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

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 bywhere 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 withwhere 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 thatwhere the last defined integral is positive, since is a concerned nonconstant meromorphic function. Further, by (6), when , we conclude thatTherefore, we can obtain thatHence, when , we have thatThus, 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 thatApplying Theorem 1 for any concerned sequence of specific points in , with , we getfor all , with . Let , and note that . ButHence, 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 thatIt is obvious to see that , for , and we obtainwhere . 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 , whenThus, we can find a concerned sequence of points in , for which

Theorem 4. Let . For a concerned sequence of points , whenfor the same concerned sequence , we havefor all values of,whereandas well as, with.

Proof. Assume that condition (25) holds. Then, we have , , such thatThus, we can find a concerned subsequence of , for whichthis can be verfied for sufficiently large . Let , , , which verifies thatThis implies thatwhere . Applying the theorem of Dufresngy (see [15]), we deduce thatand 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 thatThus, when and , we have that for some specific constant . Thus, for all , we conclude thatThis inequality holds for any concerned sequence of points in with . WhenTherefore, using (10), we obtainIfThus, we can consider two cases.

Case 1. In this case, we can find a concerned sequence of points in , such that , for whichor we can consider the following case.

Case 2. We can find , ; also, there exists , for whichwhere we consider all . If Case 1 is verified, by Theorem 1, for the aforementioned sequence such that , we deduce thatThis 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 thatIs 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 whichare 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.