The basic problem is too find an analytic function that approximates $ln(E(z))$ near the real line ( strip described in OP ) where $E(z)$ is a given entire function. From the Weierstrass Product Theorem we know that $E(z) = exp(F(z))(z-z_1)(z-z_2)...$
Hence if $E(z)$ has no zero's within the strip and $E(z)$ is of the form $E(z) = exp(F(z)) (z^4 + a_1)(z^4 + a_2)...$
Then we can write $ln(E(z))$ ~ $F(z) + log(z^4 + a_1) + log(z^4 + a_2) + ...$ And then we only need to find an entire function that approximates $ln(z)$ near real $x > 1$. Such an example is $-(ln(2 sinh(z)/z) - z)$. Hence this way we construct our solution if we know enough about the location of the zero's of $E(z)$.
For those not familiar with entire functions : Weierstrass Product Theorem. Notice that we can also use what we did here to give an entire function that approximates sqrt($A(z)$) for some entire $A(z)$. ( use sqrt($x$) = $exp(\frac{ln(x)}{2})$ )
It is possible to get the error term as small as we want by using better asymptotics for $ln(z)$. Im unaware of better conceptual solutions to this problem but they might exist. In case our $E(z)$ has a finite numbers of zero's then the problem is trivial ( we can compute $F(z)$ easily ). I have not seen $ln(2 sinh(z)/z) - z$ often in the literature but imho its an interesting function (or I did not read enough).