Differential equations of the form f′′+A(z)f′+B(z)f=0\documentclass[12pt]{minimal}
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\begin{document}$$f'' + A(z)f' + B(z)f = 0$$\end{document} (*) are considered, where A(z) and B(z)≢0\documentclass[12pt]{minimal}
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\begin{document}$$B(z) \not \equiv 0$$\end{document} are entire functions. The Lindelöf function is used to show that for any ρ∈(1/2,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\rho \in (1/2, \infty )$$\end{document}, there exists an equation of the form (*) which possesses a solution f with a Nevanlinna deficient value at 0 satisfying ρ=ρ(f)≥ρ(A)≥ρ(B)\documentclass[12pt]{minimal}
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\begin{document}$$\rho =\rho (f)\ge \rho (A)\ge \rho (B)$$\end{document}, where ρ(h)\documentclass[12pt]{minimal}
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\begin{document}$$\rho (h)$$\end{document} denotes the order of an entire function h. It is known that such an example cannot exist when ρ≤1/2\documentclass[12pt]{minimal}
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\begin{document}$$\rho \le 1/2$$\end{document}. For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any ρ∈(2,∞)\documentclass[12pt]{minimal}
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\begin{document}$$\rho \in (2, \infty )$$\end{document}, there exists an equation of the form (*) which possesses a solution f with a Valiron deficient value at 0 satisfying ρ=ρlog(f)≥ρlog(A)≥ρlog(B)\documentclass[12pt]{minimal}
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\begin{document}$$\rho =\rho _{\log }(f)\ge \rho _{\log }(A)\ge \rho _{\log }(B)$$\end{document}, where ρlog(h)\documentclass[12pt]{minimal}
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\begin{document}$$\rho _{\log }(h)$$\end{document} denotes the logarithmic order of an entire function h. This result is essentially sharp. In both proofs, the separation of the zeros of the indicated solution plays a key role. Observations on the deficient values of solutions of linear differential equations are also given, which include a discussion of Wittich’s theorem on Nevanlinna deficient values, a modified Wittich theorem for Valiron deficient values, consequences of Gol’dberg’s theorem, and examples to illustrate possibilities that can occur.
