Geodesic
Analog of Hayman's theorem and its application to some system of linear partial differential equations
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019) no. 2, pp. 170-191 Cet article a éte moissonné depuis la source Math-Net.Ru

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We used the analog of known Hayman's theorem to study the boundedness of $\mathbf{L}$-index in joint variables of entire solutions of some linear higher-order systems of PDE's and found sufficient conditions providing the boundedness, where $\mathbf{L}(z)=(l_1(z), \ldots, l_{n}(z)),$ $l_j:\mathbb{C}^n\to \mathbb{R}_+$ is a continuous function $j\in\{1,\ldots,n\}.$ Growth estimates of these solutions are also obtained. We proposed the examples of systems of PDE's which prove the exactness of these estimates for entire solutions. The obtained results are new even for the one-dimensional case because of the weakened restrictions imposed on the positive continuous function $l.$ 
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Andriy Bandura; Oleh Skaskiv. Analog of Hayman's theorem and its application to some system of linear partial differential equations. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019) no. 2, pp. 170-191. http://geodesic.mathdoc.fr/item/JMAG_2019_15_2_a1/

[1] A. Bandura, “New criteria of boundedness of $\mathbf{L}$-index in joint variables for entire functions”, Mat. Visn. Nauk. Tov. Shevchenka, 13 (2016), 58–67 (Ukrainian) | MR | Zbl

[2] A.I. Bandura, M.T. Bordulyak, O.B. Skaskiv, “Sufficient conditions of boundedness of L-index in joint variables”, Mat. Stud., 45:1 (2016), 12–26 | MR | Zbl

[3] A.I. Bandura, N.V. Petrechko, O.B. Skaskiv, “Analytic functions in a polydisc of bounded $\mathbf{L}$-index in joint variables”, Mat. Stud., 46:1 (2016), 72–80 | DOI | MR | Zbl

[4] A. Bandura, N. Petrechko, O. Skaskiv, “Maximum modulus in a bidisc of analytic functions of bounded $\mathbf{L}$-index and an analogue of Hayman's theorem”, Math. Bohem., 143:4 (2018), 339–354 | DOI | MR | Zbl

[5] A.I. Bandura, O.B. Skaskiv, “Entire functions of bounded $L$-index in direction”, Mat. Stud., 27:1 (2007), 30–52 (Ukrainian) | MR | Zbl

[6] A.I. Bandura, O.B. Skaskiv, “Sufficient sets for boundedness $L$-index in direction for entire functions”, Mat. Stud., 30:2 (2008), 177–182 | MR | Zbl

[7] A. Bandura, O. Skaskiv, Entire Functions of Several Variables of Bounded Index, Publisher I.E. Chyzhykov, Chyslo, Lviv, 2016

[8] J. Math. Sci. (N.Y.), 227:1 (2017), 1–12 | DOI | MR | Zbl

[9] A. Bandura, O. Skaskiv, Analytic Functions in the Unit Ball. Bounded $L$-Index in Joint Variables and Solutions of Systems of PDE's, LAP Lambert Academic Publishing, Beau–Bassin, 2017 | MR

[10] A. Bandura, O. Skaskiv, “Asymptotic estimates of entire functions of bounded $\mathbf{L}$-index in joint variables”, Novi Sad J. Math., 48:1 (2018), 103–116 | DOI | MR | Zbl

[11] A. Bandura, O. Skaskiv, P. Filevych, “Properties of entire solutions of some linear PDE's”, J. Appl. Math. Comput. Mech., 16:2 (2017), 17–28 | DOI | MR

[12] M.T. Bordulyak, “The space of entire in $\mathbb{C}^n$ functions of bounded $\mathbf{L}$-index”, Mat. Stud., 4 (1995), 53–58 (Ukrainian) | MR | Zbl

[13] M.T. Bordulyak, Boundedness of L-Index of Entire Functions of Several Complex Variables, Ph.D thesis, Lviv University, Lviv, 1995 (Ukrainian) | MR

[14] M.T. Bordulyak, “On the growth of entire solutions of linear differential equations”, Mat. Stud., 13:2 (2000), 219–223 | MR | Zbl

[15] M.T. Bordulyak, M.M. Sheremeta, “Boundedness of the $\mathbf{L}$-index of an entire function of several variables”, Dopov. Nats. Akad. Nauk Ukr., 1993, no. 9, 10–13 (Ukrainian) | MR

[16] B.C. Chakraborty, R. Chanda, “A class of entire functions of bounded index in several variables”, J. Pure Math., 12 (1995), 16–21 | MR | Zbl

[17] B.C. Chakraborty, T.K. Samanta, “On entire functions of bounded index in several variables”, J. Pure Math., 17 (2000), 53–71 | MR | Zbl

[18] B.C. Chakraborty, T.K. Samanta, “On entire functions of L-bounded index”, J. Pure Math., 18 (2001), 53–64 | MR | Zbl

[19] W.K. Hayman, “Differential inequalities and local valency”, Pacific J. Math., 44:1 (1973), 117–137 | DOI | MR | Zbl

[20] G.J. Krishna, S.M. Shah, “Functions of bounded indices in one and several complex variables”, Mathematical essays dedicated to A.J. Macintyre, Ohio Univ. Press, Athens–Ohio, 1970, 223–235 | MR

[21] V.O. Kushnir, “Analogue of Hayman theorem for analytic functions of bounded $l$-index”, Visn. Lviv Un-ty. Ser. Mekh. Math., 53 (1999), 48–51 (Ukrainian) | MR | Zbl

[22] B. Lepson, “Differential Equations of Infinite Order, Hyperdirichlet Series and Entire Functions of Bounded Index”, Entire Functions and Related Parts of Analysis (LaJolla, Calif., 1966), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I., 1968, 298–307 | DOI | MR

[23] F. Nuray, R.F. Patterson, “Entire bivariate functions of exponential type”, Bull. Math. Sci., 5:2 (2015), 171–177 | DOI | MR | Zbl

[24] F. Nuray, R.F. Patterson, “Multivalence of bivariate functions of bounded index”, Matematiche (Catania), 70:2 (2015), 225–233 | MR | Zbl

[25] Q.I. Rahman, J. Stankiewicz, “Differential inequalities and local valency”, Pacific J. Math., 54:2 (1974), 165–181 | DOI | MR | Zbl

[26] M. Salmassi, “Functions of bounded indices in several variables”, Indian J. Math., 31:3 (1989), 249–257 | MR | Zbl

[27] M. Salmassi, Some Classes of Entire Functions of Exponential Type in One and Several Complex Variables, Ph.D. thesis, University of Kentucky, Lexington, KY, 1978 | MR

[28] Russian Math. (Iz. VUZ), 36:9 (1992), 76–82 | MR | Zbl

[29] M. Sheremeta, Analytic Functions of Bounded Index, VNTL Publishers, Lviv, 1999 | MR | Zbl

[30] V. Singh, R.M. Goel, “Differential inequalities and local valency”, Houston J. Math., 18:2 (1992), 215–233 | MR | Zbl