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Analysis Mathematica - Let E be an arbitrary closed set on the unit circle ?? and u be a harmonic function on the unit disk ? satisfying |u(z)| ? (1 ?... 相似文献
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If φ: [0, 1) → (0,∞) is a non-decreasing unbounded function, then the φ-order of a meromorphic function f in the unit disc is defined as $$ \sigma _\phi (f) = \mathop {\lim \sup }\limits_{r \to 1^ - } \frac{{\log ^ + T(r,f)}} {{\log \phi (r)}}, $$ where T(r, f) is the Nevanlinna characteristic of f. In particular, $ \sigma _{\tfrac{1} {{1 - r}}} $ f is the order of f, and $ \sigma _{\log \tfrac{1} {{1 - r}}} $ f is the logarithmic order of f. Several results on the finiteness of the φ-order of solutions of $$ f^{(k)} + A_{k - 1} (z)f^{(k - 1)} + \cdots + A_1 (z)f' + A_0 (z)f = 0 $$ are obtained in the case when the coefficients A 0(z), ...,A k?1(z) are analytic functions in the unit disc. This paper completes some earlier results by various authors. 相似文献
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I. E. Chyzhykov 《Proceedings of the American Mathematical Society》2002,130(2):517-528
We obtain a sharp asymptotic relation between the infimum and the maximum on a circle of a subharmonic function of zero lower order. An example is constructed, which shows the sharpness of the relation in the class of entire functions of zero order such that , where as .
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I. E. Chyzhykov 《Ukrainian Mathematical Journal》2007,59(7):1088-1109
For arbitrary 0 ≤ σ ≤ ρ ≤ σ + 1, we describe the class A
σ
ρ
of functions g(z) analytic in the unit disk
= {z : ∣z∣ < 1} and such that g(z) ≠ 0, ρT[g] = σ, and ρM[g] = ρ, where
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 979–995, July, 2007. 相似文献
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Chyzhykov Igor; Gundersen Gary G.; Heittokangas Janne 《Proceedings London Mathematical Society》2003,86(3):735-754
We prove two sharp inequalities for the growth of solutionsof certain linear differential equations in the unit disk. Forthe proofs of these inequalities, we use the method of successiveapproximations and sharp estimates for the logarithmic derivativesof finite order meromorphic functions in the unit disk. Thesetechniques can also be used to give an alternate proof of awell-known result in the plane. The sharp logarithmic derivativeestimates are a corollary of general estimates, and all theseestimates have independent interest. 2000 Mathematics SubjectClassification 34M10, 30D35. 相似文献
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