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Growth estimates for solutions of linear complex differential equations. (English) Zbl 1057.34111

Let

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

be a complex differential equation. The coefficients A 0 (z), A 1 (z),,A k-1 (z) are analytic in the disc D R ={z:|z|<R}, 0<R+. A representation theorem for the solutions of equation (1) is given. By this theorem, for any z, z 0 D R it holds

f(z)= n=0 K-1 f (n) (z 0 ) n!(z-z 0 ) n -1 (K-1)! z 0 z (z-ξ) k-1 A(ξ)f(ξ)dξ·(2)

The representation thereom yields the growth estimates on the solutions of the general equation (1) in D R .

(a) If 0<R1, then there exist a constant c 1 >0, depending on the initial values of f at z 0 =0, and a constant c 2 >0, such that

f ( z )c 1 expc 2 j=0 n-1 n=0 j 0 r A j (n) (se iθ )(R-S) K-j+n-1 ds

for all θ[0,2π) and r[0,R).

(b) If 1<R+, then there exist a constant C 1 >0, depending on the initial values of at z 0 =e iθ , and a constant C 2 >0, such that

f ( z )C 1 r K-1 expC 2 j=0 K-1 n=0 j 0 r A j (n) (se iθ )s K-j+n-1 ds

for all θ[0,aπ) and r(1,R). The Herold’s comparison theorem yields the next growth estimates.


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