i+\frac{2}{\pi}R^{\alpha}\ln \frac{R^{\alpha}+iz^{\alpha}}{R^{\alpha}-iz^{\alpha}}). $$ However this function is not continuous at $(R, \pm \frac{\pi}{2\alpha})$. They went on to apply the strong maximum principle on $\frac{u-N}{w_R(r, \theta)}$. It sounds to me their proof works.
To sum up, I would like to make sure.
Question: In view of Theorem 2, Theorem 1 seems to hold even if we replace $\lim_{r \rightarrow \infty}$ by $\liminf_{r \rightarrow \infty} \frac{\ln M(r)}{r^{\alpha}} \leq 0$. If so, is there a proof which only makes use of holomorphic functions other than subharmonic? If this is indeed the case, then this improved version deserves to appear in complex analysis textbooks. Otherwise there must be something wrong. Thanks for the help!
Edited on June 25th, it turns out that the question has a positive answer, Such an improvement (with $\liminf$) can be found in Ahlfors (Transactions AMS 1937) or his book "Conformal Invariants" (p.40).