Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables

Vita Baksa 1 , Andriy Bandura 2 ,  and Oleh Skaskiv 1
  • 1 Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 1 Universytetska street 79000, Lviv, Ukraine
  • 2 Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska street, 76019, Ivano-Frankivsk, Lviv, Ukraine
Vita Baksa
  • Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 1 Universytetska street 79000, Lviv, Ukraine
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
, Andriy Bandura
  • Corresponding author
  • Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska street, 76019, Ivano-Frankivsk, Lviv, Ukraine
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Oleh Skaskiv
  • Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 1 Universytetska street 79000, Lviv, Ukraine
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

In this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball B2={zC2:|z|=|z1|2+|z2|2<1}, where L = (l1, l2): 𝔹2R+2 is a positive continuous vector-valued function.

Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).

  • [1]

    Baksa, V. P.: Analytic vector-functions in the unit ball having bounded L-index in joint variables, Carpathian Math. Publ. 11(2) (2019), 213–227.

    • Crossref
    • Export Citation
  • [2]

    Baksa, V. P.—Bandura, A. I.—Skaskiv, O. B.: Analogs of Fricke’s theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables, Proc. IAMM NASU 33 (2019), 16–26.

    • Crossref
    • Export Citation
  • [3]

    Bandura, A.—Skaskiv, O.: Linear directional differential equations in the unit ball: solutions of bounded L-index, Math. Slovaca 69 (2019), 1089–1098.

    • Crossref
    • Export Citation
  • [4]

    Bandura, A.—Skaskiv, O.: Sufficient conditions of boundedness of L-index and analog of Hayman’s Theorem for analytic functions in a ball, Stud. Univ. Babeş-Bolyai Math. 63 (2018), 483–501.

    • Crossref
    • Export Citation
  • [5]

    Bandura, A. I.—Skaskiv, O. B.: Analytic functions in the unit ball of bounded L-index: asymptotic and local properties, Mat. Stud. 48 (2017), 37–73.

  • [6]

    Bandura A.: New criteria of boundedness of L-index in joint variables for entire functions, Math. Bull. Shevchenko Sci. Soc. 13 (2016), 58–67.

  • [7]

    Bandura, A.—Skaskiv, O.: Asymptotic estimates of entire functions of bounded L-index in joint variables, Novi Sad J. Math. 48 (2018), 103–116.

    • Crossref
    • Export Citation
  • [8]

    Bandura, A.—Petrechko, N.—Skaskiv, O.: Maximum modulus in a bidisc of analytic functions of bounded L-index and an analogue of Hayman’s theorem, Math. Bohem. 143 (2018), 339–354.

    • Crossref
    • Export Citation
  • [9]

    Bandura, A. I.—Skaskiv, O. B.—Tsvigun, V. L. Some characteristic properties of analytic functions in 𝔻 × ℂ of bounded L-index in joint variables, Bukovynskij Matematicheskij Zhurnal 6 (2018), 21–31.

  • [10]

    Bandura, A. I.—Petrechko, N. V.—Skaskiv, O. B.: Analytic in a polydisc functions of bounded L-index in joint variables. Mat. Stud. 46(1), 72–80 (2016).

  • [11]

    Bandura, A. I.—Bordulyak, M. T.—Skaskiv, O. B.: Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016), 12–26.

  • [12]

    Bandura, A.—Skaskiv, O.: Analytic functions in the unit ball of bounded L-index in joint variables and of bounded L-index in direction: a connection between these classes, Demonstr. Math. 52 (2019), 82–87.

    • Crossref
    • Export Citation
  • [13]

    Bandura, A.—Skaskiv, O.: Boundedness of the L-index in a direction of entire solutions of second order partial differential equation, Acta Comment. Univ. Tartu. Math. 22 (2018), 223–234.

  • [14]

    Bandura, A. I.—Skaskiv, O. B.: Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables, J. Math. Sci. 239 (2019), 17–29.

    • Crossref
    • Export Citation
  • [15]

    Bandura, A. I.—Skaskiv, O. B.: Exhaustion by balls and entire functions of bounded L-index in joint variables, Ufa Math. J. 11 (2019), 100–113.

    • Crossref
    • Export Citation
  • [16]

    Bandura, A.—Skaskiv, O.: Analog of Hayman’s theorem and its application to some system of linear partial differential equations, J. Math. Phys. Anal. Geom. 15 (2019), 170–191.

  • [17]

    Bandura, A.—Skaskiv, O.: Functions Analytic in the Unit Ball Having Bounded L-Index in a Direction, Rocky Mountain J. Math. 49 (2019), 1063–1092.

    • Crossref
    • Export Citation
  • [18]

    Bordulyak, M. T.: On the growth of entire solutions of linear differential equations, Mat. Stud. 13 (2000), 219–223.

  • [19]

    Bordulyak, M. T.—Sheremeta, M. M.: Boundedness of l-index of analytic curves, Mat. Stud. 36 (2011), 152–161.

  • [20]

    Hayman, W.K.: Differential inequalities and local valency, Pacific J. Math. 44 (1973), 117–137.

    • Crossref
    • Export Citation
  • [21]

    Heath, L.F. Vector-valued entire functions of bounded index satisfying a differential equation, Journal of Research of NBS 83B(1978), 75–79.

  • [22]

    Nuray, F.—Patterson, R. F.: Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations. Mat. Stud. 49(1), (2018), 67–74.

  • [23]

    Roy, R.—Shah, S. M.: Growth properties of vector entire functions satisfying differential equations, Indian J. Math. 28 (1986), 25–35.

  • [24]

    Roy, R.—Shah, S. M.: Vector-valued entire functions satisfying a differential equation, J. Math. Anal. Appl. 116 (1986), 349–362.

    • Crossref
    • Export Citation
  • [25]

    Patterson, R. F.—Nuray, F.: A characterization of holomorphic bivariate functions of bounded index, Math. Slovaca 67 (2017), 731–736.

  • [26]

    Sheremeta, M. N.—Kuzyk, A. D.: Logarithmic derivative and zeros of an entire function of bounded l-index, Sib. Math. J. 33 (1992), 304–312.

    • Crossref
    • Export Citation
  • [27]

    Sheremeta, M.: Boundedness of lM-index of analytic curves, Visnyk of the Lviv Univ. Series Mech. Math. 75 (2011), 226–231.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process.

Search