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On a complete description of the class of functions without zeros analytic in a disk and having given orders

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Abstract

For arbitrary 0 ≤ σ ≤ ρ ≤ σ + 1, we describe the class A ρσ of functions g(z) analytic in the unit disk \(\mathbb{D}\) = {z : ∣z∣ < 1} and such that g(z) ≠ 0, ρT[g] = σ, and ρM[g] = ρ, where

$$M(r,g) = \max \left\{ {\left| {g(z)} \right|:\left| z \right| \leqslant r} \right\}, T(r,u) = \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\ln ^ + \left| {g(re^{i\varphi } )} \right|d\varphi ,} $$
$$\rho _{\rm M} [g] = \lim \sup _{r \uparrow 1} \frac{{\ln \ln ^ + M(r,g)}}{{ - \ln (1 - r)}}, \rho _{\rm T} [g] = \lim \sup _{r \uparrow 1} \frac{{\ln ^ + T(r,g)}}{{ - \ln (1 - r)}}.$$

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References

  1. C. N. Linden, “On a conjecture of Valiron concerning sets of indirect Borel points,” J. London Math. Soc., 41, 304–312 (1966).

    Article  MathSciNet  Google Scholar 

  2. M. M. Dzhrbashyan, Integral Transformations and Representations of Functions in a Complex Domain [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  3. A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge (1959).

    MATH  Google Scholar 

  4. I. E. Chyzhykov, “Growth of harmonic functions in the unit disc and an application,” Obervolfach Rep., 1, No. 1, 391–392 (2004).

    MathSciNet  Google Scholar 

  5. G. H. Hardy and J. E. Littlewood, “Some properties of fractional integrals. II,” Math. Z., 34, 403–439 (1931–1932).

    Article  MATH  MathSciNet  Google Scholar 

  6. F. A. Shamoyan, “Some remarks on the parametric representation of Nevanlinna-Dzhrbashyan classes,” Mat. Zametki, 52, No. 1, 128–140 (1992).

    MATH  MathSciNet  Google Scholar 

  7. R. Salem, “On a theorem of Zygmund,” Duke Math. J., 10, 23–31 (1943).

    Article  MATH  MathSciNet  Google Scholar 

  8. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge (1927).

    MATH  Google Scholar 

  9. M. A. Subkhankulov, Tauberian Theorems with Remainder [in Russian], Nauka, Moscow (1976).

    Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 979–995, July, 2007.

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Chyzhykov, I.E. On a complete description of the class of functions without zeros analytic in a disk and having given orders. Ukr Math J 59, 1088–1109 (2007). https://doi.org/10.1007/s11253-007-0070-8

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  • DOI: https://doi.org/10.1007/s11253-007-0070-8

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