REMARKS ON SOME CLASSES OF POSITIVE CONTINUOUS FUNCTIONS IN C^n
DOI:
https://doi.org/10.31471/2304-7399-2020-1(59)-9-15Keywords:
positive continuous function, several complex variables, partial logarithmic derivative, local behavior, complex conjugate.Abstract
Here we prove two propositions providing sufficient conditions of belonging positive continuous functions in to classes and These auxiliary classes plays important role in theory of entire functions of bounded L-index in direction and bounded L-index in joint variables, where are continuous functions. They help to constuct general theory of bounded index for very wide class of entire functions, because for every entire functions with bounded multiplicities of zero points there exists a corresponding function or providing boundedness of L-index in direction or boundedness L–index in joint variables respectively. Our result requires uniform boundedness of logarithmic derivative in all variables and for belonging the function to class Q^n. Another result requires uniform boundedness of logarithmic derivative in directions and for belonging the function to class Q^n_b where is the complex conjugate vector to b.
References
Bandura A., Skaskiv O. Entire functions of several variables of bounded index, Lviv: Publ. I.E. Chyzhykov, 2016, 128 p.
Bandura A., Skaskiv O. Asymptotic estimates positive integrals and entire functions, Lviv-Ivano-Frankivsk, Publ. O.M. Goliney, 2015, 108 p. (in Ukrainian)
Bandura A., Skaskiv O. Analytic functions in the unit Ball. Bounded L-index in joint variables and solutions of systems of PDE’s. – Beau-Bassin: LAP Lambert Academic Publishing, 2017. – 100 p.
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., 2018, 22(2), 223–234, DOI: 10.12697/ACUTM.2018.22.18
Bandura A., Skaskiv O. Asymptotic estimates of entire functions of bounded L-index in joint variables, Novi Sad J. Math., 48 (2018) (1), 103–116. doi: 10.30755/NSJOM.06997
Bandura, A.I.: Properties of positive continuous functions in C^n. Carpathian Math. Publ. 7(2), 137–147 (2015). doi: 10.15330/cmp.7.2.137-147
Downloads
Published
Versions
- 2021-02-01 (3)
- 2021-01-29 (2)
- 2021-01-28 (1)