Abstract
Entire functionsf(z), z∃C n, of exponential type at most a and bounded on subsets E of the real hyperplane, are investigated. It is known that if E is relatively dense with respect to the Lebesgue measure or it is an ɛ-net inR n, then such f(z) are bounded on all ofR n (for e-nets in the case of sufficiently small σ). It is shown that if E is close in a certain sense either to a relatively dense subset ofR n, or to an ε-net, then f(z) cannot increase fast alongR n. Similar estimates are established for integral metrics.
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Additional information
Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 74–76, 1988.
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Logvinenko, V.N. Entire functions of exponential type, increasing slowly along the real hyperplane. J Math Sci 49, 1289–1290 (1990). https://doi.org/10.1007/BF02209174
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DOI: https://doi.org/10.1007/BF02209174