On location in a half-plane of zeros of perturbed first order entire functions

Michael Gil’

We consider the entire functions $$h(z)=um_{k=0}^ıfty \frac{a_{k}z^{k}}{k!} \quad\mbox{and}\quad ilde h(z)=um_{k=0}^ıfty \frac{ilde a_kz^{k}}{k!}$$ $( a_0=\tilde a_0=1; z, a_k, \tilde a_k\in {\bf C}, k=1, 2, \dots )$, provided $$um_{k=0}^ıfty |a_{k}|^2<ıfty, um_{k=0}^ıfty |ilde a_{k}|^2<ıfty \] and all the zeros of $h(z)$ are in a half-plane. We investigate the following problem: how small should be the quantity $q:=(\sum_{k=1}^\infty |a_k-\tilde a_k|^2)^{1/2}$ in order to all the zeros of $\tilde h(z)$ lie in the same half-plane?