Question about three lines theorem

Question

I'm trying to prove that the function defined as $$F_\epsilon(z)=F(z)M_0^{z-1}M_1^{-z}e^{\epsilon z(z-1)},$$ where $F$ is an holomorphic function on $0<Re z<1$ and continuous and bounded on the clousure of the strip such as $$|F(it)|\leq M_0,\,|F(1+it)|\leq M_1,\,\forall t\in\mathbb{R},$$ satisfies that $$|F_\epsilon(it)|\leq 1,\,|F(1+it)|\leq 1.$$ I have done the following $$|F_\epsilon(it)|=|F(it)M_0^{it-1}M_1^{-it}e^{-\epsilon t^2}e^{-\epsilon it}\leq |M_0^{it}M_1^{-it}|=|e^{it\log(M_0/M_1)}|\leq 1.$$ Im not sure that im able to say that $$M_0^{it}M_1^{-it}=e^{it\log(M_0/M_1)},$$ because of the behaviour of complex logarithm. What do you think? How can i prove that it is well define? Any help will be very appreciated

Answer 1

For any $a>0$ and $t\in\Bbb{R}$, we have $a^{it}:=e^{it\log a}$, and no funny business with complex logarithms; this is the usual logarithm $(0,\infty)\to\Bbb{R}$. So, the absolute value is $|a^{it}|=|e^{it\log a}|=1$. Apply this with $M_0,M_1$.