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

S. F. Prokoptsev

Matematičeskie zametki, Tome 20 (1976) no. 2, pp. 227-240.

Résumé

The functions \begin{gather*} f_n(z)=e^{{\lambda_n}z}[1+\alpha_n(z)],\\ \varphi_n(z)=e^{{\mu_n}z}[1+\beta_n(z)]\qquad(n=1,2,\dots), \end{gather*} are considered, where $\lambda_n$ and $\mu_n$ are, respectively, the positive and negative zeros of some entire function of special type, while the functions $\alpha_n(z)$ and $\beta_n(z)$ are small in some sense. Estimates of a linear combination $P_1(z)$ of the functions $f_n(z)$ in the left half-plane, and of a linear combination $P_2(z)$ of functions $\varphi_n(z)$ in the right half-plane, are obtained in terms of the maximum modulus of $P_1(z)+P_2(z)$ in a segment of the imaginary axis.