Let $a,b >0$ and $|*|$ denote the absolute value. Let $f(z)$ be a realvalued analytic function defined for $Re(z)>1.$ For any $a,b$ we have $|f(1+a+bi)|<|f(1+a)|$.
Some questions :
$1)$ If $f(z)$ has meromorphic continuation to $Re(z)>0$ and no poles at $f(c)$ for any $0<c<1$ , does that imply
$|f(a+bi)|<|f(a)|$ ?
$2)$ Let $c_n$ be a sequence of positive reals. Is every $f(z)$ of the form $c_1 + \sum_{n>1} \dfrac{c_n}{n^z}$ ?
If not, what are the counterexamples ?