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On the proximate type and \(\lambda\) -proximate type of entire functions with index-pair (p,q). (English) Zbl 0546.30021

For an entire function \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\) of (p,q)-order \(\rho\) [this concept was introduced by O. P. Juneja, G. P. Kapoor and S. K. Bajpai in J. Reine Angew. Math. 282, 53-67 (1976; Zbl 0321.30031)] and (p,q)-type T [see ibid. 290, 180-190 (1977; Zbl 0501.30021)], the authors introduce the concept of proximate type T(r) as a real-valued, continuous, piecewise differentiable function satisfying the conditions: \[ (i)\quad\lim_{r\to\infty }T(r)=T,\quad (ii)\quad\lim_{r\to\infty }T'(r)\prod^{q-1}_{k=0}\log^{[k]}r=0 \] where \(\log^{[k]}x=\log (\log^{[k-1]}x)\), \(\log^{[0]}x=x\) and (iii) if \(M(r)=\max_{| z| =r}| f(z)|\) then \(\lim \sup_{r\to\infty }\log^{[p-2]}M(r)\exp\{-(\log^{[q- 1]}r)^{\rho}T(r)\}=a\) for some a such that \(0<a<\infty.\) They prove the existence of such functions on the lines of S. M. Shah [Bull. Am. Math. Soc. 52, 326-328 (1946; Zbl 0061.151)]. The idea is further extended by the introduction of the concept of lower proximate type. The results obtained generalize the results of R. S. L. Srivastava and the reviewer [Compos. Math. 18, 7-12 (1967; Zbl 0181.353)].
Reviewer: O.P.Juneja

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30D20 Entire functions of one complex variable (general theory)
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