Srivastava, H. M.; Kasana, H. S. A note on the construction of the proximate type of an entire Dirichlet series with index-pair (p,q). (English) Zbl 0607.30025 Rend. Semin. Mat. Univ. Padova 78, 145-154 (1987). Considering a natural extension of the notion of Lindelöf’s proximate order of an entire function f(s) \((s=\sigma +it)\), A. G. Azpeitia [Trans. Am. Math. Soc. 104, 495-501 (1962; Zbl 0114.038)] indicated how one can establish his propositions of existence of linear proximate orders R(\(\sigma)\) and linear lower proximate orders L(\(\sigma)\) for f(s) defined by a Dirichlet series. Motivated by Azpeitia’s work, R. S. L. Srivastava and P. Singh [J. Math. (Jabalpur) 2, 3-10 (1966)] proved the corresponding existence theorem for the proximate type T(\(\sigma)\) of an entire Dirichlet series representing f(s). For an interesting generalization of this theorem to hold true for an entire Dirichlet series with index-pair (p,q), which is due essentially to H. S. Kasana [J. Math. Anal. Appl. 105, 445-451 (1985; Zbl 0563.30020)], a remarkably simple (and markedly different) construction of the proximate type T(\(\sigma)\) is presented here. The main theorem established here applies to a much larger class of entire Dirichlet series with index-pair (p,q) than that considered earlier. Cited in 1 Document MSC: 30D10 Representations of entire functions of one complex variable by series and integrals 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 Keywords:Lindelöf’s proximate order; Dirichlet series Citations:Zbl 0114.038; Zbl 0563.30020 PDFBibTeX XMLCite \textit{H. M. Srivastava} and \textit{H. S. Kasana}, Rend. Semin. Mat. Univ. Padova 78, 145--154 (1987; Zbl 0607.30025) Full Text: Numdam EuDML References: [1] A.G. Azpeitia , On the proximate linear orders of entire Dirichlet series , Trans. Amer. Math. Soc. , 104 ( 1962 ), pp. 495 - 501 . MR 139725 | Zbl 0114.03802 · Zbl 0114.03802 [2] O.P. Juneja - K. Nandan - G.P. Kapoor , On the (p, q)-order and lower (p, q)-order of an entire Dirichlet series , Tamkang J. Math. , 9 ( 1978 ), pp. 47 - 63 . MR 539808 | Zbl 0415.30017 · Zbl 0415.30017 [3] O.P. Juneja - K. Nandan - G.P. Kapoor , On the (p, q)-type and lower (p, q)-type of entire Dirichlet series , Tamkang J. Math. , 11 ( 1980 ), pp. 67 - 76 . MR 622500 | Zbl 0459.30002 · Zbl 0459.30002 [4] H.S. Kasana , The proximate type and \lambda -proximate type of an entire Dirichlet series with index-pair (p, q), J. Math . Anal. Appl. , 105 ( 1985 ), pp. 445 - 451 . Zbl 0563.30020 · Zbl 0563.30020 [5] R.S.L. Srivastava - P. Singh , On the proximate type and \lambda -proximate type of an entire function represented by Dirichlet series , J. Math. ( Jabalpur ), 2 ( 1966 ), pp. 3 - 10 . [6] G. Valiron , Lectures on the General Theory of Integral Functions (Translated from the French by E. F. Collingwood), Chelsea , New York , 1949 . 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.