Research Article Some Properties of Solutions of Second-Order Linear Differential Equations

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Volume 2013, Article ID 253168, 5 pages http://dx.doi.org/10.1155/2013/253168

Research Article

Some Properties of Solutions of Second-Order

Linear Diﬀerential Equations

Zinelaâbidine Latreuch and Benharrat Belaïdi

Laboratory of Pure and Applied Mathematics, Department of Mathematics, University of Mostaganem (UMAB), BP 227, 27000 Mostaganem, Algeria

Correspondence should be addressed to Benharrat Belaïdi; [email protected] Received 13 August 2012; Accepted 25 September 2012

Academic Editor: Rabha W. Ibrahim

Copyright © 2013 Z. Latreuch and B. Belaïdi. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the growth and oscillation of 𝑔𝑔𝑓𝑓 = 𝑑𝑑1𝑓𝑓1+ 𝑑𝑑2𝑓𝑓2, where 𝑑𝑑1and 𝑑𝑑2are entire functions of �nite order not all vanishing

identically and 𝑓𝑓1and 𝑓𝑓2are two linearly independent solutions of the linear diﬀerential equation 𝑓𝑓′′+ 𝐴𝐴𝐴𝐴𝐴𝐴𝑓𝑓 = 𝐴.

1. Introduction and Main Results

roughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory (see [1–4]). In addition, we will use 𝜆𝜆𝐴𝑓𝑓𝐴 and 𝜆𝜆𝐴𝑓𝑓𝐴 to denote, respectively, the exponents of convergence of the zero sequence and distinct zeros of a meromorphic function 𝑓𝑓, 𝜌𝜌𝐴𝑓𝑓𝐴 to denote the order of growth of 𝑓𝑓.

De�nition � (see [4, 5]). Let 𝑓𝑓 be a meromorphic function.

en the hyperorder of 𝑓𝑓𝐴𝐴𝐴𝐴 is de�ned by 𝜌𝜌₂󶀡󶀡𝑓𝑓󶀱󶀱 = lim sup

𝑟𝑟 𝑟 +𝑟

log log 𝑇𝑇 󶀡󶀡𝑟𝑟𝑟 𝑓𝑓󶀱󶀱

log 𝑟𝑟 . (1)

De�nition 2 (see [4, 5]). Let 𝑓𝑓 be a meromorphic function.

en the hyper-exponent of convergence of zeros sequence of 𝑓𝑓𝐴𝐴𝐴𝐴 is de�ned by

𝜆𝜆₂󶀡󶀡𝑓𝑓󶀱󶀱 = lim sup

𝑟𝑟 𝑟 +𝑟

log log 𝑁𝑁 󶀡󶀡𝑟𝑟𝑟 1𝑟𝑓𝑓󶀱󶀱

log 𝑟𝑟 𝑟 (2)

where 𝑁𝑁𝐴𝑟𝑟𝑟 1𝑟𝑓𝑓𝐴 is the counting function of zeros of 𝑓𝑓𝐴𝐴𝐴𝐴 in {𝐴𝐴 𝑧 𝑧𝐴𝐴𝑧 𝑧 𝑟𝑟𝑧. Similarly, the hyperexponent of convergence of the sequence of distinct zeros of 𝑓𝑓𝐴𝐴𝐴𝐴 is de�ned by

𝜆𝜆₂󶀡󶀡𝑓𝑓󶀱󶀱 = lim sup

𝑟𝑟 𝑟 +𝑟

log log 𝑁𝑁 󶀡󶀡𝑟𝑟𝑟 1𝑟𝑓𝑓󶀱󶀱

log 𝑟𝑟 𝑟 (3)

where 𝑁𝑁𝐴𝑟𝑟𝑟 1𝑟𝑓𝑓𝐴 is the counting function of distinct zeros of 𝑓𝑓𝐴𝐴𝐴𝐴 in {𝐴𝐴 𝑧 𝑧𝐴𝐴𝑧 𝑧 𝑟𝑟𝑧.

Suppose that 𝑓𝑓₁ and 𝑓𝑓₂ are two linearly independent solutions of the complex linear diﬀerential equation

𝑓𝑓′′+ 𝐴𝐴 𝐴𝐴𝐴𝐴 𝑓𝑓 = 𝐴𝑟 (4) and the polynomial of solutions

𝑔𝑔_𝑓𝑓= 𝑑𝑑1𝑓𝑓1+ 𝑑𝑑2𝑓𝑓2𝑟 (5)

where 𝐴𝐴 and 𝑑𝑑_𝑗𝑗𝐴𝑗𝑗 = 1𝑟 2𝐴 are entire functions of �nite order in the complex plane. It is clear that if 𝑑𝑑𝑗𝑗𝐴𝑗𝑗 = 1𝑟 2𝐴 are

complex numbers or 𝑑𝑑1 = 𝑐𝑐𝑑𝑑2where 𝑐𝑐 is a complex number,

then 𝑔𝑔_𝑓𝑓is a solution of (4) or has the same properties of the solutions.

It is natural to ask what can be said about the properties of 𝑔𝑔_𝑓𝑓in the case when 𝑑𝑑1≠ 𝑐𝑐𝑑𝑑2where 𝑐𝑐 is a complex number

and under what conditions 𝑔𝑔_𝑓𝑓keeps the same properties of the solutions of (4).

In [6], Chen studied the �xed points and hyper-order of solutions of second-order linear diﬀerential equations with entire coeﬃcients and obtained the following results. eorem A (see [6]). For all nontrivial solutions 𝑓𝑓 of (4) the

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(i) If 𝐴𝐴 is a polynomial with deg 𝐴𝐴 𝐴 𝐴𝐴 𝐴 𝐴, then one has 𝜆𝜆 󶀡󶀡𝑓𝑓 𝑓 𝑓𝑓󶀱󶀱 𝐴 𝜌𝜌 󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝐴𝐴 𝑛 𝑛

𝑛 . (6)

(ii) If 𝐴𝐴 is transcendental and 𝜌𝜌𝜌𝐴𝐴𝜌 𝜌 𝜌, then one has 𝜆𝜆 󶀡󶀡𝑓𝑓 𝑓 𝑓𝑓󶀱󶀱 𝐴 𝜌𝜌 󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜌,

𝜆𝜆𝑛󶀡󶀡𝑓𝑓 𝑓 𝑓𝑓󶀱󶀱 𝐴 𝜌𝜌𝑛󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜌𝜌 𝜌𝐴𝐴𝜌 .

(7) �efore we state our results we de�ne ℎ and 𝜓𝜓 by

ℎ 𝐴 󶙀󶙀 󶙘󶙘 𝑑𝑑𝐴 0 𝑑𝑑𝑛 0 𝑑𝑑′ 𝐴 𝑑𝑑𝐴 𝑑𝑑′𝑛 𝑑𝑑𝑛 𝑑𝑑′′ 𝐴𝑓 𝑑𝑑𝐴𝐴𝐴 𝑛𝑑𝑑′𝐴 𝑑𝑑′′𝑛𝑓 𝑑𝑑𝑛𝐴𝐴 𝑛𝑑𝑑′𝑛 𝑑𝑑′′′ 𝐴 𝑓 3𝑑𝑑′𝐴𝐴𝐴 𝑓 𝑑𝑑𝐴𝐴𝐴′ 𝑑𝑑′′𝐴𝑓 𝑑𝑑𝐴𝐴𝐴 𝑛 𝑛𝑑𝑑′′𝐴 𝑑𝑑′′′𝑛 𝑓 3𝑑𝑑′𝑛𝐴𝐴 𝑓 𝑑𝑑𝑛𝐴𝐴′ 𝑑𝑑′′𝑛𝑓 𝑑𝑑𝑛𝐴𝐴 𝑛 𝑛𝑑𝑑′′𝑛 󶙀󶙀 󶙘󶙘 , 𝜓𝜓 𝜌𝑓𝑓𝜌 𝐴 𝑛 󶀣󶀣𝑑𝑑𝐴𝑑𝑑𝑛𝑑𝑑 ′ 𝑛𝑓 𝑑𝑑𝑛𝑛𝑑𝑑′𝐴󶀳󶀳 ℎ 𝜑𝜑𝜌3𝜌𝑛 𝜙𝜙𝑛𝜑𝜑′′𝑛 𝜙𝜙𝐴𝜑𝜑′𝑛 𝜙𝜙0𝜑𝜑, (8) where 𝜑𝜑 ̸≡ 0 is entire function of �nite order and

𝜙𝜙_𝑛𝐴 3𝑑𝑑𝑛𝑛𝑑𝑑′′𝐴 𝑓 3𝑑𝑑𝐴𝑑𝑑𝑛𝑑𝑑′′𝑛 ℎ , 𝜙𝜙_𝐴𝐴 𝑛𝑑𝑑𝐴𝑑𝑑𝑛𝑑𝑑′𝑛𝐴𝐴 𝑛 𝐴𝑑𝑑𝑛𝑑𝑑′𝐴𝑑𝑑𝑛′′𝑓 𝐴𝑑𝑑𝑛𝑑𝑑′𝑛𝑑𝑑′′𝐴𝑓 𝑛𝑑𝑑𝑛𝑛𝑑𝑑′𝐴𝐴𝐴 ℎ , 𝜙𝜙₀𝐴 𝑛𝑑𝑑𝑛𝑑𝑑′𝐴𝑑𝑑′′′𝑛 𝑓 𝑛𝑑𝑑𝐴𝑑𝑑′𝑛𝑑𝑑𝑛′′′𝑓 3𝑑𝑑_ℎ 𝐴𝑑𝑑𝑛𝑑𝑑′′𝑛𝐴𝐴 𝑓 3𝑑𝑑𝑛𝑑𝑑′′𝐴𝑑𝑑′′𝑛 𝑛𝑛𝑑𝑑𝐴𝑑𝑑𝑛𝑑𝑑 ′ 𝑛𝐴𝐴′𝑓 4𝑑𝑑𝑛𝑑𝑑′𝐴𝑑𝑑′𝑛𝐴𝐴 𝑓 𝐴𝑑𝑑𝐴′𝑑𝑑′𝑛𝑑𝑑′′𝑛 𝑛 3𝑑𝑑𝐴󶀣󶀣𝑑𝑑′′𝑛󶀳󶀳 𝑛 ℎ 𝑛4𝑑𝑑𝐴󶀣󶀣𝑑𝑑 ′ 𝑛󶀳󶀳 𝑛 𝐴𝐴 𝑛 3𝑑𝑑𝑛𝑛𝑑𝑑′′𝐴𝐴𝐴 𝑛 𝐴󶀣󶀣𝑑𝑑′𝑛󶀳󶀳 𝑛 𝑑𝑑′′𝐴𝑓 𝑛𝑑𝑑𝑛𝑛𝑑𝑑′𝐴𝐴𝐴′ ℎ . (9) e subject of this paper is to study the controllability of solutions of the diﬀerential equation (4). In fact, we study the growth and oscillation of 𝑔𝑔_𝑓𝑓𝐴 𝑑𝑑_𝐴𝑓𝑓_𝐴𝑛 𝑑𝑑_𝑛𝑓𝑓_𝑛where 𝑓𝑓_𝐴 and 𝑓𝑓_𝑛 are two linearly independent solutions of (4) and 𝑑𝑑𝐴 and 𝑑𝑑𝑛

are entire functions of �nite order not all vanishing identically and satisfying 𝑑𝑑𝐴≠ 𝑐𝑐𝑑𝑑𝑛where 𝑐𝑐 is a complex number, and we

obtain the following results.

eorem 3. Let 𝐴𝐴𝜌𝑓𝑓𝜌 be a transcendental entire function

of �nite order. Let 𝑑𝑑𝑗𝑗𝜌𝑓𝑓𝜌 𝜌𝑗𝑗 𝐴 𝐴, 𝑛𝜌 be �nite�order entire

functions that are not all vanishing identically such that

max{𝜌𝜌𝜌𝑑𝑑_𝐴𝜌, 𝜌𝜌𝜌𝑑𝑑_𝑛𝜌} 𝜌 𝜌𝜌𝜌𝐴𝐴𝜌. If 𝑓𝑓_𝐴 and 𝑓𝑓_𝑛 are two linearly independent solutions of (4), then the polynomial of solutions

(5) satis�es

𝜌𝜌 󶀢󶀢𝑔𝑔_𝑓𝑓󶀲󶀲 𝐴 𝜌𝜌 󶀢󶀢𝑓𝑓_𝑗𝑗󶀲󶀲 𝐴 𝜌 󶀡󶀡𝑗𝑗 𝐴 𝐴, 𝑛󶀱󶀱 ,

𝜌𝜌_𝑛󶀢󶀢𝑔𝑔_𝑓𝑓󶀲󶀲 𝐴 𝜌𝜌_𝑛󶀢󶀢𝑓𝑓_𝑗𝑗󶀲󶀲 𝐴 𝜌𝜌 𝜌𝐴𝐴𝜌 󶀡󶀡𝑗𝑗 𝐴 𝐴, 𝑛󶀱󶀱 . (10)

eorem 4. Under the hypotheses of eorem 3, let 𝜑𝜑𝜌𝑓𝑓𝜌 ̸≡ 0

be an entire function with �nite order such that 𝜓𝜓𝜌𝑓𝑓𝜌 ̸≡ 0. If 𝑓𝑓_𝐴

and 𝑓𝑓_𝑛are two linearly independent solutions of (4), then the

polynomial of solutions (5) satis�es

𝜆𝜆 󶀢󶀢𝑔𝑔_𝑓𝑓𝑓 𝜑𝜑󶀲󶀲 𝐴 𝜆𝜆 󶀢󶀢𝑔𝑔_𝑓𝑓𝑓 𝜑𝜑󶀲󶀲 𝐴 𝜌𝜌 󶀢󶀢𝑓𝑓_𝑗𝑗󶀲󶀲 𝐴 𝜌 󶀡󶀡𝑗𝑗 𝐴 𝐴, 𝑛󶀱󶀱 , 𝜆𝜆_𝑛󶀢󶀢𝑔𝑔_𝑓𝑓𝑓 𝜑𝜑󶀲󶀲 𝐴 𝜆𝜆_𝑛󶀢󶀢𝑔𝑔_𝑓𝑓𝑓 𝜑𝜑󶀲󶀲 𝐴 𝜌𝜌_𝑛󶀢󶀢𝑓𝑓_𝑗𝑗󶀲󶀲 𝐴 𝜌𝜌 𝜌𝐴𝐴𝜌 󶀡󶀡𝑗𝑗 𝐴 𝐴, 𝑛󶀱󶀱 .

(11) eorem 5. Let 𝐴𝐴𝜌𝑓𝑓𝜌 be a polynomial of deg 𝐴𝐴 𝐴 𝐴𝐴. Let 𝑑𝑑_𝑗𝑗𝜌𝑓𝑓𝜌 𝜌𝑗𝑗 𝐴 𝐴, 𝑛𝜌 be �nite�order entire functions that are not all

vanishing identically such that ℎ ̸≡ 0 and max{𝜌𝜌𝜌𝑑𝑑𝐴𝜌, 𝜌𝜌𝜌𝑑𝑑𝑛𝜌} 𝜌

𝜌𝐴𝐴𝑛𝑛𝜌𝑛𝑛. If 𝑓𝑓_𝐴𝑓𝑓_𝑛are two linearly independent solutions of (4), then the polynomial of solutions (5) satis�es

𝜌𝜌 󶀢󶀢𝑔𝑔_𝑓𝑓󶀲󶀲 𝐴 𝜌𝜌 󶀢󶀢𝑓𝑓_𝑗𝑗󶀲󶀲 𝐴 𝐴𝐴 𝑛 𝑛_𝑛 󶀡󶀡𝑗𝑗 𝐴 𝐴, 𝑛󶀱󶀱 . (12) eorem 6. Under the hypotheses of eorem 5, let 𝜑𝜑𝜌𝑓𝑓𝜌 ̸≡ 0

be an entire function with 𝜌𝜌𝜌𝜑𝜑𝜌 𝜌 𝜌𝐴𝐴 𝑛 𝑛𝜌𝑛𝑛 such that 𝜓𝜓𝜌𝑓𝑓𝜌 ̸≡ 0.

If 𝑓𝑓_𝐴and 𝑓𝑓_𝑛are two linearly independent solutions of (4), then

the polynomial of solutions (5) satis�es

𝜆𝜆 󶀢󶀢𝑔𝑔_𝑓𝑓𝑓 𝜑𝜑󶀲󶀲 𝐴 𝜆𝜆 󶀢󶀢𝑔𝑔_𝑓𝑓𝑓 𝜑𝜑󶀲󶀲 𝐴 𝐴𝐴 𝑛 𝑛_𝑛 . (13)

2. Auxiliary Lemmas

Lemma 7 (see [7, 8]). Let 𝐴𝐴₀, 𝐴𝐴_𝐴, … , 𝐴𝐴_{𝑘𝑘𝑓𝐴}, 𝐹𝐹 ̸≡ 0 be �nite�

order meromorphic functions. If 𝑓𝑓 is a meromorphic solution of the equation

𝑓𝑓𝜌𝑘𝑘𝜌𝑛 𝐴𝐴_{𝑘𝑘𝑓𝐴}𝑓𝑓𝜌𝑘𝑘𝑓𝐴𝜌𝑛 ⋯ 𝑛 𝐴𝐴_𝐴𝑓𝑓′𝑛 𝐴𝐴₀𝑓𝑓 𝐴 𝐹𝐹 (14)

with 𝜌𝜌𝜌𝑓𝑓𝜌 𝐴 𝑛𝜌 and 𝜌𝜌_𝑛𝜌𝑓𝑓𝜌 𝐴 𝜌𝜌, then 𝑓𝑓 satis�es

𝜆𝜆 󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜆𝜆 󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜌𝜌 󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝑛𝜌, 𝜆𝜆𝑛󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜆𝜆𝑛󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜌𝜌𝑛󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜌𝜌.

(15) Here, we give a special case of the result due to Cao et al. in [9].

Lemma 8. Let 𝐴𝐴0, 𝐴𝐴𝐴, … , 𝐴𝐴𝑘𝑘𝑓𝐴, 𝐹𝐹 ̸≡ 0 be �nite�order mero�

morphic functions. If 𝑓𝑓 is a meromorphic solution of (14) with

max 󶁂󶁂𝜌𝜌 󶀢󶀢𝐴𝐴_𝑗𝑗󶀲󶀲 󶀡󶀡𝑗𝑗 𝐴 0, 𝐴, … , 𝑘𝑘 𝑓 𝐴󶀱󶀱 , 𝜌𝜌 𝜌𝐹𝐹𝜌󶁒󶁒 𝜌 𝜌𝜌 󶀡󶀡𝑓𝑓󶀱󶀱 𝜌 𝑛𝜌, (16)

then

𝜆𝜆 󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜆𝜆 󶀡󶀡𝑓𝑓󶀱󶀱 𝐴 𝜌𝜌 󶀡󶀡𝑓𝑓󶀱󶀱 . (17)

3. Proofs of the Theorems

Proof of eorem 3. Suppose that 𝑓𝑓_𝐴and 𝑓𝑓_𝑛are two linearly

independent solutions of (4). en by eorem A, we have 𝜌𝜌 󶀡󶀡𝑓𝑓_𝐴󶀱󶀱 𝐴 𝜌𝜌 󶀡󶀡𝑓𝑓_𝑛󶀱󶀱 𝐴 𝜌,

(3)

Suppose that 𝑑𝑑1 = 𝑐𝑐𝑑𝑑2, where 𝑐𝑐 is a complex number. en,

by (5) we obtain

𝑔𝑔_𝑓𝑓= 𝑐𝑐𝑑𝑑₂𝑓𝑓₁+ 𝑑𝑑₂𝑓𝑓₂= 󶀡󶀡𝑐𝑐𝑓𝑓₁+ 𝑓𝑓₂󶀱󶀱 𝑑𝑑2. (19)

Since 𝑓𝑓 = 𝑐𝑐𝑓𝑓₁+𝑓𝑓₂is a solution of (4) and 𝜌𝜌𝜌𝑑𝑑2) < 𝜌𝜌𝜌𝜌𝜌), then

we have

𝜌𝜌 󶀢󶀢𝑔𝑔_𝑓𝑓󶀲󶀲 = 𝜌𝜌 󶀡󶀡𝑐𝑐𝑓𝑓₁+ 𝑓𝑓₂󶀱󶀱 = ∞,

𝜌𝜌₂󶀢󶀢𝑔𝑔_𝑓𝑓󶀲󶀲 = 𝜌𝜌₂󶀡󶀡c𝑓𝑓₁+ 𝑓𝑓₂󶀱󶀱 = 𝜌𝜌 𝜌𝜌𝜌) . (20) Suppose now that 𝑑𝑑1 ̸≡ 𝑐𝑐𝑑𝑑2 where 𝑐𝑐 is a complex number.

Diﬀerentiating both sides of (5), we obtain

𝑔𝑔′_𝑓𝑓= 𝑑𝑑′₁𝑓𝑓₁+ 𝑑𝑑₁𝑓𝑓₁′ + 𝑑𝑑′₂𝑓𝑓₂+ 𝑑𝑑₂𝑓𝑓′₂. (21) Diﬀerentiating both sides of (21), we obtain

𝑔𝑔′′_𝑓𝑓= 𝑑𝑑′′₁𝑓𝑓₁+ 2𝑑𝑑′₁𝑓𝑓′₁+ 𝑑𝑑₁𝑓𝑓₁′′+ 𝑑𝑑′′₂𝑓𝑓₂+ 2𝑑𝑑′₂𝑓𝑓′₂+ 𝑑𝑑₂𝑓𝑓′′₂. (22) Substituting 𝑓𝑓′′_𝑗𝑗 = −𝜌𝜌𝑓𝑓_𝑗𝑗 𝜌𝑗𝑗 = 1, 2) into (22), we obtain

𝑔𝑔′′_𝑓𝑓= 󶀣󶀣𝑑𝑑′′₁− 𝑑𝑑₁𝜌𝜌󶀳󶀳 𝑓𝑓₁+ 2𝑑𝑑′₁𝑓𝑓₁′ + 󶀣󶀣𝑑𝑑′′₂− 𝑑𝑑₂𝜌𝜌󶀳󶀳 𝑓𝑓₂+ 2𝑑𝑑′₂𝑓𝑓′₂. (23) Diﬀerentiating both sides of (23) and by substituting 𝑓𝑓′′_𝑗𝑗 = −𝜌𝜌𝑓𝑓_𝑗𝑗𝜌𝑗𝑗 = 1, 2), we obtain 𝑔𝑔′′′_𝑓𝑓 = 󶀣󶀣𝑑𝑑′′′1 − 3𝑑𝑑′1𝜌𝜌 − 𝑑𝑑1𝜌𝜌′󶀳󶀳 𝑓𝑓1+ 󶀣󶀣𝑑𝑑′′1 − 𝑑𝑑1𝜌𝜌 + 2𝑑𝑑′′1󶀳󶀳 𝑓𝑓′1 + 󶀣󶀣𝑑𝑑′′′2 − 3𝑑𝑑′2𝜌𝜌 − 𝑑𝑑2𝜌𝜌′󶀳󶀳 𝑓𝑓2+ 󶀣󶀣𝑑𝑑′′2 − 𝑑𝑑2𝜌𝜌 + 2𝑑𝑑′′2󶀳󶀳 𝑓𝑓′2. (24) By (5), (21), (23), and (24) we have 𝑔𝑔_𝑓𝑓= 𝑑𝑑1𝑓𝑓1+ 𝑑𝑑2𝑓𝑓2, 𝑔𝑔′_𝑓𝑓= 𝑑𝑑′₁𝑓𝑓₁+ 𝑑𝑑₁𝑓𝑓₁′ + 𝑑𝑑′₂𝑓𝑓₂+ 𝑑𝑑₂𝑓𝑓′₂, 𝑔𝑔′′_𝑓𝑓= 󶀣󶀣𝑑𝑑′′₁− 𝑑𝑑₁𝜌𝜌󶀳󶀳 𝑓𝑓₁+ 2𝑑𝑑′₁𝑓𝑓₁′ + 󶀣󶀣𝑑𝑑′′₂− 𝑑𝑑₂𝜌𝜌󶀳󶀳 𝑓𝑓₂+ 2𝑑𝑑′₂𝑓𝑓′₂, 𝑔𝑔′′′_𝑓𝑓 = 󶀣󶀣𝑑𝑑′′′₁ − 3𝑑𝑑′₁𝜌𝜌 − 𝑑𝑑₁𝜌𝜌′󶀳󶀳 𝑓𝑓₁+ 󶀣󶀣𝑑𝑑′′₁− 𝑑𝑑₁𝜌𝜌 + 2𝑑𝑑′′₁󶀳󶀳 𝑓𝑓′₁ + 󶀣󶀣𝑑𝑑′′′₂ − 3𝑑𝑑′₂𝜌𝜌 − 𝑑𝑑₂𝜌𝜌′󶀳󶀳 𝑓𝑓₂+ 󶀣󶀣𝑑𝑑′′₂− 𝑑𝑑₂𝜌𝜌 + 2𝑑𝑑′′₂󶀳󶀳 𝑓𝑓′₂. (25)

To solve this system of equations, we need �rst to prove that ℎ ̸≡ 0. By simple calculations we obtain

ℎ = 󶙀󶙀󶙐󶙐 󶙘󶙘 𝑑𝑑1 0 𝑑𝑑2 0 𝑑𝑑′ 1 𝑑𝑑1 𝑑𝑑′2 𝑑𝑑2 𝑑𝑑′′ 1− 𝑑𝑑1𝜌𝜌 2𝑑𝑑′1 𝑑𝑑′′2− 𝑑𝑑2𝜌𝜌 2𝑑𝑑′2 𝑑𝑑′′′ 1 − 3𝑑𝑑′1𝜌𝜌 − 𝑑𝑑1𝜌𝜌′ 𝑑𝑑′′1− 𝑑𝑑1𝜌𝜌 + 2𝑑𝑑′′1 𝑑𝑑′′′2 − 3𝑑𝑑′2𝜌𝜌 − 𝑑𝑑2𝜌𝜌′ 𝑑𝑑′′2− 𝑑𝑑2𝜌𝜌 + 2𝑑𝑑′′2 󶙀󶙀󶙐󶙐 󶙘󶙘 = 󶀤󶀤4𝑑𝑑2₁󶀣󶀣𝑑𝑑′₂󶀳󶀳2+ 4𝑑𝑑2₂󶀣󶀣𝑑𝑑′₁󶀳󶀳2− 8𝑑𝑑₁𝑑𝑑₂𝑑𝑑′₁𝑑𝑑′₂󶀴󶀴 𝜌𝜌 + 2𝑑𝑑₁𝑑𝑑₂𝑑𝑑′₁𝑑𝑑′′′₂ + 2𝑑𝑑₁𝑑𝑑₂𝑑𝑑′₂𝑑𝑑′′′₁ − 6𝑑𝑑₁𝑑𝑑₂𝑑𝑑′′₁𝑑𝑑′′₂ − 6𝑑𝑑₁𝑑𝑑′₁𝑑𝑑′₂𝑑𝑑′′₂− 6𝑑𝑑₂𝑑𝑑′₁𝑑𝑑′₂𝑑𝑑′′₁ + 6𝑑𝑑₁󶀣󶀣𝑑𝑑′₂󶀳󶀳2𝑑𝑑₁′′+ 6𝑑𝑑₂󶀣󶀣𝑑𝑑′₁󶀳󶀳2𝑑𝑑₂′′− 2𝑑𝑑2₂𝑑𝑑′₁𝑑𝑑₁′′′− 2𝑑𝑑2₁𝑑𝑑′₂𝑑𝑑′′′₂ + 3𝑑𝑑2₁󶀣󶀣𝑑𝑑′′₂󶀳󶀳2+ 3𝑑𝑑2₂󶀣󶀣𝑑𝑑′′₁󶀳󶀳2. (26) To show that 4𝑑𝑑21𝜌𝑑𝑑′2) 2 +4𝑑𝑑2₂𝜌𝑑𝑑′₁)2−8𝑑𝑑₁𝑑𝑑₂𝑑𝑑′₁𝑑𝑑′₂ ̸≡ 0, we suppose that 𝑑𝑑21󶀣󶀣𝑑𝑑′2󶀳󶀳 2 + 𝑑𝑑22󶀣󶀣𝑑𝑑′1󶀳󶀳 2 − 2𝑑𝑑1𝑑𝑑2𝑑𝑑′1𝑑𝑑′2= 0. (27)

Dividing both sides of (27) by 𝜌𝑑𝑑1𝑑𝑑2)2, we obtain

󶀧󶀧𝑑𝑑′2 𝑑𝑑2󶀷󶀷 2 + 󶀧󶀧𝑑𝑑′1 𝑑𝑑1󶀷󶀷 2 − 2𝑑𝑑′1 𝑑𝑑1 𝑑𝑑′₂ 𝑑𝑑2 = 0 (28) equivalent to 󶀧󶀧𝑑𝑑_𝑑𝑑′1 1 − 𝑑𝑑 ′ 2 𝑑𝑑₂󶀷󶀷 2 = 0, (29)

which implies that 𝑑𝑑1 = 𝑐𝑐𝑑𝑑2 where 𝑐𝑐 is a complex number

and this is a contradiction. Since max{𝜌𝜌𝜌𝑑𝑑1), 𝜌𝜌𝜌𝑑𝑑2)} < 𝜌𝜌𝜌𝜌𝜌)

and 4𝑑𝑑21𝜌𝑑𝑑′2)2+4𝑑𝑑22𝜌𝑑𝑑1′)2−8𝑑𝑑1𝑑𝑑2𝑑𝑑′1𝑑𝑑′2 ̸≡ 0, we can deduce from

(26) that

𝜌𝜌 𝜌ℎ) = 𝜌𝜌 𝜌𝜌𝜌) > 0. (30) Hence ℎ ̸≡ 0. By Cramer’s method we have

𝑓𝑓₁= 󶙀󶙀󶙐󶙐󶙐󶙐 󶙘󶙘 𝑔𝑔𝑓𝑓 0 𝑑𝑑2 0 𝑔𝑔′𝑓𝑓 𝑑𝑑1 𝑑𝑑′2 𝑑𝑑2 𝑔𝑔′′ 𝑓𝑓 2𝑑𝑑′1 𝑑𝑑′′2−𝑑𝑑2𝜌𝜌 2𝑑𝑑′2 𝑔𝑔′′′ 𝑓𝑓 𝑑𝑑′′1−𝑑𝑑1𝜌𝜌+2𝑑𝑑′′1𝑑𝑑′′′2−3𝑑𝑑′2𝜌𝜌−𝑑𝑑2𝜌𝜌′𝑑𝑑′′2−𝑑𝑑2𝜌𝜌+2𝑑𝑑′′2 󶙀󶙀󶙐󶙐󶙐󶙐 󶙘󶙘 ℎ = 2 󶀣󶀣𝑑𝑑1𝑑𝑑2𝑑𝑑 ′ 2− 𝑑𝑑22𝑑𝑑′1󶀳󶀳 ℎ 𝑔𝑔𝜌3)𝑓𝑓 + 𝜙𝜙2𝑔𝑔′′𝑓𝑓+ 𝜙𝜙1𝑔𝑔′𝑓𝑓+ 𝜙𝜙0𝑔𝑔𝑓𝑓, (31)

where 𝜙𝜙_𝑗𝑗𝜌𝑗𝑗 = 0, 1, 2) are meromorphic functions of �nite order de�ned in (9). Suppose now that 𝜌𝜌𝜌𝑔𝑔𝑓𝑓) < ∞, then by

(31) we obtain 𝜌𝜌𝜌𝑓𝑓₁) < ∞, which is a contradiction, hence 𝜌𝜌𝜌𝑔𝑔_𝑓𝑓) = ∞. By (5) we have 𝜌𝜌₂𝜌𝑔𝑔_𝑓𝑓) ≤ 𝜌𝜌𝜌𝜌𝜌). Suppose that 𝜌𝜌₂𝜌𝑔𝑔_𝑓𝑓) < 𝜌𝜌𝜌𝜌𝜌), then by (31) we obtain 𝜌𝜌₂𝜌𝑓𝑓₁) < 𝜌𝜌𝜌𝜌𝜌), which is a contradiction. Hence 𝜌𝜌₂𝜌𝑔𝑔_𝑓𝑓) = 𝜌𝜌𝜌𝜌𝜌).

(4)

Proof of eorem 4. By eorem 3 we have 𝜌𝜌𝜌𝜌𝜌_𝑓𝑓) = ∞ and 𝜌𝜌₂𝜌𝜌𝜌_𝑓𝑓) = 𝜌𝜌𝜌𝜌𝜌). Set 𝑤𝑤𝜌𝑤𝑤) = 𝑤𝑤1𝑓𝑓1+ 𝑤𝑤2𝑓𝑓2− 𝜑𝜑. Since 𝜌𝜌𝜌𝜑𝜑) 𝜌 ∞,

then we have 𝜌𝜌𝜌𝑤𝑤) = 𝜌𝜌𝜌𝜌𝜌_𝑓𝑓) = ∞ and 𝜌𝜌₂𝜌𝑤𝑤) = 𝜌𝜌₂𝜌𝜌𝜌_𝑓𝑓) = 𝜌𝜌𝜌𝜌𝜌). In order to prove that 𝜆𝜆𝜌𝜌𝜌_𝑓𝑓−𝜑𝜑) = 𝜆𝜆𝜌𝜌𝜌_𝑓𝑓−𝜑𝜑) = ∞, 𝜆𝜆₂𝜌𝜌𝜌_𝑓𝑓−𝜑𝜑) = 𝜆𝜆₂𝜌𝜌𝜌_𝑓𝑓− 𝜑𝜑) = 𝜌𝜌𝜌𝜌𝜌) we need to prove only that 𝜆𝜆𝜌𝑤𝑤) = 𝜆𝜆𝜌𝑤𝑤) = ∞ and 𝜆𝜆₂𝜌𝑤𝑤) = 𝜆𝜆₂𝜌𝑤𝑤) = 𝜌𝜌𝜌𝜌𝜌). By 𝜌𝜌_𝑓𝑓 = 𝑤𝑤 + 𝜑𝜑 we get from (31) 𝑓𝑓₁=2 󶀣󶀣𝑤𝑤1𝑤𝑤2𝑤𝑤 ′ 2− 𝑤𝑤22𝑤𝑤′1󶀳󶀳 ℎ 𝑤𝑤𝜌3)+ 𝜙𝜙2𝑤𝑤′′+ 𝜙𝜙1𝑤𝑤′+ 𝜙𝜙0𝑤𝑤 + 𝑤𝑤𝑤 (32) where 𝑤𝑤 = 2 󶀣󶀣𝑤𝑤1𝑤𝑤2𝑤𝑤 ′ 2− 𝑤𝑤22𝑤𝑤′1󶀳󶀳 ℎ 𝜑𝜑𝜌3)+ 𝜙𝜙2𝜑𝜑′′+ 𝜙𝜙1𝜑𝜑′+ 𝜙𝜙0𝜑𝜑𝜑 (33)

Substituting (32) into (4), we obtain 2 󶀣󶀣𝑤𝑤1𝑤𝑤2𝑤𝑤′2− 𝑤𝑤22𝑤𝑤′1󶀳󶀳 ℎ 𝑤𝑤𝜌5)+ 4 󵠈󵠈 𝑗𝑗=0𝛽𝛽𝑗𝑗𝑤𝑤 𝜌𝑗𝑗)_{= − 󶀣󶀣𝑤𝑤}′′_{+ 𝜌𝜌𝑤𝑤󶀳󶀳 = 𝐵𝐵𝑤} (34) where 𝛽𝛽_𝑗𝑗𝜌𝑗𝑗 = 0𝑤 𝑗 𝑤 4) are meromorphic functions of �nite order. Since 𝑤𝑤 ̸≡ 0 and 𝜌𝜌𝜌𝑤𝑤) 𝜌 ∞, it follows that 𝑤𝑤 is not a solution of (4), which implies that 𝐵𝐵 ̸≡ 0. en by applying Lemma 7 we obtain (11).

Proof of eorem 5. Suppose that 𝑓𝑓₁and 𝑓𝑓₂are two linearly

independent solutions of (4). en by eorem A

𝜌𝜌 󶀡󶀡𝑓𝑓₁󶀱󶀱 = 𝜌𝜌 󶀡󶀡𝑓𝑓₂󶀱󶀱 = 𝑛𝑛 + 2₂ 𝜑 (35) By the same reasoning as in eorem 3, we have

ℎ = 󶙀󶙀 󶙘󶙘 𝑤𝑤1 0 𝑤𝑤2 0 𝑤𝑤′ 1 𝑤𝑤1 𝑤𝑤′2 𝑤𝑤2 𝑤𝑤′′ 1− 𝑤𝑤1𝜌𝜌 2𝑤𝑤′1 𝑤𝑤′′2− 𝑤𝑤2𝜌𝜌 2𝑤𝑤′2 𝑤𝑤′′′ 1 − 3𝑤𝑤′1𝜌𝜌 − 𝑤𝑤1𝜌𝜌′ 𝑤𝑤′′1− 𝑤𝑤1𝜌𝜌 + 2𝑤𝑤′′1 𝑤𝑤′′′2 − 3𝑤𝑤′2𝜌𝜌 − 𝑤𝑤2A′ 𝑤𝑤′′2− 𝑤𝑤2𝜌𝜌 + 2𝑤𝑤′′2 󶙀󶙀 󶙘󶙘 𝜑 (36) Since ℎ ̸≡ 0 and by Cramer’s method we have

𝑓𝑓₁= 󶙀󶙀󶙐󶙐󶙐󶙐 󶙘󶙘 𝜌𝜌𝑓𝑓 0 𝑤𝑤2 0 𝜌𝜌′ 𝑓𝑓 𝑤𝑤1 𝑤𝑤′2 𝑤𝑤2 𝜌𝜌′′ 𝑓𝑓 2𝑤𝑤′1 𝑤𝑤′′2−𝑤𝑤2𝜌𝜌 2𝑤𝑤′2 𝜌𝜌′′′𝑓𝑓 𝑤𝑤′′1−𝑤𝑤1𝜌𝜌+2𝑤𝑤′′1𝑤𝑤′′′2−3𝑤𝑤′2𝜌𝜌−𝑤𝑤2𝜌𝜌′𝑤𝑤′′2−𝑤𝑤2𝜌𝜌+2𝑤𝑤′′2 󶙀󶙀󶙐󶙐󶙐󶙐 󶙘󶙘 ℎ = 2 󶀣󶀣𝑤𝑤1𝑤𝑤2𝑤𝑤 ′ 2− 𝑤𝑤22𝑤𝑤′1󶀳󶀳 ℎ 𝜌𝜌𝜌3)𝑓𝑓 + 𝜙𝜙2𝜌𝜌′′𝑓𝑓+ 𝜙𝜙1𝜌𝜌′𝑓𝑓+ 𝜙𝜙0𝜌𝜌𝑓𝑓𝑤 (37)

where 𝜙𝜙_𝑗𝑗𝜌𝑗𝑗 = 0𝑤 1𝑤 2) are meromorphic functions with 𝜌𝜌𝜌𝜙𝜙_𝑗𝑗) 𝜌 𝜌𝑛𝑛 + 2)𝑛2 𝜌𝑗𝑗 = 0𝑤 1𝑤 2) de�ned in (9). By (5) we have 𝜌𝜌𝜌𝜌𝜌_𝑓𝑓) ≤ 𝜌𝑛𝑛 + 2)𝑛2. Suppose that 𝜌𝜌𝜌𝜌𝜌_𝑓𝑓) 𝜌 𝜌𝑛𝑛 + 2)𝑛2, then by (37) we obtain 𝜌𝜌𝜌𝑓𝑓₁) 𝜌 𝜌𝑛𝑛 + 2)𝑛2, which is a contradiction. Hence, 𝜌𝜌𝜌𝜌𝜌_𝑓𝑓) = 𝜌𝑛𝑛 + 2)𝑛2.

Proof of eorem 6. By eorem 5 we have 𝜌𝜌𝜌𝜌𝜌_𝑓𝑓) = 𝜌𝑛𝑛+2)𝑛2.

Set 𝑤𝑤𝜌𝑤𝑤) = 𝑤𝑤1𝑓𝑓1+ 𝑤𝑤2𝑓𝑓2− 𝜑𝜑. Since 𝜌𝜌𝜌𝜑𝜑) 𝜌 𝜌𝑛𝑛 + 2)𝑛2, then

we have 𝜌𝜌𝜌𝑤𝑤) = 𝜌𝜌𝜌𝜌𝜌_𝑓𝑓) = 𝜌𝑛𝑛 + 2)𝑛2. In order to prove that 𝜆𝜆𝜌𝜌𝜌_𝑓𝑓− 𝜑𝜑) = 𝜌𝑛𝑛 + 2)𝑛2, we need to prove only that 𝜆𝜆𝜌𝑤𝑤) = 𝜌𝑛𝑛 + 2)𝑛2. By 𝜌𝜌_𝑓𝑓= 𝑤𝑤 + 𝜑𝜑 we get from (37) 𝑓𝑓₁= 2 󶀣󶀣𝑤𝑤1𝑤𝑤2𝑤𝑤 ′ 2− 𝑤𝑤22𝑤𝑤′1󶀳󶀳 ℎ 𝑤𝑤𝜌3) + 𝜙𝜙₂𝑤𝑤′′+ 𝜙𝜙₁𝑤𝑤′+ 𝜙𝜙₀𝑤𝑤 + 𝑤𝑤𝑤 (38) where 𝑤𝑤 = 2 󶀣󶀣𝑤𝑤1𝑤𝑤2𝑤𝑤 ′ 2− 𝑤𝑤22𝑤𝑤′1󶀳󶀳 ℎ 𝜑𝜑𝜌3) + 𝜙𝜙₂𝜑𝜑′′+ 𝜙𝜙₁𝜑𝜑′+ 𝜙𝜙₀𝜑𝜑𝜑 (39) Substituting (38) into (4), we obtain

2 󶀣󶀣𝑤𝑤₁𝑤𝑤₂𝑤𝑤′₂− 𝑤𝑤2₂𝑤𝑤′₁󶀳󶀳 ℎ 𝑤𝑤𝜌5)+ 4 󵠈󵠈 𝑗𝑗=0𝛽𝛽𝑗𝑗𝑤𝑤 𝜌𝑗𝑗)_{= − 󶀣󶀣𝑤𝑤}′′_{+ 𝜌𝜌𝑤𝑤󶀳󶀳 = 𝐵𝐵𝑤} (40) where 𝛽𝛽_𝑗𝑗 𝜌𝑗𝑗 = 0𝑤 𝑗 𝑤 4) are meromorphic functions with 𝜌𝜌𝜌𝛽𝛽_𝑗𝑗) 𝜌 𝜌𝑛𝑛 + 2)𝑛2. Since 𝑤𝑤 ̸≡ 0 and 𝜌𝜌𝜌𝑤𝑤) 𝜌 𝜌𝑛𝑛 + 2)𝑛2, it follows that 𝑤𝑤 is not a solution of (4), which implies that 𝐵𝐵 ̸≡ 0. en by applying Lemma 8 we obtain (13).

Acknowledgments

e authors would like to thank the referees for their helpful remarks and suggestions to improve the paper. is research is supported by Agence Nationale pour le Développement de la Recherche Universitaire (ANDRU) and University of Mostaganem (UMAB), (PNR Project Code 8/u27/3144).

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