On the growth of an entire function of exponential type on a sequence of points

A. F. Leont'ev

Sbornik. Mathematics, Tome 25 (1975) no. 4, pp. 567-578

Résumé

We consider a function $F(\lambda)=\int_Ce^{\lambda t}d\sigma(t)$, where $C$ is an analytic arc whose tangent at each point makes an angle with the real axis of less than $\pi/4$ radians, and $\sigma(t)$ is a function of bounded variation on $C$ which is continuous from the left on $C$ and nonconstant in any neighborhood of the right end-point $b$. Suppose that $0<\lambda_k\uparrow\infty$, $\lambda_{k+1}-\lambda_k\geqslant h>0$ ($k\geqslant1$) and $\sum_1^\infty\lambda_k^{-1}=\infty$. We show that $$ \varlimsup_{k\to\infty}\frac{\ln|F(\lambda_k)|}{\lambda_k}=\operatorname{Re}b. $$ When $C$ is a segment of the real axis, this result is well known. Bibliography: 3 titles.

Bibliographie

[1] R. P. Boas, Entire Functions, Academic Press, New York, 1954 | MR | Zbl

[2] A. F. Leontev, “O polnote sistemy eksponent na krivoi”, Sib. matem. zh., XV:5 (1974), 1103–1114 | MR

[3] A. F. Leontev, “O polnote sistemy pokazatelnykh funktsii v krivolineinoi polose”, Matem. sb., 36(78) (1955), 555–568 | MR