Journal of Mathematics
Volume 2021 (2021), Article ID 5588626, 9 pages
https://doi.org/10.1155/2021/5588626

Research Article

On Some Classes with Norms of Meromorphic Function Spaces Defined by General Spherical Derivatives

A. El-Sayed Ahmed¹ and S. Attia Ahmed2,3

¹Mathematics Department, Faculty of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia
²Mathematics Department, Faculty of Science, Assiut University, Assiut, Egypt
³Umm Al Qura University, AL-Qunfudah University College, Makka, Saudi Arabia

Correspondence should be addressed to A. El-Sayed Ahmed; ahsayed80@hotmail.com

Received 10 February 2021; Revised 1 March 2021; Accepted 28 March 2021; Published 14 April 2021

Academic Editor: Zakia Hammouch

Copyright © 2021 A. El-Sayed Ahmed and S. Attia Ahmed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main concerned target of this article is to define and study some concerned classes of meromorphic function spaces using the general spherical derivatives. The general Besov-type classes of meromorphic functions as well as the general normal functions are considered intensively and both are compared deeply with each other. Specifically, multiple results concerning general meromorphic-type classes as well as non-normal classes are obtained by the help of general spherical derivatives. The concerned results are proved by constructing some specific mild conditions on the sequences of points belonging to the concerned meromorphic-type classes. The obtained results generalize and improve the corresponding previous results in some concerned respects. The concerned proofs and methods are simply presented.

1. Introduction

The area of complex function spaces is fundamental and essential in many branches of pure and applied mathematics. Some decades ago, there have been obvious interests on meromorphic function classes, from concerned point of view of their singularities. For various studies on meromorphic function spaces, we may refer to all citations therein. As a concerned result, some new general classes of meromorphic functions shall be introduced by using the general spherical derivatives, which will be associated to obtain the new classes of meromorphic function spaces. Fundamental concerned properties of these concerned aforementioned meromorphic-type classes which include generalizations of meromorphic Besov spaces as well as normal function classes shall be studied and intensively discussed. As a concerned consequence of our investigation, some relevant special cases can be pointed out. Furthermore, to capture some new generalized results under the current concerned proofs, some new concepts and definitions are introduced. Let be the open unit disk in the complex plane ℂ and let be the usual Euclidean area element on . The symbol stands for the concerned class of all meromorphic functions in . The pseudohyperbolic metric between the points and is defined by . For , assume that defines the concerned pseudohyperbolic disc which is centered with the specific radius . For and , the classes are defined by (see [1] pp.10)

where is the usual spherical derivative of . The meromorphic classes are called the meromorphic Besov classes and denoted by , for which

For the analytic corresponding classes of Besov spaces, we cite [2–7]. In this article, the general meromorphic Besov-type classes always refer to the concerned classes . Using the general spherical derivative (see [8]), we give the following general meromorphic spaces.

Let , , . Then, the general meromorphic Besov-type spaces are defined by

where the concerned weight function is and . Here, denotes the usual Möbius transformation . Also,

The concerned meromorphic counterpart of the Bloch-type space is the class of all concerned normal functions (see [1, 9]); this class of meromorphic functions can be extended to the following concerned class.

Definition 1. Assume that is a meromorphic function in . When

of concerned normal functions.

Definition 2. Suppose that the function stands for a concerned meromorphic function in . The concerned sequence of points in is called a -sequence if

In Definitions 2, by letting , we obtain the class of all usual normal functions (see [1, 9]). For more interesting various studies on different meromorphic function classes, we refer to [10–15] and others. The following definitions can be introduced.

Definition 3. Assume that is a meromorphic function in . For and , the concerned sequence of points in is called a -sequence if

2. Families of - and -Type Sequences

Theorem 1. Let . Suppose that defines the type sequence, thus any sequence of points in , such that is a -type sequence for all values of with .

Proof. In view of [16], we can find two concerned sequences and , with with

where the concerned sequence of functions has uniformly converging type on each concerned compact subset of to a concerned nonconstant meromorphic function . Thus,

Using the uniform convergence techniques, we deduce that

where the last defined integral is positive, since is a concerned nonconstant meromorphic function. Further, by (6), when , we conclude that

Therefore, we can obtain that

Hence, when , we have that

Thus, is a -type sequence for all , with . The proof of Theorem 1 is completely finished.

Theorem 2. We can find a concerned non-normal function and a concerned sequence in which is a -sequence for all , with , whereas is not a -sequence.

Proof. Assume that the function is a non-normal function where . Considering the concerned sequence , after simple computation, we deduce that

Applying Theorem 1 for any concerned sequence of specific points in , with , we get

for all , with . Let , and note that . But

Hence, the concerned sequence is our needed sequence of points.

Theorem 3. Let and suppose that and . For a concerned sequence of points in the disc , when

Proof. When condition (17) holds, then for and , using the known inequality of Hölder, we conclude that

It is obvious to see that , for , and we obtain

where . Therefore, .
Thus, the following inequality can be followed:

Then, condition (18) must be verified. Thus, the proof is established completely.

Remark 1. Using the specific condition (17), we deduce that the function not in the classes , this because the concerned meromorphic classes have a specific nesting property and the meromorphic function is not belong to the meromorphic classes when and . Nevertheless, Theorem 3 shows further details on this case which clearing that the similar concerned sequence of points , for which -condition can be excluded, also it excludes the condition.

Remark 2. From the concerned proof of Theorem 3, we can clearly show that for a fixed , and , when

Thus, we can find a concerned sequence of points in , for which

Theorem 4. Let . For a concerned sequence of points , when

for the same concerned sequence , we have

for all values of,whereandas well as, with.

Proof. Assume that condition (25) holds. Then, we have , , such that

Thus, we can find a concerned subsequence of , for which

this can be verfied for sufficiently large . Let , , , which verifies that

This implies that

where . Applying the theorem of Dufresngy (see [15]), we deduce that

and this is a contradiction of the concerned assumption. Therefore, the concerned proof of Theorem 4 is finished.

Theorem 5. Suppose that . For , we can find a concerned sequence of points , for which

Hence, for any concerned sequence of points in such that , we have

Proof. Let and be two specific positive constants satisfying . Suppose that

Thus, when and , we have that for some specific constant . Thus, for all , we conclude that

This inequality holds for any concerned sequence of points in with . When

Therefore, using (10), we obtain

If

Thus, we can consider two cases.

Case 1. In this case, we can find a concerned sequence of points in , such that , for which

or we can consider the following case.

Case 2. We can find , ; also, there exists , for which

where we consider all . If Case 1 is verified, by Theorem 1, for the aforementioned sequence such that , we deduce that

This is because . Also, when Case 2 holds, using the same concerned conclusions for the concerned weight functions, we obtain that necessarily condition for any concerned sequence of points such that ,

This is the end of the concerned proof.
Now we are dealing with the following interesting question:
Assume that for any concerned sequence and assume also that

Is the following equation correct?

for with .
We give the answer of this important question by introducing Theorem 6 with its concerned proof.

Definition 4. Let be any concerned sequence of points in ; then, is said to be a sequence, when

Theorem 6. Let and assume that

When the concerned sequence of points in is not a concerned sequence, for any with , we conclude that

Proof. As in [1], we can deduce that . Also, we have

(i) for all , with and .(ii)For all values of , , , with , we have that

Therefore, it is obvious to get that

Remark 3. The recent developments of fractional calculus as well as its applications are more essential to complex function spaces with the specific arbitrary fractional order derivatives. For some recent interesting studies on the subject of fractional calculus, we can here refer to [17–19] and others. To the best of our knowledge, a few number of manuscripts researched some certain classes of analytic function spaces by the help of general fractional derivatives (see [20]). For further research work, the following specific interesting question can be considered:
How one can define and study the Besov spaces of general meromorphic functions by using the general fractional derivatives?

3. Conclusions

Certain concerned weighted classes of meromorphic function spaces using the general spherical derivatives are studied and discussed in this article. The general Besov-type classes of meromorphic functions as well as the general normal functions are considered intensively and both are compared deeply with each other. For a concerned non-normal function , the concerned families of points and , for which

are introduced and discussed. Several connections between families (sequences) of and type are established. The obtained results improve, extend, and generalize numerous results in [21–23].

Remark 4. Quite recently there are some important enjoyable research studies on hyperbolic function classes (see [24, 25]). For more interesting research, how we can construct some workable conditions on some hyperbolic-type sequences of points that make guarantee to belong to some specific hyperbolic-type classes?

Data Availability

No data were applied or considered to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author would like to thank Taif University Researchers concerning the support of project no. TURSP-2020/159, Taif University, Saudi Arabia.

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