Properties of the Riemann zeta function

Question

I have begun studying the Riemann zeta function $\zeta(s)$, particularly Titchmarsh's book, and have come up with two questions that I have been unable to find an answer to. They are:

1. Let $c>0$ be a constant and $\sigma_1=\sigma_2>1/2$. Is the function $\dfrac{\zeta(\sigma_1+i(t+c))}{\zeta(\sigma_2+it)}$ analytic?

2. Again, let $\sigma_1=\sigma_2>1/2$. Is there a constant $A>0$ such that $\dfrac{\zeta(\sigma_1+i(t+c))}{\zeta(\sigma_2+it)}<A$ for all $t>t_0>0$?

I believe the answer to be yes to both. The first because $\zeta(s)$ is analytic and the second because by varying $t$ by a bounded amount, I think that the function should be bounded itself.