Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded L-Index in Joint Variables

CONSTRUCTIVE MATHEMATICAL ANALYSIS 3 (2020), No. 1, pp. 9-19 http://dergipark.gov.tr/en/pub/cma ISSN 2651 - 2939

Research Article Growth Estimates for Analytic Vector-Valued Functions in the Unit Ball Having Bounded L-index in Joint Variables

VITA BAKSA,ANDRIY BANDURA*, AND OLEH SKASKIV

ABSTRACT. Our results concern growth estimates for vector-valued functions of L-index in joint variables which are analytic in the unit ball. There are deduced analogs of known growth estimates obtained early for functions analytic in the unit ball. Our estimates contain logarithm of sup-norm instead of logarithm modulus of the function. They describe the behavior of logarithm of norm of analytic vector-valued function on a skeleton in a bidisc by behavior of the function L. These estimates are sharp in a general case. The presented results are based on bidisc exhaustion of a unit ball. Keywords: Bounded index, bounded L-index in joint variables, analytic function, unit ball, growth estimates, maximum norm. 2010 Mathematics Subject Classiﬁcation: 32A10, 32A17, 32A37.

1. INTRODUCTION In this paper, we consider vector-valued functions of bounded L-index in joint variables which are analytic in the unit ball. This paper is a continuation of investigations initiated in [1,2,3]. There was proposed the deﬁnition of L-index boundedness in joint variables and obtained some criteria of L-index boundedness in joint variables for vector-valued analytic functions in the unit ball. Here, we pose the following goal: to obtain growth estimates of analytic functions having bounded L-index in joint variables. It is important because functions of bounded index has many applications in analytic theory of linear differential equations. Moreover, vector-valued entire functions of bounded index in joint variables have applications to some system of partial differential equations [19]. Therefore, combination of sufﬁcient conditions of L-index boundedness for analytic solutions of the system with growth estimates of functions from this class will give a priori estimates of growth for all analytical solutions of the system. Other applications of concept of bounded index in analytic theory of differential equations were considered for various function classes: entire functions of bounded L-index in direction [12], entire functions of bounded L-index in joint variables [15], analytic functions in the unit ball having bounded L-index in joint variables [4], entire bivariate vector-valued function of bounded index [19].

Received: 26.11.2019; Accepted: 14.01.2020; Published Online: 28.01.2020 *Corresponding author: Andriy Bandura; [email protected] DOI: 10.33205/cma.650977

9 10 Vita Baksa, Andriy Bandura and Oleh Skaskiv

2. NOTATIONS, DEFINITIONS AND AUXILIARY PROPOSITIONS

We need some standard notations (for example see [5,4,6]). Let R+ = [0; +∞), 0 = (0, 0) ∈ 2 2 2 p 2 2 2 R+, 1 = (1, 1) ∈ R+, R = (r1, r2) ∈ R+, |(z, ω)| = |z| + |ω| . For A = (a1, a2) ∈ R , 2 B = (b1, b2) ∈ R , we will use formal notations without violation of the existence of these B b1 b2 expressions: AB = (a1b1, a2b2), A/B = (a1/b1, a2/b2), A = (a1 , a2 ), and the notation A < B means that aj < bj, j ∈ {1, 2}; the relation A ≤ B is defined in the similar way. For K = 2 (k1, k2) ∈ Z+, let us denote K! = k1! · k2!. Addition, multiplication by scalar and conjugation 2 2 2 in C is defined componentwise. For z ∈ C , w ∈ C we define hz, wi = z1w1 + z2w2, where w1, w2 is the complex conjugate of w1, w2. 2 2 The bidisc {(z, ω) ∈ C : |z − z0| < r1, |ω − ω0| < r2} is denoted by D ((z0, ω0),R), its 2 2 skeleton {(z, ω) ∈ C : |z − z0| = r1, |ω − ω0| = r2} is denoted by T ((z0, ω0),R), the closed 2 2 2 2 polydisc {(z, ω) ∈ C : |z − z0| ≤ r1, |ω − ω0| ≤ r2} is denoted by D [(z0, ω0),R], D = D (0; 1), 2 p 2 2 D = {z ∈ C : |z| < 1}. The open ball {(z, ω) ∈ C : |z − z0| + |ω − ω0| < r} is de- 2 2 p 2 2 noted by B ((z0, ω0), r), the sphere {(z, ω) ∈ C : |z − z0| + |ω − ω0| = r} is denoted 2 2 p 2 2 by S ((z0, ω0), r), and the closed ball {z ∈ C : |z − z0| + |ω0 − ω0| ≤ r} is denoted by 2 2 2 1 B [(z0, ω0), r], B = B (0, 1), D = B = {z ∈ C : |z| < 1}. 2 Let F (z, ω) = (f1(z, ω), f2(z, ω)) be an analytic vector-function in B . Then at a point (a, b) ∈ B2, the function F (z, ω) has a bivariate Taylor expansion: ∞ ∞ X X k m F (z, ω) = Ckl(z − a) (ω − b) , k=0 m=0 k+m k+m 1 ∂ f1(z,ω) ∂ f2(z,ω) 1 (k,m) where Ckm = k!m! ∂zk∂ωm , ∂zk∂ωm z=a,ω=b = k!m! F (a, b). 2 2 Let L(z, ω) = (l1(z, ω), l2(z, ω)), where lj(z, ω): B → R+ is a positive continuous function such that 2 β (2.1) ∀(z, ω) ∈ B : lj(z, ω) > , 1 − p|z|2 + |ω|2 √ j ∈ {1, 2}, where β > 2 is a some constant. The norm for the vector-function F : B2 → C2 is defined as the sup-norm:

kF (z, ω)k = max {|fj(z, ω)|}. 1≤j≤2 We write ∂i+jF (z, ω) ∂i+jf (z, ω) ∂i+jf (z, ω) F (i,j)(z, ω) = = 1 , 2 . ∂zi∂ωj ∂zi∂ωj ∂zi∂ωj An analytic vector-function F : B2 → C2 is said to be of bounded L-index (in joint variables), if there exists n0 ∈ Z+ such that 2 2 ∀(z, ω) ∈ B ∀(i, j) ∈ Z+ : kF (i,j)(z, ω)k kF (k,m)(z, ω)k ≤ max : k, m ∈ , k + m ≤ n . (2.2) i j k m Z+ 0 i!j!l1(z, ω)l2(z, ω) k!m!l1 (z, ω)l2 (z, ω) The least such integer n0 is called the L-index in joint variables of the vector-function F and is denoted by N(F, L, B2). The concept of boundedness of L-index in joint variables were considered for other classes of analytic functions. They are differed domains of analyticity: the unit ball [5,4, 11, 13], the polydisc [8, 10], the Cartesian product of the unit disc and complex plane [9], n-dimensional complex space [7, 11, 14]. Vector-valued functions of one and several complex variables having bounded index were considered in [18, 20, 17, 23, 21, 19]. Growth estimates for analytic vector-valued functions in the unit ball having bounded L-index in joint variables 11

2 2 The function class Q(B ) is deﬁned as following: ∀R ∈ R+, |R| ≤ β, j ∈ {1, 2} :

0 < λ1,j(R) ≤ λ2,j(R) < ∞, where lj(z, ω) 2 (2.3) λ1,j(R) = inf inf :(z, ω) ∈ [(z0, ω0),R/L(z0, ω0)] , 2 D (z0,ω0)∈B lj(z0, ω0) lj(z, ω) 2 (2.4) λ2,j(R) = sup sup :(z, ω) ∈ D [(z0, ω0),R/L(z0, ω0)] . 2 l (z , ω ) (z0,ω0)∈B j 0 0 We need some propositions from [1,2]. For an analytic vector-function F : B2 → C2, we put

2 M(R, (z0, ω0),F ) = max kF (z, ω)k :(z, ω) ∈ T ((z0, ω0),R) ,

2 2 where (z0, ω0) ∈ B , R ∈ R+. Then

2 M(R, (z0, ω0),F ) = max kF (z, ω)k :(z, ω) ∈ D ((z0, ω0),R) , because the maximum modulus of the analytic vector-function in a closed bidisc is attained on its skeleton. To prove an growth estimates, we need the following theorem. The theorem gives sufﬁcient conditions by the estimate of maximum modulus on the skeleton of bidisc.

Theorem 2.1 ([2]). Let L ∈ Q(B2). If analytic vector-function F : B2 → C2 has bounded L-index in 0 00 2 0 00 00 0 00 joint variables, then for all R ,R ∈ R+, R < R , |R | ≤ β there exists p1 = p1(R ,R ) ≥ 1 such 2 that for every (z0, ω0) ∈ B inequality R00 R0 (2.5) M , (z0, ω0),F ≤ p1M , (z0, ω0),F L(z0, ω0) L(z0, ω0) holds.

3. GROWTH ESTIMATES OF ANALYTIC VECTOR-VALUEDFUNCTIONSINTHEUNITBALL 2 2 2 We put [0, 2π] = [0, 2π]×[0, 2π]. For R = (r1, r2) ∈ R+, Θ = (θ1, θ2) ∈ [0, 2π] , A = (a1, a2) ∈ C2, we will write

iΘ iθ1 iθ2 Re = (r1e , r2e ), arg A = (arg a1, arg a2).

2 Denote by K(B ) the class of positive continuous vector-valued functions L = (l1, l2) , where 2 2 every lj : B → R+ obeys inequality (2.1) and there exists c ≥ 1 such that for all R ∈ R+ with |R| < 1 and j ∈ {1, 2},

l (ReiΘ2 ) max j ≤ c. 2 iΘ1 Θ1,Θ2∈[0,2π] lj(Re ) L(z, w) = (l (|z|, |w|), l (|z|, |w|)) , L ∈ K( 2) β = √β , √β In the case 1 2 we have that B . Put 2 2 . 12 Vita Baksa, Andriy Bandura and Oleh Skaskiv √ 2 T 2 2 2 Theorem 3.2. Let L ∈ Q(B ) K(B ), β > c 2. If an analytic vector-function F : B → C has bounded L-index in joint variables, then 2 ln max{|F (z, w)|:(z, w) ∈ T (0,R)} = Z r1 Z r2 iθ1 iθ2 0 = O min min l1(te , r2e )dt + l2(r , t)dt ; 2 1 Θ∈[0,2π] 0 0 Z r1 Z r2 iθ1 iθ2 0 (3.6) min l1(te , r2e )dt + l2(r , t)dt , 2 1 Θ∈[0,2π] 0 0 0 0 0 with |R| → 1 − 0, R = (r1, r2) is a ﬁxed radius.

2 ∗ ∗ 2 β Proof. Let R > 0 , |R| > 1, Θ ∈ [0, 2π] , and a point (z , w ) ∈ T 0,R + L(ReiΘ) be such that β kF (z∗, w∗)k = max kF (z, w)k :(z, w) ∈ 2(0,R + ) . T L(ReiΘ) ∗ ∗ z r1 w r2 We put z0 = R+β/L(ReiΘ) , w0 = R+β/L(ReiΘ) . Thus, √ ∗ ∗ iΘ ∗ z r1 ∗ z β/(c 2l1(Re )) β |z0 − z | = − z = = √ , √ β √ β iΘ r1 + iΘ r1 + iΘ c 2l1(Re ) c 2l1(Re ) c 2l1(Re ) √ ∗ ∗ iΘ ∗ w r2 ∗ w β/(c 2l2(Re )) β |w0 − w | = − w = = √ , √ β √ β iΘ r2 + iΘ r2 + iΘ c 2l2(Re ) c 2l2(Re ) c 2l2(Re ) z∗r w∗r L(z , w ) = L 1 , 2 = 0 0 R + β/L(ReiΘ) R + β/L(ReiΘ) ∗ ∗ (R + β/L(ReiΘ))r ei arg z (R + β/L(ReiΘ))r ei arg w = L 1 , 2 = R + β/L(ReiΘ) R + β/L(ReiΘ) i arg z∗ i arg w∗ = L(r1e , r2e ). Since L ∈ K(B2), we have ∗ ∗ 1 cL(z , w ) = cL(r ei arg z , r ei arg w ) ≥ L(r eiθ1 , r eiθ2 ) ≥ L(z , w ). 0 0 1 2 1 2 c 0 0 2 e 2 β We will consider two skeletons (z0, w0), and (z0, w0), . By Theorem T L(z0,w0) T L(z0,w0) e 0 e 00 2.1, there exist p1 = p1 c , cβ ≥ 1 such that (2.5) is true for R = c , R = cβ : 2 β max kF (z, w)k :(z, w) ∈ T 0,R + iθ iθ ≤ L(r1e 1 , r2e 2 ) 2 β ≤ kF (z, w)k :(z, w) ∈ T (z0, w0), iθ iθ ≤ L(r1e 1 , r2e 2 ) 2 cβ ≤ kF (z, w)k :(z, w) ∈ T (z0, w0), ≤ L(z0, w0) 2 e ≤ p1 kF (z, w)k :(z, w) ∈ T (z0, w0), ≤ cL(z0, w0) 2 e (3.7) ≤ p1 kF (z, w)k :(z, w) ∈ T 0,R + iθ iθ . L(r1e 1 , r2e 2 ) Growth estimates for analytic vector-valued functions in the unit ball having bounded L-index in joint variables 13

+ 2 The function ln max{kF (z, w)k :(z, w) ∈ T (0,R)} is convex relative ln r1, ln r2 . Therefore, + 2 ln max{kF (z, w)k :(z, w) ∈ T (0,R)}− Z r1 + 2 0 A1(t, r2) (3.8) − ln max{kF (z, w)k :(z, w) ∈ T (0,R + (r1 − r1)e1)} = dt, 0 t r1

+ 2 ln max{kF (z, w)k :(z, w) ∈ T (0,R)}− Z r2 + 2 0 A2(r1, t) (3.9) − ln max{kF (z, w)k :(z, w) ∈ T (0,R + (r2 − r2)e2)} = dt 0 t r2

0 for each 0 < rj < rj, j{1, 2} , where functions A1(t, r2), A2(r1, t) are positive non-decreasing t. Then from (3.7), we obtain β ln p ≥ ln max kF (z, w)k :(z, w) ∈ 2 0,R + − 1 T L(ReiΘ) e − ln max kF (z, w)k :(z, w) ∈ 2 0,R + = T L(ReiΘ) ( β !) e + ( √ − 1)e1 = ln max kF (z, w)k :(z, w) ∈ 2 0,R + 2c − T L(ReiΘ) ( β !) e + ( √ − 1)e2 − ln max kF (z, w)k :(z, w) ∈ 2 0,R + 2c = T L(ReiΘ)

√ iΘ Z r1+β/(c 2l1(Re )) 1 β = A1 t, r2 + √ dt+ iΘ iΘ r1+1/l1(Re ) t c 2l2(Re ) √ iΘ Z r2+β/(c 2l2(Re )) 1 β + A2 r1 + √ dt ≥ iΘ iΘ r2+1/l2(Re ) t c 2l1(Re , t) √β − 1 ! √β − 1 ! 2c 1 2c ≥ ln 1 + iΘ A1 r1, r2 + iΘ + ln 1 + iΘ × r1l1(Re ) + 1 l2(Re ) r2l2(Re ) + 1 1 (3.10) ×A2 r1 + iΘ , r2 . l1(Re )

iΘ 0 Then, we have rjlj(Re ) −→ +∞ with |R| −→ 1 − 0 . We obtain, for j ∈ {1, 2} and rj ≥ rj :

√β − 1 ! √β − 1 √β − 1 2c 2c 2c ln 1 + iΘ ∼ iΘ ≥ iΘ . rjlj(Re ) + 1 rjlj(Re ) + 1 2rjlj(Re )

Thus, (3.10) implies that β iΘ 2 ln p1 iΘ A1 r1, r2 + √ l2(Re ) ≤ r1l1(Re ), c 2 √β − 1 c 2

β 2 ln p1 iΘ A2 r1 √ , r2 ≤ r2l2(Re ). c 2l (ReiΘ) √β − 1 1 c 2 14 Vita Baksa, Andriy Bandura and Oleh Skaskiv

0 0 0 0 Let R = (r1, r2), where rj it is chosen higher. From inequalities (3.8) and (3.9) , it follows that 2 ln max kF (z, w)k :(z, w) ∈ T (0,R) = Z r1 2 0 A1(t, r2) = ln max kF (z, w)k :(z, w) ∈ T (0,R + (r1 − r1)e1) + dt = 0 t r1 2 0 0 = ln max kF (z, w)k :(z, w) ∈ T (0,R + (r1 − r1)e1 + (r2 − r2)e2) + Z r1 A (t, r ) Z r2 A (r0, t) + 1 2 dt + 2 1 dt = 0 t 0 t r1 r2 2 0 2 ln p1 = ln max kF (z, w)k :(z, w) ∈ T (0,R ) + × √β − 1 2c Z r1 Z r2 iθ1 iθ2 0 iθ1 iθ2 × l1(te , r2e )dt + l2(r1e , te )dt ≤ 0 0 2 ln p Z r1 Z r2 1 iθ1 iθ2 0 iθ1 iθ2 ≤ (1 + O(1)) β l1(te , r2e )dt + l2(r1e , te )dt . √ − 1 0 0 c 2 Function ln max{kF (z, w)k :(z, w) ∈ T2(0,R)} is independent of Θ . We obtain 2 ln max{kF (z, w)k :(z, w) ∈ T (0,R)} = Z r1 Z r2 iθ1 iθ2 0 iθ1 iθ2 = O( min l1(te , r2e )dt + l2(r e , te )dt , 2 1 Θ∈[0,2π] 0 0 with |R| −→ 1 − 0 . Theorem is proved. 2 T 2 iΘ Corollary 3.1. If L ∈ Q(B ) K(B ), minΘ∈[0,2π]2 lj(Re ) is non-decreasing in each variable rk, k ∈ {1, 2}, j ∈ {1, 2}, k 6= j, an analytic vector-function F : B2 → C2 has bounded L-index in joint variables, then  2  Z rj 2 X (j) iΘ ln max{kF (z, w)k :(z, w) ∈ T (0,R)} = O  min lj(R e )dt , Θ∈[0,2π]2 j=1 0

(1) (2) as |R| −→ 1 − 0, with R = (t, r2), R = (r1, t). + tR iΘ We denote a = max{a, 0}, uj(t) = uj(t, R, Θ) = lj r∗ e , with a ∈ R, t ∈ R+, j ∈ {1, 2} , ∗ t r = max1≤j≤2 rj 6= 0 and r∗ |R| < 1. iΘ Theorem 3.3. Let L(Re ) be a positive continuously differentiable function in each variable rk, k ∈ {1, 2}, |R| < 1, Θ ∈ [0, 2π]2 . If the function L obeys inequality (2.1) and an analytic vector-function 2 2 2 2 F : B → C has bounded L-index in joint variables, then for each Θ ∈ [0, 2π] and for all R ∈ R+, |R| < 1 and (s, p) ∈ Z2, kF (s,p)(ReiΘ)k ln max s iΘ p iΘ : s + p ≤ N ≤ s!p!l1(Re )l2(Re ) kF (s,p)(0)k ≤ ln max s p : s + p ≤ N + s!p!l1(0)l2(0) Z r∗ n τ iΘ τ iΘo max (s + 1)l1 ∗ Re + (p + 1)l2 ∗ Re + 0 s+p≤N r r 0 + 0 + s(−u1(τ)) p(−u2(τ)) (3.11) + max τ + τ dτ. s+p≤N iΘ iΘ l1( r∗ Re ) l2( r∗ Re ) Growth estimates for analytic vector-valued functions in the unit ball having bounded L-index in joint variables 15

2 2 rj Proof. Let R ∈ R \{0}, Θ ∈ [0, 2π] . We put aj = r∗ , j ∈ {1, 2} and A = (a1, a2). Consider the function

kF (s,p)(AeiΘ)k (3.12) g(t) = max s iΘ p iΘ : s + p ≤ N , s!p!l1(Ae )l2(Ae )

iΘ iθ1 iθ2 where At = (a1t, a2t), Ate = (a1t , a2t ). Since the function

(s,p) iθ1 iθ2 kF (a1e , a2e )k p s iθ1 iθ2 iθ1 iθ2 s!p!l1(a1e , a2e )l2(a1e , a2e )

is continuously differentiable function of real variable t ∈ [0; +∞), outside the zero set of func- ∗ (s,p) iθ1 iθ2 r tion kF (a1e , a2e )k, then g(t) is also a continuously differentiable function on [0, |R| ) except for a countable set of points. d 0 Hence, in view of dr |g(r)| ≤ |g (r)| , which holds everywhere except r = t, where g(t) = 0 we obtain that:

(s,p) iθ1 iθ2 d kF (a1e , a2e )k p = s iθ1 iθ2 iθ1 iθ2 dt s!p!l1(a1e , a2e )l2(a1e , a2e ) 1 d (s,p) iθ1 iθ2 (s,p) iθ1 iθ2 = p kF (a1e , a2e )k + kF (a1e , a2e )k× s iθ1 iθ2 iθ1 iθ2 s!p!l1(a1e , a2e )l2(a1e , a2e ) dt d 1 1 × p ≤ p × s iθ1 iθ2 iθ1 iθ2 s iθ1 iθ2 iθ1 iθ2 dt s!p!l1(a1e , a2e )l2(a1e , a2e ) s!p!l1(a1e , a2e )l2(a1e , a2e ) (s+1,p) iθ1 iθ2 iθ1 (s,p+1) iθ1 iθ2 iθ2 × kF (a1e , a2e )a1e k + kF (a1e , a2e )a2e k −

(s,p) iθ1 iθ2 0 0 kF (a1e , a2e )k su1(t) pu2(t) − p + ≤ s iθ1 iθ2 iθ1 iθ2 iΘ iΘ s!p!l1(a1e , a2e )l2(a1e , a2e ) l1(Ate ) l2(Ate ) kF (s+1,p)(a eiθ1 , a eiθ2 )k 1 2 iθ1 iθ2 ≤ p a1(s + 1)l1(a1e , a2e )+ s+1 iθ1 iθ2 iθ1 iθ2 (s + 1)!p!l1 (a1e , a2e )l2(a1e , a2e ) kF (s,p+1)(a eiθ1 , a eiθ2 )k 1 2 iθ1 iθ2 + p+1 a2(p + 1)l2(a1e , a2e )+ s iθ1 iθ2 iθ1 iθ2 s!(p + 1)!l1(a1e , a2e )l2 (a1e , a2e ) (s,p) iθ1 iθ2 0 + 0 + kF (a1e , a2e )k s(−u1(t)) p(−u2(t)) (3.13) + p + . s iθ1 iθ2 iθ1 iθ2 iΘ iΘ s!p!l1(a1e , a2e )l2(a1e , a2e ) l1(Ate ) l2(Ate ) 16 Vita Baksa, Andriy Bandura and Oleh Skaskiv

For absolutely continuous functions h1, h2 and h(x) := max{hj(z, w) : 1 ≤ j ≤ 2}, one has 0 0 h (x) ≤ max{hj(z, w) : 1 ≤ j ≤ 2}, x ∈ [a, b]. The function g is absolutely continuous. There- fore, (3.13) implies that

(s,p) iθ1 iθ2 0 d kF (a1e , a2e )k g (t) ≤ max p : s + p ≤ N ≤ s iθ1 iθ2 iθ1 iθ2 dt s!p!l1(a1e , a2e )l2(a1e , a2e ) iΘ (s+1,p) iΘ a1(s + 1)l1(Ae )kF (Ae )k ≤ max p + s+p≤N s+1 iΘ iΘ (s + 1)!p!l1 (Ae )l2(Ae ) a (p + 1)l (AeiΘ)kF (s,p+1)(AeiΘ)k + 2 2 + s iΘ p+1 iΘ s!(p + 1)!l1(Ae )l2 (Ae ) (s,p) iΘ 0 + 0 + kF (Ae )k s(−u1(t)) p(−u2(t)) + s iΘ p iΘ iΘ + iΘ ≤ s!p!l1(Ae )l2(Ae ) l1(Ae ) l2(Ae ) iΘ iΘ ≤ g(t)( max {a1(s + 1)l1(Ae ) + a2(p + 1)l2(Ae )}+ s+p≤N 0 + 0 + s(−u1(t)) p(−u2(t)) + max iΘ + iΘ ) = s+p≤N l1(Ae ) l2(Ae ) = g(t)(β(t) + γ(t)), with iΘ iΘ β(t) = max a1(s + 1)l1(Ae ) + a2(p + 1)l2(Ae ) , s+p≤N 0 + 0 + s(−u1(t)) p(−u2(t)) γ(t) = max iΘ + iΘ . s+p≤N l1(Ae ) l2(Ae )

d Then, dt ln g(t) ≤ β(t) + γ(t) and Z t (3.14) g(t) ≤ g(0) exp (β(τ) + γ(τ))dτ, 0 because g(0) 6= 0. But, one has r∗A = R. It follows from (3.14) and (3.12) that kF (s,p)(ReiΘ)k kF (s,p)(0)k ln max s iΘ p iΘ : s + p ≤ N ≤ ln max s p : s + p ≤ N + s!p!l1(Re )l2(Re ) s!p!l1(0)l2(0) Z r∗ iΘ iΘ + max a1(s + 1)l1(Aτe ) + a2(p + 1)l2(Aτe ) dτ+ 0 s+p≤N Z r∗ 0 + 0 + s(−u1(τ)) p(−u2(τ)) + max iΘ + iΘ dτ. 0 s+p≤N l1(Aτe ) l2(Aτe ) Inequality (3.14) is true.

iΘ Proposition 3.1. Let L(Re ) be a positive continuously differentiable function in each variable rk,k ∈ {1, 2}, |R| < 1, Θ ∈ [0, 2π]2. If the function L obeys inequality (2.1) and an analytic vector-function F : B2 → C2 has bounded L-index N = N(F, L, B2) in joint variables and there exists C > 0 such that the function L satisﬁes the inequality

0 + (−(uj(t, R, Θ))t) (3.15) sup max max max rj t ≤ C t∈[0,r ] Θ∈[0,2π]2 1≤j≤2 2 iΘ |R|<1 ∗ r∗ lj ( r∗ Re ) Growth estimates for analytic vector-valued functions in the unit ball having bounded L-index in joint variables 17 and ln max{kF (z, w)k :(z, w) ∈ T2(0,R)} (3.16) lim|R|−→1−0 ≤ (C + 1)N + 1. R 1 iΘ maxΘ∈[0,2π]2 0 hR, L(τ, Re )idτ Proof. If the function L satisﬁes inequality (2.1), then Z 1 (3.17) max hR, L(τReiΘ)idτ −→ +∞, as|R| −→ 1 − 0. 2 Θ∈[0,2π] 0

P2 iΘ ∗ ∗ ∗ ∗ We put βe(t) = j=1 ajlj(Ate ). If in addition (3.15) holds, then for some s , p , s + p ≤ N and se, pe, se+ pe ≤ N, s∗(−u0 (t))+ p∗(−u0 (t))+ 1 2 0 + 0 + γ(t) iΘ + iΘ (−u (t)) (−u (t)) = l1(Ate ) l2(Ate ) ≤ s∗ 1 + p∗ 2 ≤ P2 iΘ a l2(AteiΘ) a l2(AteiΘ) βe(t) j=1 ajlj(Ate ) 1 1 2 2 ≤ (s∗ + p∗)C ≤ NC and β(t) a (s + 1)l (AteiΘ) + a (p∗ + 1)l (AteiΘ) a sl (AteiΘ) = 1 e 1 2 2 = 1 + 1e 1 + P2 iΘ a l (AteiΘ) βe(t) j=1 ajlj(Ate ) 1 1 a pl (AteiΘ) + 2 e 2 ≤ 1 + s + p ≤ 1 + N. iΘ e e a2l2(Ate )

iΘ R t ∗ ∗ But, kF (Ate )k ≤ g(t) ≤ g(0) exp 0 (β(τ)+γ(τ))dτ and r A = R. Put t = r . In view of (3.17), we have 2 iΘ ln max{kF (z, w)k :(z, w) ∈ T (0,R)} = ln max kF (Re )k ≤ Θ∈[0,2π]2 Z r∗ ≤ ln max g(r∗) ≤ ln g(0) + max (β(τ) + γ(τ))dτ ≤ 2 2 Θ∈[0,2π] Θ∈[0,2π] 0 Z r∗ ≤ ln g(0) + (NC + N + 1) max (βe(τ))dτ = 2 Θ∈[0,2π] 0 Z r∗ 2 X iΘ = ln g(0) + (NC + N + 1) max ajlj(Aτe )dτ = Θ∈[0,2π]2 0 j=1 Z r∗ 2 X rj τ iΘ = ln g(0) + (NC + N + 1) max lj( Re )dτ = Θ∈[0,2π]2 r r∗ 0 j=1 Z 1 2 X iΘ = ln g(0) + (NC + N + 1) max rjlj(τRe )dτ. Θ∈[0,2π]2 0 j=1

Then, (3.16) is true. The Proposition 3.1 is proved.

iΘ Proposition 3.2. Let L(Re ) be a positive continuously differentiable function in each variable rk,k ∈ {1, 2}, |R| < 1, Θ ∈ [0, 2π]2. If the function L obeys inequality (2.1) and an analytic vector-function F : B2 → C2 has bounded L-index N = N(F, L) in joint variables and ∗ 0 + 2 iΘ (3.18) r (−(uj(t, R, Θ))t t=r∗ ) /(rjlj (Re )) −→ 0 18 Vita Baksa, Andriy Bandura and Oleh Skaskiv for all Θ ∈ [0, 2π]2, j ∈ {1, 2}, with |R| −→ 1 − 0 , then ln max{kF (z, w)k :(z, w) ∈ T2(0,R)} (3.19) lim|R|−→1−0 ≤ N + 1. R 1 iΘ maxΘ∈[0,2π]2 0 hR, L(τ, Re )idτ

If L(z, w) = L(r1, r2) = L(R), then (3.18) can be rewritten in another form.

Corollary 3.2. Let L(R) be a positive continuously differentiable function in each variable rk, k ∈ {1, 2}, |R| < 1. If the function L obeys inequality (2.1) and an analytic vector-function F : B2 → C2 has bounded L-index N = N(F, L) in joint variables and for each j ∈ {1, 2}

hR, ∇lj(R)i 2 −→ 0, rjlj (R) with |R| −→ 1 − 0, then ln max{kF (z, w)k :(z, w) ∈ 2(0,R)} lim T ≤ N + 1, |R|−→1−0 R 1 0 hR, L(τR)idτ

∂l1(R) ∂l2(R) where ∇lj(R) = , . ∂r1 ∂r2 The main result in this section is following:

Theorem 3.4. Let L(R) = (l1(R), l2(R)), lj(R) be a positive continuously differentiable non-decreasing function in each variable rk, k ∈ {1, 2},|R| < 1. If the function L obeys inequality (1) and an analytic vector-function F : B2 → C2 has bounded L-index N = N(F, L) in joint variables, then ln max{kF (z, w)k :(z, w) ∈ 2(0,R)} lim T ≤ N + 1. |R|−→1−0 R 1 0 hR, L(τR)idτ iΘ Proof. Note that L(Re ) ≡ L(R) in this theorem. Since lj(R) is a positive continuously dif- tR 0 ferentiable non-decreasing function and uj(t) = uj(t, R) = lj r∗ , one has (uj(t, R))t ≤ 0. ∗ 0 + 2 Therefore, we obtain r (−(uj(t, R))t t=r∗ ) /(rjlj (R)) = 0. Thus, condition (3.18) is satisﬁed. Thus, the theorem is a direct consequence of the Proposition 3.2.

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IVAN FRANKO NATIONAL UNIVERSITYOF LVIV DEPARTMENT OF MECHANICSAND MATHEMATICS 1 UNIVERSYTETSKASTREET, 79000, LVIV,UKRAINE Email address: [email protected]

IVANO-FRANKIVSK NATIONAL TECHNICAL UNIVERSITYOF OILAND GAS DEPARTMENT OF ADVANCED MATHEMATICS 15 KARPATSKA STREET, 76019, IVANO-FRANKIVSK,UKRAINE ORCID: 0000-0003-0598-2237 Email address: [email protected]

IVAN FRANKO NATIONAL UNIVERSITYOF LVIV DEPARTMENT OF MECHANICSAND MATHEMATICS 1 UNIVERSYTETSKASTREET, 79000, LVIV,UKRAINE ORCID: 0000-0001-5217-8394 Email address: [email protected]