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  Volume 8, Issue 4, Article 107
 
Some Precise Estimates of the Hyper Order of Solutions of Some Complex Linear Differential Equations
    Authors: Benharrat Belaidi,  
    Keywords:  
    Date Received: 05/03/07  
    Date Accepted: 30/11/07  
    Subject Codes:

34M10, 30D35.

  
    Editors: Doru Stefanescu,  
 
    Abstract:

Let $ rho left( fright) $ and $ rho _{2}left( fright) $ denote respectively the order and the hyper order of an entire function$ f.$ In this paper, we obtain some precise estimates of the hyper order of solutions of the following higher order linear differential equations

$displaystyle f^{left( kright) }+sum_{j=0}^{k-1}A_{j}left( zright) e^{P_{j}left( zright) }f^{left( jright) }=0$

and
$displaystyle f^{left( kright) }+sum_{j=0}^{k-1}left( A_{j}left( zright) e^{P_{j}left( zright) }+B_{j}left( zright) right) f^{left( jright) }=0$

where $ kgeq 2,$ $ P_{j}left( zright) $ $ left( j=0,dots ,k-1right) $ are nonconstant polynomials such that $ deg P_{j}=n$ $ left( j=0,dots ,k-1right) $ and $ A_{j}left( zright) $ $ left( notequiv 0right) ,$ $ B_{j}left( zright) $ $ left( notequiv 0right) $ $ left( j=0,dots ,k-1right) $ are entire functions with $ left( j=0,dots ,k-1right) $. Under some conditions, we prove that every solution $ fleft( zright) notequiv 0$ of the above equations is of infinite order and $ rho _{2}left( fright) =n$.

         
       
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