Rocky Mountain Journal of Mathematics

Functions analytic in the unit ball having bounded $L$-index in a direction

Andriy Bandura and Oleh Skaskiv

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We propose a generalization of a concept of bounded index for analytic functions in the unit ball. Use of directional derivatives gives us a possibility to deduce the necessary and sufficient conditions of boundedness of $L$-index in a direction for analytic functions of several variables, namely, we obtain an analog of Hayman's theorem and a logarithmic criteria for this class. The criteria describe the behavior of the directional logarithmic derivative outside the zero set and a uniform distribution of zeros in some sense. The criteria are useful for studying analytic solutions of partial differential equations and estimating their growth. We present a scheme of this application.

Article information

Rocky Mountain J. Math., Volume 49, Number 4 (2019), 1063-1092.

First available in Project Euclid: 29 August 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32A10: Holomorphic functions
Secondary: 32A17: Special families of functions 35B08: Entire solutions

Analytic function unit ball bounded $L$-index in direction growth estimates partial differential equation several complex variables


Bandura, Andriy; Skaskiv, Oleh. Functions analytic in the unit ball having bounded $L$-index in a direction. Rocky Mountain J. Math. 49 (2019), no. 4, 1063--1092. doi:10.1216/RMJ-2019-49-4-1063.

Export citation


  • A. Bandura and O. Skaskiv, Directional logarithmic derivative and the distribution of zeros of an entire function of bounded $L$-index along the direction, Ukr. Math. J., 69 (2017), no. 3, 500–508.
  • A.I. Bandura, Some improvements of criteria of $L$-index boundedness in direction, Mat. Stud. 47 (2017), no. 1, 27–32.
  • A.I. Bandura, M.T. Bordulyak and O.B. Skaskiv, Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016), no. 1, 12–26.
  • A. Bandura, O. Skaskiv and P. Filevych, Properties of entire solutions of some linear PDE's, J. Appl. Math. Comput. Mech. 16 (2017), no. 2, 17–28.
  • A. Bandura and O. Skaskiv, Functions analytic in a unit ball of bounded $L$-index in joint variables, J. Math. Sci. 227 (2017), no. 1, 1–12.
  • A.I. Bandura and O.B. Skaskiv, Analytic functions in the unit ball of bounded $\mathbf{L}$-index: asymptotic and local properties, Mat. Stud. 48 (2017), no. 1, 37–73.
  • A. Bandura, N. Petrechko and O. Skaskiv, Maximum modulus in a bidisc of analytic functions of bounded ${\mathbf L}$-index and an analogue of Hayman's theorem, Mat. Bohemica 143 (2018), no. 4, 339–354.
  • A.I. Bandura and N.V. Petrechko, Properties of power series of analytic in a bidisc functions of bounded $\mathbf{L}$-index in joint variables, Carpathian Math. Publ. 9 (2017), no. 1, 6–12.
  • A.I. Bandura and O.B. Skaskiv, Entire functions of bounded $L$-index in direction, Mat. Stud. 27 (2007), no. 1, 30–52.
  • A. Bandura and O. Skaskiv, Entire functions of bounded $\mathbf{L}$-Index: its zeros and behavior of partial logarithmic derivatives, J. Complex Analysis 2017 (2017), art. ID 3253095.
  • A. Bandura and O. Skaskiv, Entire functions of several variables of bounded index, I.E. Chyzhykov, Lviv (2016).
  • M.T. Bordulyak, A proof of Sheremeta conjecture concerning entire function of bounded $l$-index, Mat. Stud. 11 (1999), no. 2, 108–110.
  • W.K. Hayman, Differential equations and local valency, Pacific J. Math. 44 (1973), no. 1, 117–137.
  • Gopala J. Krishna and S.M. Shah, Functions of bounded indices in one and several complex variables, pp. 223–235 in Mathematical essays dedicated to A.J. Macintyre, Ohio Univ. Press, Athens, Ohio (1970).
  • A.D. Kuzyk and M.M. Sheremeta, Entire functions of bounded $l$-distribution of values, Math. Notes 39 (1986), no. 1, 3–8.
  • B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, pp. 298–307 in Entire functions and related parts of analysis (La Jolla, CA, 1966), Proc. Sympos. Pure Math. \bf11, Amer. Math. Soc., Providence, RI (1968).
  • W. Rudin, Function theory in the unit ball of $\mathbb{C}^n$, Grundlehren der Math. Wiss. \bf241, Springer (1980).
  • M. Salmassi, Functions of bounded indices in several variables, Indian J. Math. 31 (1989), no. 3, 249–257.
  • S.M. Shah, Entire function of bounded index, Lect. Notes in Math. 599 (1977), 117–145.
  • M. Sheremeta, Analytic functions of bounded index, VNTL Publishers, Lviv (1999).
  • S.N. Strochyk and M.M. Sheremeta, Analytic in the unit disc functions of bounded index, Dopov. Akad. Nauk Ukr. 1 (1993), 19–22.
  • M.N. Sheremeta and A. D. Kuzyk, Logarithmic derivative and zeros of an entire function of bounded l-index, Sib. Math. J. 33 (1992), 12, 304–312.