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Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

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Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction axioms Article Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction Andriy Bandura 1,* and Oleh Skaskiv 2 1 Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, 76019 Ivano-Frankivsk, Ukraine 2 Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine * Correspondence: [email protected] Received: 25 June 2019; Accepted: 25 July 2019; Published: 26 July 2019 Abstract: In this paper, for a given direction b 2 Cn n f0g we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line fz0 + tb : t 2 Cg for any z0 2 Cn. Unlike to quaternionic analysis, we fix the direction b. The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable z1 and continuous in variable z2. For this class of functions there is introduced a concept of boundedness of L-index n in the direction b where L : C ! R+ is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index n in direction in any bounded domain and for any continuous function L : C ! R+. Keywords: bounded index; bounded L-index in direction; slice function; entire function; bounded l-index MSC: 32A10; 32A17; 32A37; 30H99; 30A05 1. Introduction In recent years, analytic functions of several variables with bounded index have been intensively investigated. The main objects of investigations are such function classes: entire functions of several variables [1–3], functions analytic in a polydisc [4], in a ball [5] or in the Cartesian product of the complex plane and the unit disc [6]. For entire functions and analytic function in a ball there were proposed two approaches to introduce a concept of index boundedness in a multidimensional complex space. They generate so-called functions of bounded L-index in a direction, and functions of bounded L-index in joint variables. Let us introduce some notations and definitions. ∗ n Let R+ = (0, +¥), R+ = [0, +¥), 0 = (0, ... , 0), b = (b1, ... , bn) 2 C n f0g be a given direction, n n L : C ! R+ be a continuous function, F : C ! C an entire function. The slice functions on a line 0 0 n 0 0 fz + tb : t 2 Cg for fixed z 2 C we will denote as gz0 (t) = F(z + tb) and lz0 (t) = L(z + tb). Definition 1 ([7]). An entire function F : Cn ! C is called a function of bounded L-index in a direction b, n if there exists m0 2 Z+ such that for every m 2 Z+ and for all z 2 C one has j¶mF(z)j j¶k F(z)j b ≤ b m max k , (1) m!L (z) 0≤k≤m0 k!L (z) Axioms 2019, 8, 88; doi:10.3390/axioms8030088 www.mdpi.com/journal/axioms Axioms 2019, 8, 88 2 of 12 n 0 ¶F(z) k k−1 where ¶bF(z) = F(z), ¶bF(z) = ∑ ¶z bj, ¶bF(z) = ¶b ¶b F(z) , k ≥ 2. j=1 j The least such integer number m0, obeying (1), is called the L-index in the direction b of the function F and is denoted by Nb(F, L). If such m0 does not exist, then we put Nb(F, L) = ¥, and the function F is said to be of unbounded L-index in the direction b in this case. If L(z) ≡ 1, then the function F is said to be of bounded index in the direction b and Nb(F) = Nb(F, 1) is called the index in the direction b. Let l : C ! R+ be a continuous function. For n = 1, b = 1, L(z) ≡ l(z), z 2 C inequality (1) defines a function of bounded l-index with the l-index N(F, l) ≡ N1(F, l) [8,9], and if in addition l(z) ≡ 1, then we obtain a definition of index boundedness with index N(F) ≡ N1(F, 1) [10,11]. It is also worth to mention paper [12], which introduces the 0 concept of generalized index. It is quite close to the bounded l-index. Let Nb(F, L, z ) stands for the 0 L-index in the direction b of the function F at the point z , i.e., it is the least integer m0, for which inequality (1) is satisfied at this point z = z0. By analogy, the notation N( f , l, z0) is defined if n = 1, i.e., in the case of functions of one variable. The concept of L-index boundedness in direction requires to consider a slice fz0 + tb : t 2 Cg. We fixed z0 2 Cn and used considerations from one-dimensional case. Then we construct uniform estimates above all z0. This is a nutshell of the method. In view of this, Prof S. Yu. Favorov (2015) posed the following problem in a conversation with one of the authors. n n Problem 1 ([13]). Let b 2 C n f0g be a given direction, L : C ! R+ be a continuous function. Is it possible to replace the condition “F is holomorphic in Cn” by the condition “F is holomorphic on all slices z0 + tb” and to deduce all known properties of entire functions of bounded L-index in direction for this function class? There is a negative answer to Favorov’s question [13]. This relaxation of restrictions by the function F does not allow the proving of some theorems. Here by D we denote a closure of domain D. There was proved the following proposition. Proposition 1 ([13], Theorem 5). For every direction b 2 Cn n f0g there exists a function F(z) and a bounded domain D ⊂ Cn with following properties: (1) F is holomorphic function of bounded index on every slice fz0 + tb : t 2 Cg for each fixed z0 2 Cn; (2) F is not entire function in Cn; (3) F does not satisfy (1) in D, i.e., for any p 2 Z+ there exists m 2 Z+ and zp 2 D m ( k ) j¶ F(zp)j j¶ F(zp)j b > max b : 0 ≤ k ≤ p . m! k! Let D be a bounded domain in Cn. If inequality (1) holds for all z 2 D instead Cn, then F is called function of bounded L-index in the direction b in the domain D. The least such integer m0 is called the n L-index in the direction b 2 C n f0g in the domain D and is denoted by Nb(F, L, D) = m0. Proposition 2 ([13], Theorem 2). Let D be a bounded domain in Cn, b 2 Cn n f0g be arbitrary direction. n 0 0 If L : C ! R+ is continuous function and F(z) is an entire function such that (8z 2 D) : F(z + tb) 6≡ 0, then Nb(F, L, D) < ¥. Hence, if we replace holomorphy in Cn by holomorphy on the slices fz0 + tb : t 2 Cg, then conclusion of Proposition2 is not valid. Thus, Proposition1 shows impossibility to replace joint holomorphy by slice holomorphy without additional hypothesis. The proof of Proposition2 uses Axioms 2019, 8, 88 3 of 12 continuity in joint variables (see [13], Equation (6)). It leads to the following question (see [14], where it is also formulated. There was considered a case L(z) ≡ 1). Problem 2. What are additional conditions providing validity of Proposition2 for slice holomorphic functions? A main goal of this investigation is to deduce an analog of Proposition2 for slice holomorphic functions. Please note that the positivity and continuity of the function L are weak restrictions to deduce constructive results. Thus, we assume additional restrictions by the function L. Let us denote L(z + t1b) h lb(h) = sup sup : jt1 − t2j ≤ . n L(z + t b) minfL(z + t b), L(z + t b)g z2C t1,t22C 2 1 2 n n By Qb we denote a class of positive continuous function L : C ! R+, satisfying the condition (8h ≥ 0) : lb(h) < +¥, (2) Moreover, it is sufficient to require validity of (2) for one value h > 0. b For a positive continuous function l(t), t 2 C, and h > 0 we define l(h) ≡ l1 (h) in the cases 1 when b = 1, n = 1, L ≡ l. As in [15], let Q ≡ Q1 be a class of positive continuous functions l(t), t 2 C, obeying the condition 0 < l(h) < +¥ for all h > 0. n n n Besides, we denote by ha, ci = ∑ ajcj the scalar product in C , where a, c 2 C . j=1 n 0 Let Heb be a class of functions which are holomorphic on every slices fz + tb : t 2 Cg for each 0 n n n z 2 C and let Hb be a class of functions from Heb which are joint continuous. The notation ¶bF(z) p (p) stands for the derivative of the function gz(t) at the point 0, i.e., for every p 2 N ¶bF(z) = gz (0), n where gz(t) = F(z + tb) is entire function of complex variable t 2 C for given z 2 C . In this research, we will often call this derivative as directional derivative because if F is entire function in Cn then the derivatives of the function gz(t) matches with directional derivatives of the function F.
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