У цiй статтi отримано необхiднi i достатнi умови обмеженостi $\mathbb{L}$-iндексу за сукупнiстю змiнних для векторнозначних функiй, аналiтичних в одиничнiй кулi, де $\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$— додатна неперервна вектор-функцiя, $\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$ Цi умови описують локальне поводження однорiдних багаточленiв (так званих головних багаточленiв) з розвинення у степеневий ряд аналiтичних в одиничнiй кулi векторнозначних функцiй. Отриманi результати використовують бiкругове вичерпання одиничної кулi.

[1] Baksa V.P., Analytic vector-functions in the unit ball having bounded $L$-index in joint variables, Carpathian Mathematical Publications (in print).

[2] Baksa V.P., Bandura A.I., Skaskiv O.B., Analogs of Fricke's theorems for analytic vector-valued functions in the unit ball having bounded $L$-index in joint variables, submitted to Proceedings of IAMM of NASU.

[3] Baksa V.P., Bandura A.I., Skaskiv O.B., Analogs of Hayman's theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded $L$-index in joint variables. submitted to Matematica Slovaca.

[4] Bandura A. I., Skaskiv O. B. Analytic functions in the unit ball of bounded $L$-index: asymptotic and local properties. Mat. Stud. 2017, 48 (1), 37-73. doi: 10.15330/ms.48.1.37-73 

[5] Bandura A., Skaskiv O. Sufficient conditions of boundedness of $L$-index and analog of Hayman's Theorem for analytic functions in a ball. Stud. Univ. Babe¸s-Bolyai Math. 2018, 63 (4), 483-501. doi:10.24193/subbmath.2018.4.06

[6] Bandura A., Skaskiv O. Functions analytic in the unit ball having bounded $L$-index in a direction. Rocky Mountain J. Math. 2019, 49 (4), 1063-1092. doi: 10.1216/RMJ-2019-49-4-1063 

[7] 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. Mat. Bohemica 2018, 143 (4), 339-354. doi: 10.21136/MB.2017.0110-16

[8] Bandura A.I., Skaskiv O.B., Tsvigun V.L. Some characteristic properties of analytic functions in $\mathbb{D}×\mathbb{C}$ of bounded $L$-index in joint variables. Bukovyn. Mat. Zh. 2018, 6 (1-2), 21-31.

[9] Bandura A., Petrechko N. Properties of power series expansion of entire function of bounded $L$-index in joint variables. Visn. Lviv. Un-ty. Ser. Mech.-Math. 2016, Iss. 82, 27-33.

[10] Bandura A.I., Petrechko N.V. Properties of power series of analytic in a bidisc functions of bounded $L$-index in joint variables. Carpathian Math. Publ. 2017, 9 (1), 6-12. doi: 10.15330/cmp.9.1.6-12

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

[12] 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. 2019, 52 (1), -87. doi: 10.1515/dema-2019-0008

[13] Bandura A.I., Skaskiv O.B. Partial logarithmic derivatives and distribution of zeros of analytic functions in the unit ball of bounded $L$-index in joint variables. J. Math. Sci. 2019, 239 (1), 17-29. doi:10.1007/s10958-019-04284-z

[14] Bandura A.I., Skaskiv O.B. Exhaustion by balls and entire functions of bounded $L$-index in joint variables, Ufa Math. J. 2019, 11 (1), 100-113. doi: 10.13108/2019-11-1-100

[15] Bordulyak M.T., Sheremeta M.M. Boundedness of $l$-index of analytic curves. Mat. Stud. 2011, 36 (2), 152-161.

[16] Heath L.F. Vector-valued entire functions of bounded index satisfying a differential equation. Journal of Research of NBS 1978, 83B (1), 75-79.

[17] Nuray F., Patterson R.F. Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations. Mat. Stud. 2018, 49 (1), 67-74. doi: 10.15330/ms.49.1.67-74

[18] Roy R., Shah S.M. Growth properties of vector entire functions satisfying differential equations. Indian J. Math. 1986, 28 (1), 25-35.

[19] Roy R., Shah S.M. Vector-valued entire functions satisfying a differential equation. J. Math. Anal. Appl. 1986, 116 (2), 349-362.

[20] Sheremeta M. Boundedness of $l − M$-index of analytic curves. Visnyk Lviv Un-ty. Ser. Mech.-Math. Iss. 75, 226-231 (2011)