### Boundedness of $\mathbf{L}$-Index in Joint Variables for Sum of Entire Functions

Authors: A. BANDURA

DOI: 10.46793/KgJMat2204.595B

Abstract:

In the paper, we present suﬃcient conditions of boundedness of L-index in joint variables for a sum of entire functions, where L : n +n is a continuous function, + = (0, +). They are applicable to a very wide class of entire functions because for every entire function F in n with bounded multiplicities of zero points there exists a positive continuous function L such that F has bounded L-index in joint variables. Our propositions are generalizations of Pugh’s result obtained for entire functions of one variable of bounded index.

Keywords:

Entire function of several variables, bounded L-index in joint variables, sum of entire functions.

References:

[1]   A. I. Bandura, O. B. Skaskiv and V. L. Tsvigun, Some characteristic properties of analytic functions in ???? × of bounded L-index in joint variables, Bukovyn. Mat. Zh. 6 (2018), 21–31.

[2]   A. Bandura, N. Petrechko and O. Skaskiv, 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.

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

[4]   A. Bandura and O. Skaskiv, 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.

[5]    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. Babeş-Bolyai Math. 63 (2018), 483–501.

[6]   A. Bandura and O. Skaskiv, 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. Mathem. 52 (2019), 82–87.

[7]   A. Bandura, O. Skaskiv and P. Filevych, Properties of entire solutions of some linear pde’s, J. Appl. Math. Comput. Mech. 16 (2017), 17–28.

[8]   A. I. Bandura, Sum of entire functions of bounded l-index in direction, Mat. Stud. 45 (2016), 149–158.

[9]   A. I. Bandura, M. T. Bordulyak and O. B. Skaskiv, Suﬃcient conditions of boundedness of l-index in joint variables, Mat. Stud. 45 (2016), 12–26.

[10]   A. I. Bandura and N. V. Petrechko, Sum of entire functions of bounded index in joint variables, Electr. J. Math. Anal. Appl. 6(2) (2018), 60–67.

[11]   A. I. Bandura, Product of two entire functions of bounded L-index in direction is a function with the same class, Bukovyn. Mat. Zh. 4(1–2) (2016), 8–12.

[12]   A. I. Bandura and O. B. Skaskiv, Iyer’s metric space, existence theorem and entire functions of bounded L-index in joint variables, Bukovyn. Mat. Zh. 5(3–4) (2017), 8–14 (in Ukrainian).

[13]   M. T. Bordulyak, A proof of Sheremeta conjecture concerning entire function of bounded l-index, Mat. Stud. 12 (1999), 108–110.

[14]   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 University Press, Athens, Ohio, 1970, 223–235.

[15]   A. D. Kuzyk and M. N. Sheremeta, Entire functions of bounded l-distribution of values, Math. Notes. 39 (1986), 3–8.

[16]   B. Lepson, Diﬀerential equations of inﬁnite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math. 11 (1968), 298–307.

[17]   F. Nuray and R. F. Patterson, Entire bivariate functions of exponential type, Bull. Math. Sci. 5 (2015), 171–177.

[18]   F. Nuray and R. F. Patterson, Multivalence of bivariate functions of bounded index, Le Matematiche 70 (2015), 225–233.

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

[20]   R. F. Patterson and F. Nuray, A characterization of holomorphic bivariate functions of bounded index, Math. Slovaca 67 (2017), 731–736.

[21]   W. J. Pugh, Sums of functions of bounded index, Proc. Amer. Math. Soc. 22 (1969), 319–323.

[22]   W. J. Pugh and S. M. Shah, On the growth of entire functions of bounded index, Paciﬁc J. Math. 33 (1970), 191–201.

[23]   R. Roy and S. M. Shah, Sums of functions of bounded index and ordinary diﬀerential equations, Complex Var. Elliptic Equ. 12 (1989), 95–100.

[24]   M. Salmassi, Functions of bounded indices in several variables, Indian J. Math. 31 (1989), 249–257.

[25]   M. Sheremeta, Analytic Functions of Bounded Index, Mathematical Studies, Monograph Series 6, VNTL Publishers, Lviv, 1999.