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 ana- lytic 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, maxi- mum 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 appli- cations in analytic theory of linear differential equations. Moreover, vector-valued entire func- tions of bounded index in joint variables have applications to some system of partial differen- tial 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; +1), 0 = (0; 0) 2 2 2 2 p 2 2 2 R+, 1 = (1; 1) 2 R+, R = (r1; r2) 2 R+, j(z; !)j = jzj + j!j . For A = (a1; a2) 2 R , 2 B = (b1; b2) 2 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 2 f1; 2g; the relation A ≤ B is defined in the similar way. For K = 2 (k1; k2) 2 Z+, let us denote K! = k1! · k2!. Addition, multiplication by scalar and conjugation 2 2 2 in C is defined componentwise. For z 2 C , w 2 C we define hz; wi = z1w1 + z2w2; where w1; w2 is the complex conjugate of w1; w2. 2 2 The bidisc f(z; !) 2 C : jz − z0j < r1; j! − !0j < r2g is denoted by D ((z0;!0);R), its 2 2 skeleton f(z; !) 2 C : jz − z0j = r1; j! − !0j = r2g is denoted by T ((z0;!0);R), the closed 2 2 2 2 polydisc f(z; !) 2 C : jz − z0j ≤ r1; j! − !0j ≤ r2g is denoted by D [(z0;!0);R], D = D (0; 1), 2 p 2 2 D = fz 2 C : jzj < 1g: The open ball f(z; !) 2 C : jz − z0j + j! − !0j < rg is de- 2 2 p 2 2 noted by B ((z0;!0); r), the sphere f(z; !) 2 C : jz − z0j + j! − !0j = rg is denoted 2 2 p 2 2 by S ((z0;!0); r), and the closed ball fz 2 C : jz − z0j + j!0 − !0j ≤ rg is denoted by 2 2 2 1 B [(z0;!0); r], B = B (0; 1), D = B = fz 2 C : jzj < 1g: 2 Let F (z; !) = (f1(z; !); f2(z; !)) be an analytic vector-function in B . Then at a point (a; b) 2 B2, the function F (z; !) has a bivariate Taylor expansion: 1 1 X X k m F (z; !) = Ckl(z − a) (! − b) ; k=0 m=0 k+m k+m 1 @ f1(z;!) @ f2(z;!) 1 (k;m) where Ckm = k!m! @zk@!m ; @zk@!m z=a;!=b = k!m! F (a; b): 2 2 Let L(z; !) = (l1(z; !); l2(z; !)), where lj(z; !): B ! R+ is a positive continuous function such that 2 β (2.1) 8(z; !) 2 B : lj(z; !) > ; 1 − pjzj2 + j!j2 p j 2 f1; 2g; where β > 2 is a some constant. The norm for the vector-function F : B2 ! C2 is defined as the sup-norm: kF (z; !)k = max fjfj(z; !)jg: 1≤j≤2 We write @i+jF (z; !) @i+jf (z; !) @i+jf (z; !) F (i;j)(z; !) = = 1 ; 2 : @zi@!j @zi@!j @zi@!j An analytic vector-function F : B2 ! C2 is said to be of bounded L-index (in joint variables), if there exists n0 2 Z+ such that 2 2 8(z; !) 2 B 8(i; j) 2 Z+ : kF (i;j)(z; !)k kF (k;m)(z; !)k ≤ max : k; m 2 ; k + m ≤ n : (2.2) i j k m Z+ 0 i!j!l1(z; !)l2(z; !) k!m!l1 (z; !)l2 (z; !) 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 con- sidered 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: 8R 2 R+; jRj ≤ β; j 2 f1; 2g : 0 < λ1;j(R) ≤ λ2;j(R) < 1; where lj(z; !) 2 (2.3) λ1;j(R) = inf inf :(z; !) 2 [(z0;!0); R=L(z0;!0)] ; 2 D (z0;!0)2B lj(z0;!0) lj(z; !) 2 (2.4) λ2;j(R) = sup sup :(z; !) 2 D [(z0;!0); R=L(z0;!0)] : 2 l (z ;! ) (z0;!0)2B j 0 0 We need some propositions from [1,2]. For an analytic vector-function F : B2 ! C2, we put 2 M(R; (z0;!0);F ) = max kF (z; !)k :(z; !) 2 T ((z0;!0);R) ; 2 2 where (z0;!0) 2 B , R 2 R+: Then 2 M(R; (z0;!0);F ) = max kF (z; !)k :(z; !) 2 D ((z0;!0);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 2 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 2 R+, R < R , jR j ≤ β there exists p1 = p1(R ;R ) ≥ 1 such 2 that for every (z0;!0) 2 B inequality R00 R0 (2.5) M ; (z0;!0);F ≤ p1M ; (z0;!0);F L(z0;!0) L(z0;!0) holds. 3. GROWTH ESTIMATES OF ANALYTIC VECTOR-VALUED FUNCTIONS IN THE UNIT BALL 2 2 2 We put [0; 2π] = [0; 2π]×[0; 2π]. For R = (r1; r2) 2 R+, Θ = (θ1; θ2) 2 [0; 2π] , A = (a1; a2) 2 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 2 R+ with jRj < 1 and j 2 f1; 2g; l (ReiΘ2 ) max j ≤ c: 2 iΘ1 Θ1;Θ22[0;2π] lj(Re ) L(z; w) = (l (jzj; jwj); l (jzj; jwj)) ; L 2 K( 2) β = pβ ; pβ In the case 1 2 we have that B . Put 2 2 . 12 Vita Baksa, Andriy Bandura and Oleh Skaskiv p 2 T 2 2 2 Theorem 3.2. Let L 2 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 maxfjF (z; w)j:(z; w) 2 T (0;R)g = Z r1 Z r2 iθ1 iθ2 0 = O min min l1(te ; r2e )dt + l2(r ; t)dt ; 2 1 Θ2[0;2π] 0 0 Z r1 Z r2 iθ1 iθ2 0 (3.6) min l1(te ; r2e )dt + l2(r ; t)dt ; 2 1 Θ2[0;2π] 0 0 0 0 0 with jRj ! 1 − 0, R = (r1; r2) is a fixed radius.