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A note on entire functions of irregular (p,q)-growth. (English) Zbl 0588.30031

For a given entire function f the lower (p,q)-order is defined as \[ \lambda =\lambda (p,q)=\lim \inf_{r\to \infty}\log^{(p)}M(r,f)/\log^{(q)}r, \] where \(M(r,f)=\max_{| z| =r}| f(z)|\) and \(\log^{(q)}r\) denotes the qth iterate of log r. If f has lower (p,q)-order \(\lambda\) (where \(\lambda >0\) if \(p>q\) and \(\lambda >1\) if \(p=q)\), a lower (p,q)-proximate order \(\lambda\) (r) is defined if \[ (i)\quad \lim_{r\to \infty}\lambda (r)=\lambda,\quad (ii)\quad \lim_{r\to \infty}\lambda (r)(\log^{(q)}r)(\log^{(q- 1)}r)...(\log r)r=0, \] and \[ (iii)\quad \lim \inf_{r\to \infty}(\log^{(p-1)}M(r,f))/(\log^{(q-1)}r)^{\lambda (r)}=t_{\lambda} \] where \(0<t_{\lambda}<\infty\). Then \(t_{\lambda}\) is defined as the generalized \(\lambda\) (p,q)-type of f with respect to \(\lambda\) (r). A necessary and sufficient condition for f to be of given generalized \(\lambda\) (p,q)-type is found in terms of the Taylor coefficients of f under certain restrictions.
Reviewer: C.N.Linden

MSC:

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