Abstract:
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.
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