Some Results on Composition of Analytic Functions in a Unit Polydisc

Year 2024, Volume: 7 Issue: 3, 121 - 128, 21.09.2024

Andriy Bandura , Petro Kurliak , Oleh Skaskiv

https://doi.org/10.32323/ujma.1444221

Abstract

The manuscript is an attempt to consider all methods which are applicable to investigation a directional index for composition of an analytic function in some domain and an entire function. The approaches are applied to find sufficient conditions of the $L$-index boundedness in a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$, where the continuous function $L$ satisfies some growth condition and the condition of positivity in the unit polydisc. The investigation is based on a counterpart of the Hayman Theorem for the class of analytic functions in the polydisc and a counterpart of logarithmic criterion describing local conduct of logarithmic derivative modulus outside some neighborhoods of zeros. The established results are new advances for the functions analytic in the polydisc and in multidimensional value distribution theory.

Keywords

Analytic function, Finite directional $L$-index, Boundedness of $L$-index in a direction, $L$-index in direction, Composition, Directional derivative, Entire function, Several complex variables, Unit polydisc

References

• [1] A. Bandura, T. Salo, Analytic in a unit polydisc functions of bounded L-index in direction, Mat. Stud., 60(1) (2023), 55–78.
• [2] V. P. Baksa, A. I. Bandura, T. M. Salo, Skaskiv O.B., Note on boundedness of the L-index in the direction of the composition of slice entire functions, Mat. Stud., 58 (1) (2022), 58–68.
• [3] A. I. Bandura, M. M. Sheremeta, Bounded l-index and l􀀀M-index and compositions of analytic functions, Mat. Stud., 48(2) (2017), 180-188.
• [4] M. M. Sheremeta, On the l-index boundedness of some composition of functions, Mat. Stud., 47(2) (2017), 207–210.
• [5] B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, in: Entire Functions and Related Parts of Analysis, J. Korevaar (ed.), Proceedings of Symposia in Pure Math., 11, Am. Math. Soc., Providence (1968), 298–307.
• [6] A. D. Kuzyk, M. M. Sheremeta, Entire functions of bounded l-distribution of values, Math. Notes, 39(1) (1986), 3–8.
• [7] A. I. Bandura, Composition, product and sum of analytic functions of bounded L-index in direction in the unit ball, Mat. Stud., 50(2) (2018), 115–134.
• [8] A. Bandura, Composition of entire functions and bounded L-index in direction, Mat. Stud., 47(2) (2017), 179–184.
• [9] A. I. Bandura, O. B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27(1) (2007), 30–52. (in Ukrainian)
• [10] W. K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1) (1973), 117-137.
• [11] A. I. Bandura, O. B. Skaskiv, I. R. Tymkiv, Composition of entire and analytic functions in the unit ball, Carpathian Math. Publ., 14 (1) (2022), 95–103.
• [12] M. M. Sheremeta, Y.S. Trukhan, Boundedness of the l-index of the Naftalevich-Tsuji product, Ukr. Math. J., 56(2) (2004), 305–317.
• [13] A. Bandura, O. Skaskiv, L. Smolovyk, Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction, Demonstr. Math., 52(1) (2019), 482–489.
• [14] A. A. Goldberg, M. N. Sheremeta, Existence of an entire transcendental function of bounded l-index, Math. Notes, 57(1) (1995), 88–90.
• [15] I. M. Hural, About some problem for entire functions of unbounded index in any direction, Mat. Stud., 51(1) (2019), 107–110.
• [16] M. M. Sheremeta, Y. S. Trukhan, Properties of analytic solutions of three similar differential equations of the second order, Carp. Math. Publ., 13(2) (2021), 413–425.
• [17] M. M. Sheremeta, Y. S. Trukhan, Properties of analytic solutions of a differential equation, Mat. Stud., 52 (2) (2019), 138–143.
• [18] A. Bandura, O. Skaskiv, Analog of Hayman’s Theorem and its Application to Some System of Linear Partial Differential Equations, J. Math. Phys., Anal., Geom., 15(2) (2019), 170–191.
• [19] F. Nuray, R.F. Patterson, Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49(1) (2018), 67–74.
• [20] A. I. Bandura, Some improvements of criteria of L-index boundedness in direction, Mat. Stud., 47(1) (2017), 27–32.
• [21] A. I. Bandura, Analytic functions in the unit ball of bounded value L-distribution in a direction, Mat. Stud., 49 (1) (2018), 75–79.
• [22] G. H. Fricke, A note on bounded index and bounded value distribution, Indian J. Pure Appl. Math. 11 (4) (1980), 428–432.
• [23] S. Shah, Entire functions of bounded value distribution and gap power series, In: Studies in Pure Mathematics To the Memory of Paul Tur´an, (P. Erd˝os, L. Alp´ar, G. Hal´asz, A. S´ark¨ozy, eds.). Birkhauser Basel, Basel, 1983. pp. 629-634.
• [24] R. Roy, S. M. Shah, The product of two functions of bounded value distribution, Indian J. Pure Appl. Math. 17(5) (1986), 690–693.
• [25] R. Roy, S. M. Shah, Functions of bounded index, bounded value distribution and v-bounded index, Nonlinear Analysis 11 (1987), 1383–1390.
• [26] M. M. Sheremeta, On the univalence of entire functions of bounded l-index, Mat. Stud., 43(2) (2015), 185–188.
• [27] F. Nuray, R. F. Patterson, Multivalence of bivariate functions of bounded index, Le Matematiche, 70(2) (2015), 225–233.
• [28] A. Bandura, T. Salo, O. Skaskiv, L-Index in Joint Variables: Sum and Composition of an Entire Function with a Function With a Vanished Gradient, Fractal and Fractional, 7(8) (2023), Article ID: 593.
• [29] F. Nuray, R. F. Patterson, Entire bivariate functions of exponential type, Bull. Math. Sci. 2015, 5 () (2015), 171–177.
• [30] F. Nuray, Bounded index and four dimensional summability methods, Novi Sad J. Math., 49(2) (2019), 73–85.
• [31] R. F. Patterson, F. A. Nuray, A characterization of holomorphic bivariate functions of bounded index, Math. Slov., 67(3) (2017), 731–736.