A connection between L-index of vector-valued entire function and L-index of each its component

Vita Baksa (Ivan Franko National University of Lviv, Lviv, Ukraine) E-mail: [email protected] Andriy Bandura (Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine) E-mail: [email protected] Oleh Skaskiv (Ivan Franko National University of Lviv, Lviv, Ukraine) E-mail: [email protected]

The present talk is devoted to the properties of entire vector-valued functions of bounded L-index n n in join variables. We need some notations and definitions. Let L : C → R+ be any fixed continuous n p function. We consider a class of vector-valued entire functions F = (f1, . . . , fp): C → C . For this class of functions there was introduced a concept of boundedness of L-index in joint variables. p n Let k · k0 be a norm in C . Let L(z) = (l1(z), . . . , ln(z)), where lj(z): C → R+ is a positive n p continuous function. An entire vector-valued function F : C → C is said to be of bounded L-index n n in joint variables, if there exists n0 ∈ Z+ such that (∀z = (z1, . . . , zn) ∈ C )(∀J ∈ Z+): ( ) kF (J)(z)k kF (K)(z)k 0 ≤ max 0 : K ∈ n , kKk ≤ n , J!LJ (z) K!LK (z) Z+ 0 kJk (J) (J) (J) (J) ∂ where F (z) = (f (z), . . . , fp (z)), f (z) = fk(z), kJk = j1 +...+jn, J! = j1!·...·jn! 1 k j1 jn ∂z1 . . . ∂zn for J = (j1, . . . , jn), k ∈ {1, . . . , p}. The least such integer n0 is called the L-index in joint variables and is denoted by N(F, L). n  n Denote by D [z0,R/L(z0)] = z = (z1, . . . , zn) ∈ C : |zj −zj0| ≤ rj/lj(z0) for every j ∈ {1, . . . , n} n n n n the closed polydisc in C . Let Q be a class of continuous functions L: C → R+ such that 0 < n λ1,j(R) ≤ λ2,j(R) < ∞ for any j ∈ {1, 2, . . . , n} and ∀R = (r1, . . . , rn) ∈ R+, where λ1,j(R) = n inf inf {lj(z)/lj(z0): z ∈ D [z0,R/L(z0)]} , λ2,j(R) is defined analogously with replacement inf by z ∈ n sup0 C. n p For F : C → C let us introduce the sup-norm |F (z)|p = max1≤j≤p{|Fj(z)|}. The notation A ≤ B n for A = (a1, . . . , an),B = (b1, . . . , bn) ∈ R means that aj ≤ bj for every j ∈ {1, . . . , n}. The following proposition was firstly deduced for analytic curves in [1]. Similar proposition was also obtained for 2 2 2 2 2 2 analytic vector-valued functions F : B → C in the unit ball B = {z ∈ C : |z1| + |z2| < 1}[2]. Here n p we present it for vector-valued entire functions F : C → C . n Proposition 1. Let L = (l1(z), . . . , ln(z)) be a positive continuous function in C . If each component n p fs of an entire vector-valued function F = (f1, . . . , fp): C → C is of bounded L-index N(L, fs) in joint variables then F is of bounded L-index in joint variables in every norm, in particular, in the sup- norm and N(L; F ) ≤ max{N(L, f ): 1 ≤ s ≤ p}, and also F is of bounded L -index in the Euclidean √ s ∗ norm with L∗(z, w) ≥ pL(z, w) and NE(L∗,F ) ≤ max{N(L, fs) : 1 ≤ s ≤ p}. (Here N(L,F ) and NE(L∗,F ) are the L-index and the L∗-index in joint variables with the sup-norm and the Euclidean norm, respectively.)

1 2 A. Bandura, V. Baksa, O. Skaskiv

n n p Theorem 2 ([3]). Let L ∈ Q . An entire vector-valued function F : C → C has bounded L-index in n n joint variables if and only if for every R ∈ R+ there exist n0 ∈ Z+, p0 > 0 such that for all z0 ∈ C n there exists K0 ∈ Z+, kK0k ≤ n0, satisfying inequality (K) (K0) n|F (z)|p n o |F (z0)|p max K : kKk ≤ n0, z ∈ D [z0,R/L(z0)] ≤ p0 K . K!L (z) K0!L 0 (z0) This theorem is basic in the theory of functions of bounded index. Theorem 2 implies also the following corollary. n n p Corollary 3. Let L ∈ Q . An entire vector-function F : C → C has bounded L-index in joint variables in the sup-norm if and only if it has bounded L-index in joint variables in the norm k · k0. n n p Theorem 4. Let L ∈ Q . In order that an entire vector-valued function F : C → C be of bounded n L-index in joint variables it is necessary that for all R ∈ R+ there exist n0 ∈ Z+, p1 ≥ 1 such that for n n all z0 ∈ C there exists K0 ∈ Z+, kK0k ≤ n0, satisfying inequality

(K0) n (K0) max{|F (z)|p : z ∈ D [z0,R/L(z0)]}≤p1|F (z0)|p (1) n n 0 0 and it is sufficiently that for all R ∈ R+ there exist n0 ∈ Z+, p1 ≥ 1 ∀z0 ∈ C ∃K1 = (k1, 0,..., 0), 0 0 0 0 0 ∃K2 = (0, k2, 0,..., 0),..., ∃Kn = (0,..., 0, kn) : kj ≤ n0, and (K0) n (K0) (∀j ∈ {1, . . . , n}): max{|F j (z)|p : z ∈ D [z0,R/L(z0)]} ≤ p1|F j (z0)|p (2) References [1] Bordulyak M.T., Sheremeta M.M. Boundedness of l-index of analytic curves. Mat. Stud. 36 (2) : 152–161 (2011) [2] Baksa V.P. Analytic vector-functions in the unit ball having bounded L-index in joint variables. Carpathian Math. Publ., 11 (2) : 213–227 (2019). https://doi.org/10.15330/cmp.11.2.213-227 [3] Bandura A.I., Baksa V.P. Entire multivariate vector-valued functions of bounded -index: analog of Fricke’s theorem. Mat. Stud. 54 (1) : 56–63 (2020). https://doi.org/10.30970/ms.54.1.56-63