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Maximum term of an entire Dirichlet series. (English) Zbl 0569.30007

The authors consider the general entire Dirichlet series \[ f(s)=\sum^{\infty}_{1}a_ n\exp (\lambda_ ns),\quad (s=\sigma +it,\quad 0\leq \lambda_ 1<...<\lambda_ n<\lambda_{n+1}\to \infty \quad as\quad n\to \infty) \] with \(\lim_{n\to \infty}(\log n/\lambda_ n)=D<\infty\). They take the maximum term \(\mu\) (\(\sigma)\) of the sequence \(\{| a_ n| e^{\lambda_ n\sigma}\}\) and its nth derivative \(\mu^{(n)}(\sigma)\) and express the (p,q)-orders of an entire Dirichlet series [introduced by O. P. Juneja, K. Nandan and G. P. Kapoor, Tamkang J. Math. 9, 47-63 (1978; Zbl 0415.30017)] in terms of \(\mu^{(n)}(\sigma)\) and \(\mu\) (\(\sigma)\). They also establish conditions under which two entire Dirichlet series have the same growth constants.
Reviewer: R.Parvatham

MSC:

30B50 Dirichlet series, exponential series and other series in one complex variable
30D15 Special classes of entire functions of one complex variable and growth estimates
30D10 Representations of entire functions of one complex variable by series and integrals

Citations:

Zbl 0415.30017
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