I have the complex function $\frac{C^{1+iz}}{iz(1+iz)}$ for $C$ a positive constant and $z$ a complex variable for which $\Im(z)>1$. I am looking for a bound of this function. Any suggestions?
I forgot to mention a very important fact $\Im(z)>1$ and also $\Im(z)$ is constant.
Edit My reasoning up to this point is a follows:
Denote $z=x+iy$ where $x,y$ are real numbers and $y>1$.Then I have $\frac{C^{1+ix-y}}{(ix-y)(1+ix-y)}$.
$\frac{C^{1+ix-y}}{(ix-y)(1+ix-y)}\leq \frac{|C^{1+ix-y}|}{|ix-y||1+ix-y|}\leq\frac{|e^{(1+ix-y)\ln C}|}{\sqrt{x^2+y^2}*\sqrt{x^2+(1-y)^2}}\leq\frac{C^{1-y}}{\sqrt{x^2+y^2}*\sqrt{x^2+(1-y)^2}}\leq\frac{C^{1-y}}{y|1-y|}$ where the last step follows by observing that the maximum is achieved at $x=0$.
Your function blows up as $z\to i$ from above, so there is no bound.