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**Maximum modulus, characteristic, deficiency and growth of solutions of second order linear differential equations.**
*(English)*
Zbl 1005.34079

This paper is devoted to considering the growth of solutions to \(f''+A(z)f'+B(z)f=0\) with entire coefficients, still defectively understood. In a previous paper due to the reviewer and P. Wu [Proc. Am. Math. Soc. 128, No. 9, 2693-2703 (2000; Zbl 0952.34070)], it was proved that all nontrivial solutions to the linear differential equation above are of infinite order of growth, provided \(\rho(B)<\rho(A)<\infty\) and \(T(r,A)\sim\log M(r,A)\) as \(t\to\infty\) outside a set of finite logarithmic measure.

In the present paper, the latter condition will be improved to \(T(r,A)\sim\log M(r,A)\) as \(t\to\infty\) outside a set of upper logarithmic density \(<(\rho(A)-\rho(B))/\rho(A)\). Another condition to the same result offered here assumes that \(\rho(B)\leq 1/2\), \(\rho(B)<\rho(A)\) and that \(A(z)\) has a finite deficient value. The proofs offer a nice analysis of logarithmic derivatives, relying on estimates given by W. Fuchs in [Ill. J. Math. 7, 661-667 (1963; Zbl 0113.28702)] and G. G. Gundersen in [J. Lond. Math. Soc., II. Ser. 37, No. 1, 88-104 (1988; Zbl 0638.30030)].

In the present paper, the latter condition will be improved to \(T(r,A)\sim\log M(r,A)\) as \(t\to\infty\) outside a set of upper logarithmic density \(<(\rho(A)-\rho(B))/\rho(A)\). Another condition to the same result offered here assumes that \(\rho(B)\leq 1/2\), \(\rho(B)<\rho(A)\) and that \(A(z)\) has a finite deficient value. The proofs offer a nice analysis of logarithmic derivatives, relying on estimates given by W. Fuchs in [Ill. J. Math. 7, 661-667 (1963; Zbl 0113.28702)] and G. G. Gundersen in [J. Lond. Math. Soc., II. Ser. 37, No. 1, 88-104 (1988; Zbl 0638.30030)].

Reviewer: Ilpo Laine (Joensuu)

### MSC:

34M10 | Oscillation, growth of solutions to ordinary differential equations in the complex domain |