# Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded L-index in joint variables

Article

First Online:

- 2 Downloads

## Abstract

We obtain the sufficient conditions of boundedness of **L**-index in joint variables for analytic functions in the unit ball, where \( L:{\mathbb{C}}^n\to {\mathbb{R}}_{+}^n \) is a continuous positive vector-function. They give an stimate of the maximum modulus of an analytic function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives outside some exceptional set and the distribution of zeros. The deduced results are also new for analytic functions in the unit disc of bounded index and *l*-index. They generalize known results by G. H. Fricke, M. M. Sheremeta, A. D. Kuzyk, and V. O. Kushnir.

## Keywords

Analytic function unit ball bounded L-index in joint variables maximum modulus partial derivative, minimum modulus, distribution of zeros, skeleton of polydisc.## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Bandura and O. Skaskiv, “Functions analytic in a unit ball of bounded
*L*-index in joint variables,”*J. Math. Sci.*,**227**, No. 1, 1–12 (2017), doi:10.1007/s10958-017-3570-6.Google Scholar - 2.A. Bandura and O. Skaskiv, “Sufficient conditions of boundedness of
**L**-index and analog of Hayman’s theorem for analytic functions in a ball,”*Stud. Univ. Babe¸s-Bolyai Math.*,**63**, No. 4, 483–501 (2018).Google Scholar - 3.A. I. Bandura and O. B. Skaskiv, “Analytic functions in the unit ball of bounded
**L**-index: asymptotic and local properties,”*Mat. Stud.*,**48**, No. 1, 37–73 (2017), doi: 10.15330/ms.48.1.37-73.Google Scholar - 4.A. I. Bandura and O. B. Skaskiv, “Entire functions of bounded
*L*-index in direction,”*Mat. Stud.*,**27**, No. 1, 30–52 (2007).Google Scholar - 5.A. Bandura and O. Skaskiv, “Entire functions of bounded
**L**-index: its zeros and behavior of partial logarithmic derivatives,”*J. Complex Analysis*,**2017**, Article ID 3253095, 10 p., (2017), doi:10.1155/2017/3253095.Google Scholar - 6.A. I. Bandura, “Some improvements of criteria of
*L*-index boundedness in direction,”*Mat. Stud.*,**47**, No. 1, 27–32 (2017), doi: 10.15330/ms.47.1.27-32.Google Scholar - 7.A. Bandura and O. Skaskiv,
*Entire Functions of Several Variables of Bounded Index*, Chyzhykov, Lviv, 2016, http://chyslo.com.ua.Google Scholar - 8.A. I. Bandura and O. B. Skaskiv, “Directional logarithmic derivative and the distribution of zeros of an entire function of bounded
*L*-index along the direction,”*Ukrain. Mat. J.*,**69**, No. 1, 500–508 (2017), doi:10.1007/s11253-017-1377-8.Google Scholar - 9.A. Bandura, O. Skaskiv, and P. Filevych, “Properties of entire solutions of some linear PDE’s,”
*J. Appl. ath. Comput. Mech.*,**16**, No. 2, 17–28 (2017), doi:10.17512/jamcm.2017.2.02.Google Scholar - 10.A. I. Bandura, M. T. Bordulyak, and O. B. Skaskiv, “Sufficient conditions of boundedness of L-index in joint variables,”
*Mat. Stud.*,**45**, No. 1, 12–26 (2016), doi:10.15330/ms.45.1.12-26.Google Scholar - 11.A. I. Bandura, N. V. Petrechko, and O. B. Skaskiv, “Maximum modulus in a bidisc of analytic functions of bounded
**L**-index and an analogue of Hayman’s theorem,”*Math. Bohemica*,**143**, No. 4, 339–354 (2018).Google Scholar - 12.A. I. Bandura and N. V. Petrechko, “Properties of power series of analytic in a bidisc functions of bounded
**L**-index in joint variables,”*Carpathian Math. Publ.*,**9**, No. 1, 6–12 (2017), doi:10.15330/cmp.9.1.6-12.Google Scholar - 13.T. O. Banakh and V. O. Kushnir, “On growth and distribution of zeros of analytic functions of bounded
*l*-index in arbitrary domains,”*Mat. Stud.*,**14**, No. 2, 165–170 (2000).Google Scholar - 14.M. T. Bordulyak, “A proof of Sheremeta conjecture concerning entire function of bounded
*l*-index,”*Mat. Stud.*,**11**, No. 2, 108–110 (1999).Google Scholar - 15.I. E. Chyzhykov and N. S. Semochko, “Generalization of the Wiman–Valiron method for fractional derivatives,”
*Int. J. Appl. Math.*,**29**, No. 2, 19–30 (2016).Google Scholar - 16.P. C. Fenton, “Wiman–Valiron theory in several variables,”
*Ann. Acad. Sci. Fenn. Math.*,**38**, No. 1, 29–47 (2013).Google Scholar - 17.G. H. Fricke, “Entire functions of locally slow growth,”
*J. Anal. Math.*,**28**, No. 1, 101–122 (1975).Google Scholar - 18.G. H. Fricke and S. M. Shah, “On bounded value distribution and bounded index,”
*Nonlinear Anal.*,**2**, No. 4, 423–435 (1978).Google Scholar - 19.G. H. Fricke, “A note on bounded index and bounded value distribution,”
*Indian J. Pure Appl. Math.*,**11**, No. 4, 428–432 (1980).Google Scholar - 20.W. K. Hayman, “Differential inequalities and local valency,”
*Pacific J. Math.*,**44**, No. 1, 117–137 (1973).Google Scholar - 21.G. J. Krishna and S. M. Shah, “Functions of bounded indices in one and several complex variables,” in:
*Mathematical Essays Dedicated to A.J. Macintyre*, Ohio Univ. Press, Athens, Ohio, 1970, pp. 223–235.Google Scholar - 22.A. D. Kuzyk and M. M. Sheremeta, “Entire functions of bounded
*l*-distribution of values,”*Math. Notes*,**39**, No. 1, 3–8 (1986), doi: 10.1007/BF01647624.Google Scholar - 23.V. O. Kushnir and M. M. Sheremeta, “Analytic functions of bounded
*l*-index,”*Mat. Stud.*,**12**, No. 1, 59–66 (1999).Google Scholar - 24.B. Lepson, “Differential equations of infinite order, hyperdirichlet series and analytic in 𝔹
^{n}functions of bounded index,”*Proc. Sympos. Pure Math.*,**2**, 298–307 (1968).Google Scholar - 25.F. Nuray and R. F. Patterson, “Multivalence of bivariate functions of bounded index,”
*Le Matem.*,**70**, No. 2, 225–233 (2015), doi: 10.4418/2015.70.2.14.Google Scholar - 26.R. Patterson and F. Nuray, “A characterization of holomorphic bivariate functions of bounded index,”
*Math. Slovaca*,**67**, No. 3, 731–736 (2017), doi: 10.1515/ms-2017-0005.Google Scholar - 27.N. Petrechko, “Bounded
**L**-index in joint variables and analytic solutions of some systems of PDE’s in bidisc,”*Visn. Lviv Univ. Ser. Mech. Math.*, Issue 83, 100–108 (2017).Google Scholar - 28.L. I. Ronkin,
*Introduction to Theory of Entire Functions of Several Variables*, AMS, Providence, RI, 1974.Google Scholar - 29.R. Roy and S. M. Shah, “Functions of bounded index, bounded value distribution and
*v*-bounded index,”*Nonlinear Anal.*,**11**, 1383–1390 (1987).Google Scholar - 30.M. Salmassi, “Functions of bounded indices in several variables,”
*Indian J. Math.*,**31**, No. 3, 249–257 (1989).Google Scholar - 31.T. M. Salo, O. B. Skaskiv, and O. M. Trakalo, “On the best possible description of exceptional set in Wiman–Valiron theory for entire function,”
*Mat.Stud.*,**16**, No. 2, 131–140 (2001).Google Scholar - 32.S. M. Shah, “Entire function of bounded index,”
*Lect. Notes in Math.*,**599**, 117–145 (1977).Google Scholar - 33.M. Sheremeta,
*Analytic Functions of Bounded Index*, VNTL , Lviv, 1999.Google Scholar - 34.M. N. Sheremeta, “An
*l*-index and an*l*-distribution of the values of entire functions,”*Soviet Math. (Izv. VUZ)*,**34**, No. 2, 115–117 (1990).Google Scholar - 35.S. N. Strochyk and M. M. Sheremeta, “Analytic in the unit disc functions of bounded index,”
*Dopov. Akad. Nauk Ukr.*, No. 1, 19–22 (1993).Google Scholar - 36.M. N. Sheremeta and A. D. Kuzyk, “Logarithmic derivative and zeros of an entire function of bounded l-index,”
*Siber. Math. J.*,**33**, No. 2, 304–312 (1992), doi: 10.1007/BF00971102.Google Scholar - 37.Sh. Strelitz, “Asymptotic properties of entire transcendental solutions of algebraic differential equations,”
*Contemp. Math*.,**25**, 171–214 (1983).Google Scholar

## Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019