Abstract
In this paper, we obtain new sufficient conditions of boundedness of -index in joint variables for entire function in functions. They give an estimate of maximum modulus of an entire function by its minimum modulus on a skeleton in a polydisc and describe the behavior of all partial logarithmic derivatives and the distribution of zeros. In some sense, the obtained results are new for entire functions of bounded index and -index in too. They generalize known results of Fricke, Sheremeta, and Kuzyk.
1. Introduction
In this paper, we find multidimensional sufficient conditions of boundedness of -index in joint variables, which describe distribution of zeros and behavior of partial logarithmic derivatives. Recently, we published a paper [1] where some similar restrictions are established. Another approach was used by a slice function , where is a given direction in is an entire function. It is a background for concept of function of bounded -index in direction (see definition and properties in [2, 3]). We proved that if an entire function in function is of bounded -index in every direction , then is of bounded -index in joint variables for (Theorem , [1]). It helped us to find restrictions by directional logarithmic derivatives and distribution zeros in every direction . We assumed that the logarithmic derivative in direction is bounded by a function outside some exceptional set, which contains all zeros of entire function (see definition of below). Prof. Chyzhykov paid attention in conversation with authors that this exceptional set is too small because it does not contain neighborhoods of some zeros of the function in Thus, it leads to the following question: is there sufficient conditions of boundedness of -index in joint variables with larger exceptional sets? We give a positive answer to this question (Theorem 10). Moreover, we obtain sufficient conditions of boundedness of -index in joint variables by estimating the maximum modulus of an entire function on the skeleton in polydisc by minimum modulus (Theorem 7). Theorems 9 and 10 present restrictions by a measure of zero set of an entire function , under which has bounded -index in joint variables. Nevertheless, we do not know whether the obtained conditions in Theorems 7–10 are necessary too in Note that these propositions are new even for entire functions of bounded index in joint variables, i. e. (see definition and properties in [4–8]).
It is known [9] that for every entire function with bounded multiplicities of zeros there exists a positive continuity on function () such that is of bounded -index. This result can be easily generalized for entire functions in Thus, the concept of bounded -index in joint variables allows the study of growth properties of any entire functions with bounded multiplicities of zero points.
It should be noted that the concepts of bounded -index in a direction and bounded -index in joint variables have few advantages in the comparison with traditional approaches to study properties of entire solutions of differential equations. In particular, if an entire solution has bounded index [10], then it immediately yields its growth estimates, a uniform distribution of its zeros, a certain regular behavior of the solution, and so forth. A full bibliography about application in theory of ordinary and partial differential equations is in [3, 11, 12].
The paper is devoted to two old problems in theory of entire and meromorphic functions. The first problem is the establishment of sharp estimates for the logarithmic derivatives of the functions in the unit disc outside some exceptional set. Chyzhykov et al. [13–16] considered various formulations of the problem. The obtained estimates were used to study properties of holomorphic solutions of differential equations. Instead, the authors assume that partial logarithmic derivative in every variable satisfies some inequalities (28) or (45).
Another interesting considered problem concerns zero sets of holomorphic function in The different estimates of measure of zero set and its geometrical properties are investigated in [17–22]. We suppose that zero points of entire functions admit uniform distribution in some sense, that is, (29).
Below we use results from Ukrainian papers [23, 24], but they are also included in English monographs [3, 11].
2. Main Definitions and Notations
We need some standard notations. Let . Denote , , .
For and denote , . For , , , we will use formal notations without violating the existence of these expressions:If the notation means that ; similarly, the relation is defined.
The polydisc is denoted by , its skeleton is denoted by , and the closed polydisc is denoted by For and partial derivatives of entire function we will use the notation
Let , where are positive continuous functions of , . An entire function, , is called a function of bounded -index in joint variables [1] if there exists a number such that for all and
If then we obtain a concept of entire functions of bounded -index in a sense of definition given in [24]. If , then the entire function is called a function of bounded index in joint variables [4–8, 25].
The least integer for which inequality (3) holds is called -index in joint variables of the function and is denoted by .
For , and we define
By we denote a class of functions which for every and satisfy the conditionIf then .
Let . A notation means that there exist , such that .
3. Auxiliary Propositions
We need the following theorems.
Theorem 1 ([11, p. 158, Th. 4.2], see also [23]). Let and . An entire function has bounded -index in joint variables if and only if has bounded -index in joint variables.
Theorem 2 (see [1]). Let . An entire function is of bounded -index in joint variables if and only if, for any , , , there exists a number such that for every inequalityholds.
Remark 3. It was also proved that the condition “for any , , , there exists a number ” in Theorem 2 can be replaced by the condition “there exist , , , and ”. It is Theorem in [1].
Now we relax the restriction in sufficient conditions.
Theorem 4. Let , be an entire function. If there exist , , , and such that for every inequality (6) holds; then the function has bounded -index in joint variables.
Proof. From (6) with it follows thatDenoting , we obtain where . In view of Remark 3, has bounded -index in joint variables. By Theorem 1, the function is bounded -index in joint variables.
Note that Theorem 4 is new even if .
Lemma 5. If is a continuous function such that then .
Proof. Let i.e. . Hence, we have for . This means that . Using definition of , we deduceThus,
Remark 6. By Lemma 5 the left inequality in (5) is excessive because the condition implies . But in our considerations we will use so as It is convenient.
4. Estimate Maximum Modulus on a Skeleton in Polydisc
Let be a zero set of entire function We denote
Theorem 7. Let be an entire in function. If , , , ( for ) such that , , for whichthen the function has bounded -index in joint variables ( is the Lebesgue measure on the skeleton in the polydisc).
Proof. By Theorem 4, we will show that Denote , , , . The following estimate holds Hence, there exists independent of such thatfor some because . Indeed,Thus, , where is the integer part of .
Let and be such a point in thatwhere , and be the intersection point in of the segment with . We construct a sequence of polydisc with , and (see Figures 1 and 2).
Denote . Hence, for every and . Thus, for some , , we deduce To deduce (18), we implicitly used thatCondition (11) provides (19). Indeed, we will find a lower estimate of measure of the set and will show that the measure is not lesser than a left part of inequality (11).
The set is a Cartesian product of the following arcs on circles: for every (see Figure 3)and for (see Figure 4)It is easy to prove that the length of arc equals But for and the argument in arccosine from (23) and (22) does not exceed . This means that the length of arc is not lesser thanbecause . Accordingly, the measure of the seton the skeleton of polydisc is always not lesser than . Assuming a strict inequality in (11), we deduce that (19) is valid.
Applying (18) th times in every variable , we obtainBy Theorem 2 the function has bounded -index in joint variables.
Let us denote . For Theorem 7 implies the following corollary.
Corollary 8. Let be an entire function. If , , , such that , and and then the function has bounded -index (here means the Lebesgue measure on the circle).
In a some sense, this corollary is new even for an entire function of one variable because the circle can contain zeros of the function Meanwhile, in corresponding theorem from [26, 27] the circle is chosen such that for all
5. Behavior of Partial Logarithmic Derivatives
Denote .
Theorem 9. Let . If an entire function satisfies the following conditions (1)for every there exists such that for all and for all where is the principal value of logarithm.(2)for every and exists that for all such that , where the sets are connected disjoint sets, and either (a) , or (b) , or (c) , , and belong to the same set (3)for every there exists such that for all then has bounded -index in joint variables (here is -dimensional of the Lebesgue measure).
Proof. Let be arbitrarily chosen point. In view of Theorem 7 we need to prove thatLet be arbitrary radius. We choose , such that , Let , , be a volume measure in . Clearly, (see [28, p. 75-76])where is plurisubharmonic function. Hence, Obviously, there can exist points such that Let be the intersection point of the segment and the circle . Then and . Using , we estimate maximum distance between and Denote . Let be a -dimensional volume, a characteristic function of zero set of the function . Now we replace the measure in (33) by integrating on zero set in polydisc :Besides, we have thatHence, the following difference is positivebecause . From (36) it follows thatBy mean value theorem there exists with such thatHence, in view of (39) we obtain a desired inequalityClearly, for every point we have , where are connected disjoint sets, and is defined in condition (2). Let be such that . Then there exists such that . Let be such that . We choose , whereand deduceHence, By Theorem 7 the function has bounded -index in joint variables.
Let us to denote as Laplace operator. We will consider as generalized function. Using some known results from potential theory, we can rewrite Theorem 9 as follows.
Theorem 10. Let . If an entire function satisfies the following conditions (1)for every there exists such that for all and for every where is the principal value of logarithm.(2)for every and exists that for all such that , where the sets are connected disjoint sets, and either (a) , or (b) , or (c) , , and , belong to the same set (3)for every there exists such that for all then has bounded -index in joint variables.
Proof. Ronkin [28, p. 230] deduced the following formula for entire function:where is a multiplicity of zero point of the function at point , is arbitrary radius. Let be a characteristic function of zero set of . Then . Hence, that is, inequality (29) holds.
Now we want to prove that (45) implies (28). For every and , Cauchy’s integral formula can be written in the following formwhere is such that .
If and , then for every Let us consider the set . We want to find the greatest radius such that :Thus, for . Using (45), we obtain that for every where . Thus, we proved that inequality (28) is valid.
For Theorem 9 implies the following corollary.
Corollary 11. Let be an entire in function, a number of zeros of the in the disc . If the function satisfies the following conditions: (1)for every there exists such that for all (2)for every and exists that for all such that , where the sets are connected disjoint sets, and either (a) , or (b) , or (c) , , and , belong to the same set (3)for every there exists such that for all , then has bounded -index.
It is known (see [12, 27, 29]) that in one-dimensional case conditions (1) and (3) of Corollary 11 are necessary and sufficient for boundedness of -index or index. Thus, condition (2) is excessive in the case. But for , it is required because is a multiply connected domain, when contains zeros of the function .
We need some notations from [1]. Let be a given direction. For a given we denote . If one has for all , then ; if , then . And if and are zeros of the function , then , . Let
Remark 12. In [1, Theorem ], sufficient conditions of boundedness of -index in joint variables were obtained, which are similar to Theorem 10. Particularly, we assumed the validity of inequality (45) for all . However, , where . Thus, condition (1) in Theorem 10 is weaker than the corresponding assumption in Theorem from [1].
Conflicts of Interest
The authors declare that they have no conflicts of interest.