Let $f(z)$ be a holomorphic function in the angle $A=\{0<\arg z<\frac{\pi}2\}$, continuous in $\bar A$, satisfying $|f(z)|\le M$ on $\partial A$ and satysfying the following growth condition: $$ |f(x+i y)|\le Ce^{y^2}\quad\hbox{in $A$}. \label{1}\tag{$\ast$} $$ Question. Does it follow that $f$ is bounded in $A$?
If in \eqref{1} we consider $y^\rho$ with $\rho<2$, then we have $|f(z)|\le C e^{|z|^\rho}$ and by the Phragmén–Lindelöf principle $f$ would be bounded in $A$. For $\rho=2$ it's not the case as the example $f(z)=e^{-i z^2}$, where $|f(x+i y)|=e^{2xy}$ shows. But for this function the condition \eqref{1} doesn't hold.