Sufficient conditions of boundedness of L-index and analog of Hayman's Theorem for analytic functions in a ball

Andriy Bandura, Oleh Skaskiv


We generalize some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic in an unit ball functions.
Our propositions give an estimate maximum modulus of the analytic function on a skeleton in polydisc with the larger radii by
maximum modulus on a skeleton in the polydisc with the lesser radii.
An analog of Hayman's Theorem for the functions is obtained.
Also we established a connection between class of analytic in ball functions of bounded $l_j$-index in every direction $\mathbf{1}_j,$ $j\in\{1,\ldots,n\}$ and
class of analytic in ball of functions of bounded $\mathbf{L}$-index in joint variables, where
$l_j: \mathbb{B}^n\to \mathbb{R}_+$ is continuous function,
$\mathbf{1}_j=(0,\ldots,0, \underbrace{1}_{j-\mbox{th place}}, 0,\ldots,0)\in\mathbb{R}^n_{+},$ $z\in\mathbb{C}^n.$


analytic function; unit ball; bounded L-index in joint variables; maximum modulus; partial derivative; bounded L-index in direction

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Bandura, A., Skaskiv, O., Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p.

Bandura, A.I., Skaskiv, O.B., Directional logarithmic derivative and the distribution of zeros of an entire function of bounded L-index along the direction, Ukrain. Mat. J. 69 (1) (2017), 500–508, doi:10.1007/s11253-017-1377-8

Bandura, A., Skaskiv, O., Analytic in the unit ball functions of bounded L-index in direciton, (submitted in Rocky Mountain Journal of Mathematics)

Bandura, A.I., Bordulyak, M.T., Skaskiv, O.B., Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (1) (2016), 12–26,


Bandura, A., New criteria of boundedness of L-index in joint variables for entire functions, Math. Bull. Shevchenko Sci. Soc. 13 (2016), 58–67. (in Ukrainian)

Bandura, A.I., Petrechko, N.V., Skaskiv, O.B., Analytic functions in a polydisc of bounded L-index in joint variables, Mat. Stud. 46 (1) (2016), 72–80. doi:10.15330/ms.46.1.72-80

Bandura, A.I., Petrechko, N.V., Skaskiv, O.B., Maximum modulus of analytic in a bidisc functions of bounded L-index and analogue of Theorem of Hayman, Mathematica Bohemica (accepted for publication)

Bandura, A., Skaskiv, O., Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci. 227(1) (2017), 1–12, doi:10.1007/s10958-017-3570-6

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:10.17512/jamcm.2017.2.02

Chakraborty, B.C., Chanda, R., A class of entire functions of bounded index in several variables, J. Pure Math., 12 (1995), 16–21

Fricke, G.H., Entire functions of locally slow growth, J. Anal. Math., 28 (1975), no. 1, 101–122

Hayman, W.K., Differential inequalities and local valency, Pacific J. Math., 44 (1), (1973), no. 1, 117–137

Krishna, G.J., Shah, S.M., Functions of bounded indices in one and several complex variables, In: Mathematical essays dedicated to A.J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, 223–235

Kushnir, V.O., Sheremeta, M.M., Analytic functions of bounded l-index, Mat. Stud., 12 (1) (1999), no. 1, 59–66

Kushnir, V.O., Analogue of Hayman’s theorem for analytic functions of bounded l-index, Visn. Lviv Un-ty, Ser. Mekh.-Math. 53 (1999), 48–51. (in Ukrainian)

Nuray, F., Patterson, R.F., Entire bivariate functions of exponential type, Bull. Math. Sci. 5 (2) (2015), 171–177, doi:10.1007/s13373-015-0066-x

Nuray, F., Patterson, R.F., Multivalence of bivariate functions of bounded index, Le Matematiche 70 (2) (2015), 225–233, doi:10.4418/2015.70.2.14

Patterson, R.F., Nuray, F., A characterization of holomorphic bivariate functions of bounded index, Mathematica Slovaca, 67 (3) (2017), 731–736,


Salmassi, M., Functions of bounded indices in several variables, Indian J. Math.,

(3) (1989), 249–257

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

Sheremeta, M., Analytic functions of bounded index, Lviv: VNTL Publishers, 1999.

Strochyk, S.N., Sheremeta, M.M., Analytic in the unit disc functions of bounded

index, Dopov. Akad. Nauk Ukr. 1 (1993), 19–22. (Ukrainian)



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