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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)].

##### MSC:
 34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
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