### Application of Hayman's theorem to directional differential equations with analytic solutions in the unit ball

#### Abstract

In this paper, we investigate analytic solutions of higher order linear non-homogeneous directional differential equations whose coefficients are analytic functions in the unit ball. We use methods of theory of analytic functions in the unit ball having bounded \(L\)-index in direction, where \(L: \mathbb{B}^n\to\mathbb{R}_+\) is a continuous function such that

\(L(z)>\frac{\beta|\mathbf{b}|}{1-|z|}\) for all \(z\in\mathbb{B}^n,\) \(\mathbf{b}\in\mathbb{C}^n\setminus\{0\}\) be a fixed direction,

\(\beta>1\) is some constant. Our proofs are based on application of inequalities from analog of Hayman's theorem for analytic functions in the unit ball.

There are presented growth estimates of their solutions which contains parameters depending on the coefficients of the equations. Also we obtained sufficient conditions that every analytic solution of the equation has bounded \(L\)-index in the direction. The deduced results are also new in one-dimensional case, i.e. for functions analytic in the unit disc.

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Bandura, A., Composition, product and sum of analytic functions of bounded l-index in direction in the unit ball, Mat. Stud., 50(2018), no. 2, 115-134.

Bandura, A., Petrechko, N., Skaskiv, O., Maximum modulus in a bidisc of analytic functions of bounded l-index and an analogue of Hayman's theorem, Math. Bohem., 143(2018), no. 4, 339-354.

Bandura, A., Skaskiv, O., Sufficient sets for boundedness L-index in direction for entire functions, Mat. Stud., 30(2008), no. 2, 177-182.

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., 22(2018), no. 2, 223-234.

Bandura, A., Skaskiv, O., Asymptotic estimates of entire functions of bounded L-index in joint variables, Novi Sad J. Math., 48(2018), no. 1, 103-116.

Bandura, A., Skaskiv, O., Su cient conditions of boundedness of L-index and analog of Hayman's theorem for analytic functions in a ball, Stud. Univ. Babes-Bolyai Math., 63(2018), no. 4, 483-501.

Bandura, A., Skaskiv, O., Functions analytic in the unit ball having bounded l-index in a direction, Rocky Mountain J. Math., 49(2019), no. 4, 1063-1092.

Bandura, A., Skaskiv, O., 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. Math., 52(2019), no. 1, 82-87.

Bandura, A., Skaskiv, O., Linear directional differential equations in the unit ball: Solutions of bounded l-index, Math. Slovaca, 69(2019), no. 5, 1089-1098.

Bordulyak, M.T., A proof of Sheremeta conjecture concerning entire function of bounded l-index, Mat. Stud., 12(1999), no. 1, 108-110.

Bordulyak, M.T., On the growth of entire solutions of linear differential equations, Mat. Stud., 13(2000), no. 2, 219{223.

Goldberg, A.A., Sheremeta, M.N., Existence of an entire transcendental function of bounded l-index, Math. Notes, 57(1995), no. 1, 88-90.

Hayman, W.K., Di erential inequalities and local valency, Paci c J. Math., 44(1973), no. 1, 117-137.

Kuzyk, A.D., Sheremeta, M.N., Entire functions of bounded l-distribution of values, Math. Notes, 39(1986), no. 1, 3-8.AG2

Kuzyk, A.D., Sheremeta, M.N., On entire functions, satisfying linear di erential equations, (in Russian), Di . Equ., 26(1990), no. 10, 1716-1722.

Lepson, B., Di erential equations of in nite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 11(1968), 298-307.

Nuray, F., Patterson, R.F., Multivalence of bivariate functions of bounded index, Le Matematiche, 70(2015), no. 2, 225-233.

Nuray, F., Patterson, R.F., Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49(2018), no. 1, 67-74.

Sheremeta, M.N,, An l-index and an l-distribution of the values of entire functions, Soviet Math. (Iz. VUZ), 34(1990), no. 2, 115-117.

Sheremeta, M.N., Entire functions and Dirichlet series of bounded l-index, Russian Math. (Iz. VUZ), 36(1992), no. 9, 76-82.

Sheremeta, M.N., Analytic Functions of Bounded Index, Vol. 6 of Mathematical Studies. Monograph Series, VNTL Publishers, Lviv, 1999.

DOI: http://dx.doi.org/10.24193/subbmath.2024.2.06

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