Estimates of linear combinations of near-exponential functions with positive and negative exponents

Abstract

The functions

$$\begin{gathered} f_n (z) = e^{\lambda _n ^z } [1 + \alpha _n (z)], \hfill \\ \varphi _n (z) = e^{\mu _n ^z } [1 + \beta _n (z)](n = 1.2, ...), \hfill \\ \end{gathered} $$

are considered, where λ_n and Μ_n are, respectively, the positive and negative zeros of some entire function of special type, while the functions α_n(z) and Β_n(z) are small in some sense. Estimates of a linear combination P₁(z) of the functions f_n(z) in the left half-plane, and of a linear combination P₂(z) of functions ϕ_n(z) in the right half-plane, are obtained in terms of the maximum modulus of P₁(z)+ P₂(z) in a segment of the imaginary axis.