Question about $\Sigma_{i=1}^n (a_i^z-a_i^{-z})$

Question

Let $z$ be a complex number and $n$ a positive integer. Let $a_n$ be a sequence of $n$ real numbers such that $a_n > 1$ for every $n$.

Define $f_n(z;a_1,a_2,...,a_n)=\Sigma_{i=1}^n (a_i^z-a_i^{-z})$

Does there exist $f_n(z;a_1,a_2,...,a_n)$ such that :

1) $a_1 > e^{1/e}$

2) $f_n(z;a_1,a_2,...,a_n)=z$ has only one solution in $z$ : $z=0$

3) $[\dfrac{d f_n(z;a_1,a_2,...,a_n) }{dz}]_{Im(z)=0}>1$

How to find such $f_n(z;a_1,a_2,...,a_n)$ ?

Answer 1

I think the conditions cannot be satisfied. That is because of the answer of another question of me here : About the zeros of $f_n(z)=\sum_{k=1}^n k^{-z}$.

Im going to accept this as an answer, however if it turns out to be wrong or someone else gives a better answer I will reconsider.