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A theorem on means of an entire Dirichlet series with index-pair (p,q). (English) Zbl 0569.30005

For an entire Dirichlet series \(f(s)=\sum^{\infty}_{n=1}a_ n \exp (s\lambda_ n)\) where \(s=\sigma +it\), \(0\leq \lambda_ 1<\lambda_ n<\lambda_{n+1}\to \infty\) as \(n\to \infty\), the concepts of index pair, (p,q)-order, (p,q)-type etc. were introduced by the reviewer, K. Nandan and G. P. Kapoor in Tamkang J. Math. 9, 47-63 (1978; Zbl 0415.30017), ibid. 11, 67-76 (1980; Zbl 0459.30002). Using these concepts and those of generalized (p,q)-type etc. introduced by K. Nandan, R. P. Doherey and R. S. L. Srivastava in Indian. J. Pure Appl. Math. 11, 1424-1433 (1980; Zbl 0473.30020), the author obtains relations that depict the relative growth of the means \(I_{\delta}(\sigma)\) and \(m_{\delta,k}(\sigma)\) of f(s) where \[ I_{\delta}(\sigma)=\{\lim_{T\to \infty}(1/2T)\int^{T}_{-T}| f(\sigma +it)|^{\delta}dt\} \] and \[ m_{\delta,k}(\sigma)=\exp^{[p-2]}\{(1/(\log^{[q-1]}\sigma)^ k) \]
\[ \times \int^{\sigma}_{\sigma_ 0}((\log^{[p- 2]}I_{\delta}(x)(\log^{[q-1]}x)^{k-1})dx/(\prod^{q- 2}_{i=0}\log^{[i]}x)\},\quad 0<\delta,\quad k<\infty. \] For the class of Dirichlet series for which \(\log^{[p-2]}I_{\delta}(\sigma)\) is an increasing convex function of \(\log^{[q]}\sigma\), the result obtained refines an earlier result of the author obtained in Commun. Fac. Sci. Univ. Ankara 31, 135-143 (1982).
Reviewer: O.P.Juneja

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
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