On Generalized Growth of Analytic Functions Solutions of Linear Homogeneous Partial Differential Equation of Second Order
The generalized growth parameters of analytic functions solutions of linear homogeneous partial differential equations of second order have been studied. Moreover, coefficients characterizations of generalized order and generalized type of the solutions represented in convergent series of Laguerre polynomials have been obtained.
Hu and Yang [1, 2] studied the behavior of meromorphic solutions of the following homogeneous linear partial differential equations of the second order: They showed that these solutions are closely related to Bessel functions and Bessel polynomials for . Several authors such as Berenstein and Li , Hu and Yang , Hu and Li , Li , and Li and Saleeby  have investigated the global solutions of some first-order partial differential equations. McCoy  and Kumar  studied the approximation of pseudoanalytic functions on the disk and obtained some coefficient and Bernstein-type growth theorems. Kapoor and Nautiyal  and Kumar and Basu  characterized the order and type of solutions (not necessarily entire solutions) of certain linear partial differential equations in terms of rates of decay of approximation errors in various norms. These solutions are related to Jacobi polynomials. Wang et al.  obtained the growth parameters order and type of entire function solutions of the partial differential equation for a real . These solutions are related to Laguerre polynomials. In this paper, our aim is to characterize the generalized growth parameters order and type of solutions of (2) which are analytic on bidisk of finite radius . The generalized growth parameters have been studied by several authors such as Šeremeta , Shah , Srivastava and Kumar , and Vakarchuk and Zhir [16, 17]. Wang et al.  proved that the PDE (2) has an entire solution on , if and only if has a series expansion . Here are the Laguerre polynomials. Bernstein theorem identifies a real analytic function on the closed unit disk as the restriction of an analytic function defined on an open disk of radius by computing from the sequence of minimal errors generated from optimal polynomials approximates. The disk of maximum radius on which analytic function exists is denoted by . If is an entire function, it has no singularities in the finite positive plane and write . Let and be two positive, strictly increasing, and differentiable functions from , which satisfies the following conditions for every : here denotes the differential of . Now we define the generalized order and generalized type of an entire by where In view of the concept introduced by MacLane  to the measures of order and type for an analytic function on a disk , we normalize above definitions relative to the boundary under the transformation . Thus an analytic function with radial limits is said to be of generalized regular growth if it satisfies where is referred to as the -order of provided that and is referred to as the -type.
Example A. has -order 2 and -type 1 for and .
In a neighborhood of origin, the function has the local expansion where is an entire solution of the ordinary differential equation where . Following the method of Frobenius , a second independent solution of (9) can be obtained as where and are constants. So there exist and satisfying Because of the singularity of at , it leads to . Thus Since , we have the estimate for large . Set
2. Auxiliary and Main Results
First we prove the following lemma.
Lemma 1. Let and . For every , we set then
Proof. From (13), for sufficiently close to , we have or Using the result with the method of Calculus that, for every and , the maximum of the function is reached at we get Now we have Using the property of , we get Applying the limit supremum as , we obtain Now consider putting in the above, we get oror or Proceeding to limits, we obtain Combining (24) and (30), we get
Theorem 2. Let such that then is the restriction of analytic function in and its -order .
Proof. Since is an analytic function in , we have Using , we get which is necessary and sufficient condition for . So, for every , the series is convergent in when is analytic in .
Now we have to prove that is the -order of .
In order to complete the proof by Lemma 1, it is only to show that . In view of the definition of , we have, for every , that there exists such that, for every , Using Cauchy’s estimate of analytic functions, we have with the coefficients formula of the Taylor expansion,we get . Since , it gives Now using (35) in (38), we get The minimum value of right-hand side is estimated at Using the properties of functions and , for , , and the properties of logarithm, we get where is a constant. Hence, Proceeding to limit supremum as , we obtain Hence the proof is completed.
Example B. has -order for and , where is the positive integer.
Let be analytic function of -order and write
Lemma 3. Let . For every , then
Proof. Following the same reasoning as in the proof of Lemma 1, we obtain Applying the limit supremum as , we get
Theorem 4. Let be of finite generalized -order and Then is the restriction of an analytic function in and its -type .
Proof. We have proven in Theorem 2 that is the restriction of an analytic function in . Now, in order to complete the proof, first we shall prove that . In view of the definition of , for every , there exists :since From inequality (51), we get For every and sufficiently close to , we put The maximum value of is reached at Using the relation (as ) and where is a positive constant, it gives Now, from relation (53), we have or Since , it gives Then or or The inequality obviously holds for . Now we shall prove that . Let . In view of definition of , we have, for every , that there exist , such that, for every , Now, using (51), we get or or as . Now, using as , we have Set in above inequality, we get or Applying the limit supremum as , we obtain The result is obviously true for . This completes the proof.
Example C. , where and , and has -order and -type for and .
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
M. N. Šeremeta, “On the connection between the growth of the maximum modulus of an entire function and the moduli of the coefficients of its power series expansion,” in Twelve Papers on Real and Complex Function Theory, vol. 88 of American Mathematical Society Translations: Series 2, pp. 291–301, American Mathematical Society, Providence, Rhode Island, 1970.View at: Publisher Site | Google Scholar