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On the iterated order and the fixed points of entire solutions of some complex linear differential equations. (English) Zbl 1120.34070

The iterated p-order σ p (f) of an entire function f is defined as

σ p (f)=lim sup r+ log p T(r,f) logr=lim sup r+ log p+1 M(r,f) logr·

The finiteness degree of the order of an entire function f is defined as i(f)=0 if f is a polynomial or i(f)=min{j:σ j (f)<} if f is a transcendental entire function and σ j (f)< for some j, otherwise i(f)=· The iterated convergence exponent of the sequence of distinct zeros of an entire function f is defined as

λ ¯ p (f)=lim suplog p N ¯(r,1 f) logr,

where N ¯(r,1 f) is the counting function of the distinct zeros of f(z) in Δ r ={z:|z|<1}·

In this paper, the author studies the iterated order and iterated convergence exponent of the sequence of distinct zeros of the entire solutions of the following linear differential equations

f (k) +A k-1 (z)f (k) ++A 1 (z)f ' +A 0 (z)f=0,(1)
f (k) +A k-1 (z)f (k) ++A 1 (z)f ' +A 0 (z)f=F(z),(2)

where A 0 (z),,A k-1 (z) and F(z)¬0 are entire functions. The author proves that if A 0 (z),,A k-1 (z) are entire functions with σ p (A j )σ p (A s )< for some s(0sk-1), then the equation (1) has at least one solution f with i(f)=p+1 and σ p+1 (f)=σ p (A s )· Furthermore, if F(z)¬0 is an entire function with either i(F)=q<p+1 or i(F)=q=p+1 and σ p+1 (F)<σ p (A s )<+, then there exists at least one solution g of the corresponding homogeneous equation (1) of the equation (2) such that all solutions f in the solution subspace {cg+f 0 ,c} satisfying i(f)=p+1 and σ p+1 (f)=λ ¯ p+1 (f)=σ p (A s ), with at most one exception, where f 0 is a solution of (2).

MSC:
34M10Oscillation, growth of solutions (ODE in the complex domain)
30D35Distribution of values (one complex variable); Nevanlinna theory