A generalization of a theorem of \textit{R. P. Boas} [J. Indian Math. Soc., N. Ser. 16, 127--135 (1952; Zbl 0048.05501)] to functions \(f(z)\) analytic and of exponential type in a half-plane. If \(\varphi(x)\), \(\varphi(-\infty) = 0\), is increasing and convex, \(-\infty 0\) then the convergence of \(\int_0^\infty \varphi(\log \vert f(x)\vert)\,dx\) implies the convergence of \(\sum \varphi(\log\vert f(\lambda_n)\vert - \varepsilon\)), \(\varepsilon > 0\).