Kasana, H. S. On means of entire functions with index-pair \((p,q)\). (English) Zbl 0591.30025 Commun. Fac. Sci. Univ. Ankara, Sér. A1, Math. Stat. 31, 135-143 (1982). For an entire Dirichlet series \(f(s)=\sum^{\infty}_{n=1}a_ n\exp (s\lambda_ n)\) of (p,q)-order \(\rho\) and lower (p,q)-order \(\lambda\) (vide O. P. Juneja, K. Nandan and G. P. Kapoor [Tamkang J. Math. 9, 47-63 (1978; Zbl 0415.30017)] for relevant definitions), the author obtains a few growth estimates for the mean values \[ I_{\delta}(\sigma)=(\lim_{T\to \infty}\frac{1}{2T}\int^{T}_{- T}| f(\sigma +it)| \quad^{\delta}dt)^{1/\delta} \] and \[ m_{\delta,k}(\sigma)=\exp^{[p-2]}((\log^{[q-1]}\sigma)^{- k}\times \]\[ \times \int^{\sigma}_{-\sigma_ 0}(\log^{[p- 2]}I_{\delta}(x)(\log^{[q-1]}x)^{k-\quad 1}(\Lambda_{[q- 2]}(x))^{-1}dx), \] where \(0<\delta\), \(k<\infty\), \(\log^{[p]}x\) denotes the pth iterate of log x, \(\Lambda_{[q]}(x)=\prod^{q}_{i=0}\log^{[i]}x\). A typical result is: \[ \limsup_{\sigma \to \infty}\log^{[p]}m_{\delta,k}(\sigma)/\log^{[q]}\sigma =\rho, \]\[ \lim \inf_{\sigma \to \infty}\log^{[p]}m_{\delta,k}(\sigma)/\log^{[q]}\sigma =\lambda. \] Reviewer: O.P.Juneja MSC: 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 30B50 Dirichlet series, exponential series and other series in one complex variable Keywords:Dirichlet series; mean values Citations:Zbl 0415.30017 PDFBibTeX XMLCite \textit{H. S. Kasana}, Commun. Fac. Sci. Univ. Ankara, Ser. A1, Math. Stat. 31, 135--143 (1982; Zbl 0591.30025)