Kasana, H. S. The proximate type and \(\lambda\)-proximate type of an entire Dirichlet series with index-pair \((p,q)\). (English) Zbl 0563.30020 J. Math. Anal. Appl. 105, 445-451 (1985). 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 Cited in 1 ReviewCited in 1 Document 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 Keywords:entire Dirichlet series; (p,q)-order; (p,q)-type; (p,q)-proximate type Citations:Zbl 0415.30017; Zbl 0459.30002 PDFBibTeX XMLCite \textit{H. S. Kasana}, J. Math. Anal. Appl. 105, 445--451 (1985; Zbl 0563.30020) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.