Linear Differential Equations with Analytic Coefficients Having the Same Order Near a Singular Point
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Linear Differential Equations with Analytic Coefficients Having the Same Order Near a Singular Point Samir Cherief1 · Saada Hamouda1 Received: 4 April 2020 / Revised: 1 August 2020 / Accepted: 22 September 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we investigate the growth of solutions of the differential equation: f (k) + Ak−1 (z) exp
ak−1 (z 0 − z)n
f (k−1) + · · · + A0 (z) exp
a0 (z 0 − z)n
f = 0,
where A j (z) are analytic functions in the closed complex plane except at z 0 and a j ( j = 0, . . . , k − 1) are distinct complex numbers. Other linked cases have been studied. Keywords Linear differential equations · Growth of solutions · Finite singular point · Nevanlinna theory Mathematics Subject Classification 34M10 · 30D35
1 Introduction and Statement of Results Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic function on the complex plane C and in the unit disc D = {z ∈ C : |z| < 1} (see [11,17,23]). The importance of this theory has inspired many authors to find modifications and generalizations to different domains. Extensions of Nevanlinna Theory to annuli have been made by [2,13–15,18]. Recently, in [7,10], Fettouch and Hamouda
Communicated by Majid Gazor.
B
Saada Hamouda [email protected] Samir Cherief [email protected]
1
Laboratory of pure and applied mathematics, University of Mostaganem (UMAB), Mostaganem, Algeria
123
Bulletin of the Iranian Mathematical Society
investigated the growth of solutions of certain linear differential equations near a finite singular point. In this paper, we continue this investigation near a finite singular point to study other types of linear differential equations. First, we recall the appropriate definitions. Set C = C ∪ {∞}, and suppose that f (z) is meromorphic in C\ {z 0 }, where z 0 ∈ C. Define the counting function near z 0 by: r n (t, f ) − n (∞, f ) dt − n (∞, f ) log r , Nz 0 (r , f ) = − t ∞
where n (t, f ) counts the number of poles of f (z) in the region {z ∈ C : t ≤ |z − z 0 |}∪ {∞} each pole according to its multiplicity; and the proximity function by: 1 m z 0 (r , f ) = 2π
2π
ln+ f z 0 − r eiϕ dϕ.
0
The characteristic function of f is defined in the usual manner by: Tz 0 (r , f ) = m z 0 (r , f ) + Nz 0 (r , f ) . In addition, the order of meromorphic function f (z) near z 0 is defined by σT ( f , z 0 ) = lim sup r →0
log+ Tz 0 (r , f ) . − log r
For an analytic function f (z) in C\ {z 0 }, we have also the definition: σ M ( f , z 0 ) = lim sup r →0
log+ log+ Mz 0 (r , f ) , − log r
where Mz 0 (r , f ) = max {| f (z)| : |z − z 0 | = r } . When the order is infinite, we introduce the notion of hyper-order near z 0 that is defined as follows:
σ2,T ( f , z 0 ) = lim sup
log+ log+ Tz 0 (r , f ) , − log r
σ2,M ( f , z 0 ) = lim sup
log+ log+ log+ Mz 0 (r , f ) . − log r
r →0
r →0
Remark 1.1 It is shown in [7] that if f is a nonconstant
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