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On means of entire functions with index-pair \((p,q)\). (English) Zbl 0591.30025

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

Citations:

Zbl 0415.30017
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