PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 90(104) (2011), 125–133 DOI: 10.2298/PIM1104125B

COMPLEX OSCILLATION OF DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

Benharrat Belaïdi

Communicated by Miroljub Jevtić

Abstract. We investigate the complex oscillation of some differential polynomials generated by solutions of the differential equation

f′′+ A_{1}(z)f′+ A_{0}(z)f = 0, where A_{1}(z), A_{0}(z) are meromorphic functions having the same finite iterated p-order.

1. Introduction and statement of results

We assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions [7, 12]. The iterated order, an order notion for functions of fast growth, was defined by Schönhage [14] and Sato [13] (see also [6, 8, 9] for an extensive survey). For the definition of the iterated order of a meromorphic function, we use the same definition as in [8] (for iterated order of entire function see [2, p. 317], [9, p. 129]). For all r ∈ ℝ we define exp_{1} r := e^{r} and exp_{p+1} r := exp(exp_{p} r), p ∈ ℕ. We also define for all r sufficiently large log_{1} r := logr and log_{p+1} r := log(log_{p} r), p ∈ ℕ. Moreover, we denote by exp_{0} r := r, log_{0} r := r, log_{−1} r := exp_{1} r and exp_{−1} r := log_{1} r.

Definition 1.1. [8] Let f be a meromorphic function. Then the iterated p-order ρ_{p}(f) of f is defined by

ρ_{p}(f) = \overline{lim}_{r→+∞} \frac{log_{p} T (r, f)}{logr} (p ⩾ 1 is an integer),

2010 Mathematics Subject Classification: Primary 34M10; Secondary 30D35.

Key words and phrases: Linear differential equations, differential polynomials, meromorphic solutions, iterated order, iterated exponent of convergence of the sequence of distinct zeros.

Partially supported by ANDRU (Agence Nationale pour le Développement de la Recherche Universitaire) and University of Mostaganem (UMAB), (PNR Project Code 8/u27/3144).

where T (r, f) is the Nevanlinna characteristic function of f [7, 12]. For p = 1, this notation is called order and for p = 2 hyperorder.

Definition 1.2. [8] The finiteness degree of the order of a meromorphic function f is defined by

i(f) = \{ \begin{matrix}0, & if f is rational, \\ min\{j ∈ N : ρ_{j} (f) < ∞\}, & if f is transcendental with \\ & ρ_{j} (f) < ∞ for some j ∈ N, \\ ∞, & if ρ_{j} (f) = ∞ for all j ∈ N . \end{matrix}

Definition 1.3. [4] Let f be a meromorphic function. Then the iterated p-type of f, with iterated p-order 0 < ρ_{p}(f) < ∞ is defined by

σ_{p}(f) = \overline{lim}_{r→+∞} \frac{log_{p−1} T (r, f)}{r^{ρ_{p} (f)}} (p ⩾ 1 is an integer).

Definition 1.4. [3, 8] Let f be a meromorphic function. Then the iterated exponent of convergence of the sequence of distinct zeros of f(z) is defined by

λ̄_{p}(f) = \overline{lim}_{r→+∞} \frac{log_{p} N̄(r, 1/f)}{logr} (p ⩾ 1 is an integer),

where N̄(r, 1/f) is the counting function of distinct zeros of f(z) in {z : |z| < r}. For p = 1 this notation is called exponent of convergence of the sequence of distinct zeros and for p = 2 hyperexponent of convergence of the sequence of distinct zeros.

Definition 1.5. [10] Let f be a meromorphic function. Then the iterated exponent of convergence of the sequence of distinct fixed points of f(z) is defined by

τ̄_{p}(f) = λ̄_{p}(f − z) = \overline{lim}_{r→+∞} \frac{log_{p} N̄(r, 1/(F − z))}{logr} (p ⩾ 1 is an integer).

For p = 1, this notation is called exponent of convergence of the sequence of distinct fixed points and for p = 2 hyperexponent of convergence of the sequence of distinct fixed points [11]. Thus τ̄_{p}(f) = λ̄_{p}(f − z) is an indication of oscillation of distinct fixed points of f(z).

Definition 1.6. [4, 8] The finiteness degree of the iterated convergence exponent of the sequence of zeros of a meromorphic function f(z) is defined by

i_{λ}(f) = \{ \begin{matrix}0, & if n(r, 1/f) = O(logr), \\ min\{j ∈ ℕ : λ_{j} (f) < ∞\}, & if λ_{j} (f) < ∞ for some j ∈ ℕ, \\ ∞, & if λ_{j} (f) = ∞ for all j ∈ N. \end{matrix}

Remark 1.1. Similarly, we can define the finiteness degree iλ̄(f) of λ̄_{p}(f).

Consider the linear differential equation

(1.1)  f′′+ A_{1}(z)f′+ A_{0}(z)f = 0,

where A_{1}(z), A_{0}(z) are meromorphic of finite iterated p-order. For almost four decades, substantial results have been obtained on the fixed points of general transcendental meromorphic functions [16]. However, there are few studies on the fixed points of solutions of differential equations. In [5] Chen firstly pointed out the relation between the exponent of convergence of distinct fixed points and the rate of growth of solutions of second order linear differential equations with entire coefficients. In [15] Wang and Yi investigated fixed points and hyperorder of differential polynomials generated by solutions of second order linear differential equations with meromorphic coefficients. In [10] Laine and Rieppo gave an improvement of the results of [15] by considering fixed points and iterated order.

Recently, the author has studied the relation between solutions and their derivatives of the differential equation

(1.2)  f^{(k)} + A(z)f = 0,

where k ⩾ 2, A(z) is a transcendental meromorphic function of finite iterated order ρ_{p}(A) = ρ > 0 and have obtained the following result.

Theorem 1.1. [1] Let k ⩾ 2 and A(z) be a transcendental meromorphic function of finite iterated order ρ_{p}(A) = ρ > 0 such that δ(∞, A) = \underline{lim}_{r→+∞} \frac{m(r,A)}{T (r,A)} =δ >0 . Suppose, moreover, that either:

(i) all poles of f are of uniformly bounded multiplicity or (ii) δ(∞, f) > 0. If φ(z) ≢ 0 is a meromorphic function with finite iterated p-order ρ_{p}(φ) < +∞, then every meromorphic solution f(z) ≢ 0 of (1.2), satisfies

λ̄_{p}(f − φ) = λ̄_{p}(f′− φ) = ⋯= λ̄_{p}(f^{(k)} − φ) = ρ_{p}(f) = ∞,
λ̄_{p+1} (f − φ) = λ̄_{p+1} (f′− φ) = ⋯= λ̄_{p+1} (f^{(k)} − φ) = ρ_{p+1} (f) = ρ.

We know that a differential equation bears a relation to all derivatives of its solutions. Hence, linear differential polynomials generated by its solutions must have special nature because of the control of differential equations.

The main purpose of this paper is to study the growth and the oscillation of some differential polynomials generated by solutions of the second order linear differential equation (1.1). We obtain some estimates of their iterated order and fixed points.

Theorem 1.2. Let A_{1}(z), A_{0}(z) be meromorphic functions, and let i(A_{0}) = p (1 ⩽ p < +∞). Assume that either i_{λ}(1/A_{0}) < p or λ_{p}(1/A_{0}) < ρ_{p}(A_{0}) = ρ (0 < ρ < +∞) and that i(A_{1}) < p. Let d_{0}(z), d_{1}(z) be meromorphic functions such that at least one of d_{0}(z), d_{1}(z) does not vanish identically with max{ρ_{p}(d_{j}) : j = 0, 1} < ρ_{p}(A_{0}). Let φ(z) ≢ 0 be a meromorphic function with ρ_{p}(φ) < ρ_{p}(A_{0}) such that P = (d′_{1} + d_{0} − d_{1}A_{1})φ − d_{1}φ′ ≢ 0. If f ≢ 0 is a meromorphic solution of (1.1) whose poles are of uniformly bounded multiplicity, then the differential polynomial g_{f} = d_{1}f′+ d_{0}f satisfies

λ̄_{p}(g_{f} − φ) = ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞,
λ̄_{p+1} (g_{f} − φ) = ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ.

Applying Theorem 1.2 for φ(z) = z, we obtain the following result.

Corollary 1.1. Suppose that A_{1}(z), A_{0}(z), d_{0}(z), d_{1}(z) satisfy the additional hypotheses of Theorem 1.2 such that (d′_{1} + d_{0} − d_{1}A_{1})z − d_{1} ≢ 0. If f ≢ 0 is a meromorphic solution of (1.1) whose poles are of uniformly bounded multiplicity, then the differential polynomial g_{f} = d_{1}f′+d_{0}f satisfies τ̄_{p}(g_{f} ) = ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞ and τ̄_{p+1} (g_{f} ) = ρ_{p+1} (g_{f} ) = ρ_{p}(A_{0}) = ρ.

In what follows we obtain a result without the additional condition P ≢ 0.

Theorem 1.3. Let A_{1}(z), A_{0}(z) be meromorphic functions, and let i(A_{0}) = p (1 ⩽ p < +∞). Assume that either i_{λ}(1/A_{0}) < p or λ_{p}(1/A_{0}) < ρ_{p}(A_{0}) = ρ (0 < ρ < +∞) and that ρ_{p}(A_{0}) = ρ_{p}(A_{1}), σ_{p}(A_{1}) < σ_{p}(A_{0}) = σ (0 < σ < +∞). Let d_{0}(z), d_{1}(z) be meromorphic functions such that at least one of d_{0}(z), d_{1}(z) does not vanish identically with max{ρ_{p}(d_{j}) : j = 0, 1} < ρ_{p}(A_{0}). Let φ(z) ≢ 0 be a meromorphic function with ρ_{p}(φ) < ρ_{p}(A_{0}). If f ≢ 0 is a meromorphic solution of (1.1) whose poles are of uniformly bounded multiplicity, then the differential polynomial g_{f} = d_{1}f′+ d_{0}f satisfies

λ̄_{p}(g_{f} − φ) = ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞,
λ̄_{p+1} (g_{f} − φ) = ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ.

Applying Theorem 1.3 for φ(z) = z, we obtain the following result.

Corollary 1.2. Suppose that A_{1}(z), A_{0}(z), d_{0}(z), d_{1}(z) satisfy the additional hypotheses of Theorem 1.3. If f ≢ 0 is a meromorphic solution of (1.1) whose poles are of uniformly bounded multiplicity, then the differential polynomial g_{f} = d_{1}f′ + d_{0}f satisfies τ̄_{p}(g_{f} ) = ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞ and τ̄_{p+1} (g_{f} ) = ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ.

2. Auxiliary Lemmas

We need the following lemmas in the proofs of our theorems.

Lemma 2.1. [8, Remark 1.3] If f is a meromorphic function with i(f) = p ⩾ 1, then ρ_{p}(f) = ρ_{p}(f′).

Lemma 2.2. [10] If f is a meromorphic function with 0 < ρ_{p}(f) < ρ (p ⩾ 1), then ρ_{p+1} (f) = 0.

Lemma 2.3. [1] Let A_{0}, A_{1}, …, A_{k−1} , F ≢ 0 be finite iterated p-order meromorphic functions. If f is a meromorphic solution with ρ_{p}(f) = ∞ and ρ_{p+1} (f) = ρ < ∞ of the equation

f^{(k)} + A_{k−1} (z)f^{(k−1)} + ⋯+ A_{1}(z)f′+ A_{0}(z)f = F,

then λ̄_{p}(f) = λ_{p}(f) = ρ_{p}(f) = ∞ and λ̄_{p+1} (f) = λ_{p+1} (f) = ρ_{p+1} (f) = ρ.

Lemma 2.4. Let f, g be meromorphic functions with iterated p-orders 0 < ρ_{p}(f), ρ_{p}(g) < ∞ and iterated p-types 0 < σ_{p}(f), σ_{p}(g) < ∞ (1 ⩽ p < +∞). Then the following statements hold:

(i)If ρ_{p}(g) < ρ_{p}(f), then

(2.1)  σ_{p}(f + g) = σ_{p}(fg) = σ_{p}(f).

(ii) If ρ_{p}(f) = ρ_{p}(g) and σ_{p}(g) ≠ σ_{p}(f), then

ρ_{p}(f + g) = ρ_{p}(fg) = ρ_{p}(f).

Proof. By the definition of the iterated p-type, we have (2.2)

σ_{p}(f + g) = \overline{lim}_{r→+∞} \frac{log_{p−1} T (r, f + g)}{r^{ρ_{p} (f+g)}} ⩽ \overline{lim}_{r→+∞} \frac{log_{p−1} (T (r, f) + T (r, g) + O(1))}{r^{ρ_{p} (f+g)}} .

Since ρ_{p}(g) < ρ_{p}(f), then ρ_{p}(f + g) = ρ_{p}(f). Thus, from (2.2), we obtain

(2.3) σ_{p}(f + g) ⩽ \overline{lim}_{r→+∞} \frac{log_{p−1} T (r, f)}{r^{ρ_{p} (f)}} + \overline{lim}_{r→+∞} \frac{log_{p−1} T (r, g) + O(1)}{r^{ρ_{p} (f)}} = σ_{p}(f).

On the other hand since ρ_{p}(f + g) = ρ_{p}(f) > ρ_{p}(g), then by (2.3), we get

(2.4)  σ_{p}(f) = σ_{p}(f + g − g) ⩽ σ_{p}(f + g).

Hence by (2.3) and (2.4), we obtain σ_{p}(f + g) = σ_{p}(f). Now, we prove σ_{p}(fg) = σ_{p}(f). Since ρ_{p}(g) < ρ_{p}(f), then ρ_{p}(fg) = ρ_{p}(f). By the definition of the iterated p-type, we have

(2.5) σ_{p}(fg) = \overline{lim}_{r→+∞} \frac{log_{p−1} T (r, fg)}{r^{ρ_{p} (fg)}} = \overline{lim}_{r→+∞} \frac{log_{p−1} T (r, fg)}{r^{ρ_{p} (f)}}

⩽ \overline{lim}_{r→+∞} \frac{log_{p−1} (T (r, f) + T (r, g))}{r^{ρ_{p} (f)}}

⩽ \overline{lim}_{r→+∞} \frac{log_{p−1} T (r, f)}{r^{ρ_{p} (f)}} + \overline{lim}_{r→+∞} \frac{log_{p−1} T (r, g) + O(1)}{r^{ρ_{p} (f)}} = σ_{p}(f).

Since ρ_{p}(fg) = ρ_{p}(f) > ρ_{p}(g) = ρ_{p}(1/g), then by (2.5), we obtain

(2.6)  σ_{p}(f) = σ_{p}( fg \frac{1}{g}) ⩽ σ_{p}(fg).

Thus, by (2.5) and (2.6), we obtain σ_{p}(fg) = σ_{p}(f).

(ii) Without lost of generality, we suppose that ρ_{p}(f) = ρ_{p}(g) and σ_{p}(g) < σ_{p}(f). Then we have ρ_{p}(f + g) ⩽ max{ρ_{p}(f), ρ_{p}(g)} = ρ_{p}(f) = ρ_{p}(g). If we suppose that ρ_{p}(f + g) < ρ_{p}(f) = ρ_{p}(g), then by (2.1) we get

σ_{p}(g) = σ_{p}(f + g − f) = σ_{p}(f)

and this is a contradiction. Hence ρ_{p}(f + g) = ρ_{p}(f) = ρ_{p}(g).

Now, we prove that ρ_{p}(fg) = ρ_{p}(f) = ρ_{p}(g). Also we have

ρ_{p}(fg) ⩽ max{ρ_{p}(f), ρ_{p}(g)} = ρ_{p}(f) = ρ_{p}(g).

If we suppose that ρ_{p}(fg) < ρ_{p}(f) = ρ_{p}(g) = ρ_{p}(1/f), then by (2.1) we can write

σ_{p}(g) = σ_{p}( fg \frac{1}{f}) = σ_{p}( \frac{1}{f}) = σ_{p}(f)

and this is a contradiction. Hence ρ_{p}(fg) = ρ_{p}(f) = ρ_{p}(g). □

Lemma 2.5. Let A_{1}(z), A_{0}(z) be meromorphic functions, and let i(A_{0}) = p (1 ⩽ p <+∞) . Assume that either i(A_{1})<p or ρ_{p}(A_{1}) = ρ_{p}(A_{0}) = ρ (0 < ρ <+∞) and σ_{p}(A_{1}) < σ_{p}(A_{0}) = σ (0 < σ < +∞). Let d_{j} (j = 0, 1) be meromorphic functions that are not all vanish identically with max{ρ_{p}(d_{j}) : j = 0, 1} < ρ_{p}(A_{0}). Then

(2.7)  h = d_{1}(d′_{0} − d_{1}A_{0}) − d_{0}(d′_{1} + d_{0} − d_{1}A_{1}) ≢ 0.

Proof. First, we suppose that d_{1} ≢ 0. If i(A_{1}) < p, then ρ_{p}(A_{1}) = 0 and by (2.7), we obtain ρ_{p}(h) = ρ_{p}(A_{0}) = ρ > 0. If ρ_{p}(A_{1}) = ρ_{p}(A_{0}) = ρ (0 < ρ < +∞) and σ_{p}(A_{1}) < σ_{p}(A_{0}) = σ (0 < σ < +∞), then by (2.7) and Lemma 2.4 we have ρ_{p}(h) = ρ_{p}(A_{0}) = ρ > 0. Thus h ≢ 0.

Now, if d_{1} ≡ 0, d_{0} ≢ 0, then h = −d_{0}^{2} ≢ 0. □

Lemma 2.6. [4] Let A_{0}(z), …, A_{k−1} (z) be meromorphic functions, and let i(A_{0}) = p (1 ⩽ p < +∞). Assume that either i_{λ}(1/A_{0}) < p or λ_{p}(1/A_{0}) < ρ_{p}(A_{0}) and that either max{i(A_{j}) : j = 1, 2, …, k − 1} < p or

max{ρ_{p}(A_{j}) : j = 1, 2, …, k − 1} ⩽ ρ_{p}(A_{0}) = ρ (0 < ρ < +∞) and
max{σ_{p}(A_{j}) : ρ_{p}(A_{j}) = ρ_{p}(A_{0})} < σ_{p}(A_{0}) = σ (0 < σ < +∞).

Then every meromorphic solution f ≢ 0 of the equation

f^{(k)} + A_{k−1} (z)f^{(k−1)} + ⋯+ A_{1}(z)f′+ A_{0}(z)f = 0

whose poles are of uniformly bounded multiplicity satisfies i(f)=p+1 and ρ_{p}(f)=∞ , ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ.

Lemma 2.7. Let A_{1}(z), A_{0}(z) be meromorphic functions, and let i(A_{0}) = p (1 ⩽ p < +∞). Assume that either i_{λ}(1/A_{0}) < p or λ_{p}(1/A_{0}) < ρ_{p}(A_{0}) and that either i(A_{1}) < p or ρ_{p}(A_{1}) = ρ_{p}(A_{0}) = ρ (0 < ρ < +∞) and σ_{p}(A_{1}) < σ_{p}(A_{0}) = σ (0 < σ < +∞). Let d_{0}(z), d_{1}(z) be meromorphic functions such that at least one of d_{0}(z), d_{1}(z) does not vanish identically with max{ρ_{p}(d_{j}) : j = 0, 1} < ρ_{p}(A_{0}). If f ≢ 0 is a meromorphic solution of (1.1) whose poles are of uniformly bounded multiplicity, then the differential polynomial

(2.8)  g_{f} = d_{1}f′+ d_{0}f

satisfies

ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞, ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ.

Proof. Suppose that f ≢ 0 is a meromorphic solution of equation (1.1) whose poles are of uniformly bounded multiplicity. Then, by Lemma 2.6, we have ρ_{p}(f) = ∞ and ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ. Differentiating both sides of equation (2.8) and replacing f′′with f′′= −A_{1}f′− A_{0}f, we obtain

(2.9)  g′_{f} = (d′_{1} + d_{0} − d_{1}A_{1})f′+ (d′_{0} − d_{1}A_{0})f.

Set

(2.10)  α_{0} = d′_{0} − d_{1}A_{0}, α_{1} = d′_{1} + d_{0} − d_{1}A_{1}.

Then, by (2.8), (2.9) and (2.10), we have

(2.11)  d_{1}f′+ d_{0}f = g_{f} , α_{1}f′+ α_{0}f = g′_{f}.

Set

(2.12)  h = d_{1}α_{0} − d_{0}α_{1} = d_{1}(d′_{0} − d_{1}A_{0}) − d_{0}(d′_{1} + d_{0} − d_{1}A_{1}).

By Lemma 2.5 we have h ≢ 0. By h ≢ 0, (2.11) and (2.12), we obtain

(2.13)  f = \frac{d_{1}g′_{f} − α_{1}g_{f}}{h} .

If ρ_{p}(g_{f} ) < ∞, then by (2.13) and Lemma 2.1, we get ρ_{p}(f) < ∞ and this is a contradiction. Hence ρ_{p}(g_{f} ) = ∞.

Now, we prove that ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ. By (2.8), Lemma 2.1 and Lemma 2.2, we get ρ_{p+1} (g_{f} ) ⩽ ρ_{p+1} (f) and by (2.13) we have ρ_{p+1} (f) ⩽ ρ_{p+1} (g_{f} ). This yields ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ. □

3. Proof of Theorem 1.2

Suppose that f ≢ 0 is a meromorphic solution of equation (1.1) whose poles are of uniformly bounded multiplicity. Then, by Lemma 2.6, we have ρ_{p}(f) = ∞ and ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ. Set w(z) = d_{1}f′+ d_{0}f − φ. Since ρ_{p}(φ) < ρ_{p}(A_{0}), then by Lemma 2.7 we have ρ_{p}(w) = ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞ and ρ_{p+1} (w) = ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ. In order to prove λ̄_{p}(g_{f} − φ) = ∞ and λ̄_{p+1} (g_{f} − φ) = ρ, we need to prove only λ̄_{p}(w) = ∞ and λ̄_{p+1} (w) = ρ. By g_{f} = w + φ, we get from (2.13)

(3.1)  f = \frac{d_{1}w′− α_{1}w}{h} + ψ,

where

(3.2)  ψ = \frac{d_{1}φ′− α_{1}φ}{h}

and ρ_{p}(ψ) < ∞. Substituting (3.1) into equation (1.1), we obtain

\frac{d_{1}}{h}w′′′+ ϕ_{2}w′′+ ϕ_{1}w′+ ϕ_{0}w = −(ψ′′+ A_{1}(z)ψ′+ A_{0}(z)ψ) = A,

where ϕ_{j} (j = 0, 1, 2) are meromorphic functions with ρ_{p}(ϕ_{j}) ⩽ ρ_{p}(A_{0}) < ∞ (j = 0, 1, 2). Since (d′_{1} + d_{0} − d_{1}A_{1})φ − d_{1}φ′ ≢ 0, from (3.2) we have ψ(z) ≢ 0. Hence, by ψ(z) ≢ 0 and ρ_{p}(ψ) < ∞, it follows by Lemma 2.6 that A ≢ 0. Thus, by Lemma 2.3, we obtain λ̄_{p}(w) = ρ_{p}(w) = ∞, λ̄_{p+1} (w) = ρ_{p+1} (w) = ρ, i.e., λ̄_{p}(g_{f} − φ) = ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞ and λ̄_{p+1} (g_{f} − φ) = ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ.

Remark 3.1. From the proof of Theorem 1.2, we see that the condition (d′_{1} + d_{0} − d_{1}A_{1}) φ − d_{1}φ′ ≢ 0 is necessary because if (d′_{1} + d_{0} − d_{1}A_{1})φ − d_{1}φ′ ≡ 0, then ψ(z) ≡ 0 and A(z) ≡ 0.

4. Proof of Theorem 1.3

Suppose that f ≢ 0 is a meromorphic solution of equation (1.1) whose poles are of uniformly bounded multiplicity. Then, by Lemma 2.6, we have ρ_{p}(f) = ∞ and ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ. Set w(z) = d_{1}f′+ d_{0}f − φ. Since ρ_{p}(φ) < ρ_{p}(A_{0}), then by Lemma 2.7 we have ρ_{p}(w) = ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞ and ρ_{p+1} (w) = ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ. In order to prove λ̄_{p}(g_{f} − φ) = ∞ and λ̄_{p+1} (g_{f} − φ) = ρ, we need to prove only λ̄_{p}(w) = ∞ and λ̄_{p+1} (w) = ρ. Substituting g_{f} = w + φ into (2.13) and using a similar reasoning as in the proof of Theorem 1.2, we get that

\frac{d_{1}}{h}w′′′+ ϕ_{2}w′′+ ϕ_{1}w′+ ϕ_{0}w = −(ψ′+ A_{1}(z)ψ′+ A_{0}(z)ψ) = B,

where ϕ_{j} (j = 0, 1, 2) are meromorphic functions with ρ_{p}(ϕ_{j}) ⩽ ρ_{p}(A_{0}) < ∞ (j = 0, 1, 2) and

(4.1)  ψ = \frac{d_{1}φ′− α_{1}φ}{h} .

Now, we prove that ψ(z) ≢ 0. Assume that ψ(z) ≡ 0. Then from (4.1), we obtain that

(4.2)  (d′_{1} + d_{0} − d_{1}A_{1})φ = d_{1}φ′.

First, if d_{1} ≡ 0, then by (4.2), we get d_{0} ≡ 0 and this is a contradiction. Now if d_{1} ≢ 0, since ρ_{p}(φ) < ρ_{p}(A_{0}), then by (4.2), we get

ρ_{p}((d′_{1} + d_{0} − d_{1}A_{1})φ) = ρ_{p}(A_{1}) = ρ_{p}(d_{1}φ′) < ρ_{p}(A_{0}) = ρ_{p}(A_{1})

and this is a contradiction. Hence ψ(z) ≢ 0. Since ψ(z) ≢ 0 and ρ_{p}(ψ) < ∞, it follows by Lemma 2.6 that B ≢ 0. Thus, by Lemma 2.3, we obtain λ̄_{p}(w) = ρ_{p}(w) = ∞, λ̄_{p+1} (w) = ρ_{p+1} (w) = ρ, i.e., λ̄_{p}(g_{f} − φ) = ρ_{p}(g_{f} ) = ρ_{p}(f) = ∞ and λ̄_{p+1} (g_{f} − φ) = ρ_{p+1} (g_{f} ) = ρ_{p+1} (f) = ρ_{p}(A_{0}) = ρ.

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Department of Mathematics (Received 13 01 2011) Laboratory of Pure and Applied Mathematics University of Mostaganem (UMAB) B. P. 227 Mostaganem Algeria

    belaidi@univ-mosta.dz