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The proximate type and \(\lambda\)-proximate type of an entire Dirichlet series with index-pair \((p,q)\). (English) Zbl 0563.30020

Let \(f(s)=\sum^{\infty}_{n=1}a_ n\exp (s\lambda_ n)\) where \(s=\sigma +it\), \(0\leq \lambda_ n<\lambda_{n+1}\to \infty\) as \(n\to \infty\), be an entire Dirichlet series. For such functions, the concepts of (p,q)-order, (p,q)-type etc. were introduced by the reviewer, K. Nandan and G. P. Kapoor [Tamkang J. Math. 9, 47-63 (1978; Zbl 0415.30017); ibid. 11, 67-76 (1980; Zbl 0459.30002)]. In the present paper, the author introduces the concept of (p,q)-proximate type. Thus, a real-valued, continuous, piecewise differentiable function \(\tau(\sigma)\) is said to be a (p,q)-proximate type for an entire function f(s) of (p,q)-order \(\rho\) and (p,q) type \(\tau\) if \(\tau(\sigma)\to \tau\) as \(\sigma\to \infty\), \(\tau'(\sigma)\prod^{q-1}_{i=0} \log^{[i)}\sigma \to 0\) as \(\sigma\to \infty\) and for a given a \((0<a<\infty)\) \[ \limsup_{\sigma \to \infty}\log^{[p- 2]}M(\sigma)/\exp \{(\log^{[q-1]}\sigma)^{\rho}\tau (\sigma)\}=a \] where \[ M(\sigma)=\sup_{-\infty <t<\infty}| f(\sigma +it)| \quad and\quad \log^{[0]}x=x,\quad \log^{[m]}x=\log (\log^{[m- 1]}x). \] The author proves the existence of a (p,q)-proximate type for every entire Dirichlet series of (p,q)-order \(\rho\) and (p,q)-type \(\tau\) and constructs (p,q)-proximate type for a class of entire Dirichlet series.
Reviewer: O.P.Juneja

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30B50 Dirichlet series, exponential series and other series in one complex variable
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[1] Juneja, O. P.; Nandan, K.; Kapoor, G. P., On the (p, q)-order and lower (p, q)-order of an entire Dirichlet series, Tamkang J. Math., 9, 47-63 (1978) · Zbl 0415.30017
[2] Juneja, O. P.; Nandan, K.; Kapoor, G. P., On the (p, q)-type and lower (p, q)-type of an entire Dirichlet series, Tamkang J. Math., 11, 67-76 (1980) · Zbl 0459.30002
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