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

Authors: A. El-Sayed Ahmed; S. Attia Ahmed
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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. LetU=w:w<1 be the open unit disk in the complex plane ℂ and let dmw be the usual Euclidean area element on U. The symbol MU stands for the concerned class of all meromorphic functions in U. The pseudohyperbolic metric between the points w and z is defined by dz,w=φzw. For 0<R<1, assume that Ua,R=w∈U:dw,a<r defines the concerned pseudohyperbolic disc which is centered a∈U with the specific radius R. For 0<q<∞ and 0<s<∞, the classes M#p,q,s are defined by (see [1] pp.10)(1)M#p,q,s=h∈MU:supa∈U∬Uh#wp1−w2q1−φaw2sdmw<∞,where h#w=h′w/1+hw2 is the usual spherical derivative of h. The meromorphic M#q,q−2,0 classes are called the meromorphic Besov classes and denoted by Bq#, for which(2)Bq#=h∈MU:supa∈U∬Uh#wq1−w2q−2dmw<∞.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 B#q,q−2,s;n. Using the general spherical derivative hnw/1+hwn+1 (see [8]), we give the following general meromorphic spaces.Letn∈ℕ, 0<q<∞, 0<s<∞. Then, the general meromorphic Besov-type spaces are defined by(3)B#q,q−2,s;n=h∈MU:supa∈U∫Uhnw1+hwn+1q1−w2q−21−φaw2sdmw<∞,where the concerned weight function is 1−w2q−21−φaw2s and w∈U. Here, φaw denotes the usual Möbius transformation φaw=a−w/1−a¯w. Also,(4)hnw=dnhwdwn,that is, we have“n”times derivatives,n∈ℕ.The concerned meromorphic counterpart of the Bloch-type space is the class of all concerned normal functionsN (see [1, 9]); this class of meromorphic functions can be extended to the following concerned class.Definition 1.
Assume thath is a meromorphic function in U. When(5)hNn=supw∈U1−w2hnw1+hwn+1<∞,h∈Nn of concerned normal functions.Definition 2.
Suppose that the functionh stands for a concerned meromorphic function in U. The concerned sequence of points amam⟶1 in U is called a qN,n-sequence if(6)limm⟶∞hnw1+hwn+11−am2=+∞.In Definitions2, by letting n=1, we obtain the class of all usual normal functions N (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 thath is a meromorphic function in U. For 2<q<∞ and 0<s<∞, the concerned sequence of points amam⟶1 in U is called a bq,n-sequence if(7)limm⟶∞∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞.


## 2. Families ofbq,n- and qN,n-Type Sequences

Theorem 1.
Leth∈MU. Suppose that am defines the qN,n−type sequence, thus any sequence of points cm in U, such that dam,cm⟶0 is a bq,n-type sequence for all values of q with 2<q<∞.Proof.
In view of [16], we can find two concerned sequences cm⊂U and dm⊂ℝ+, with dam,cm⟶0 with(8)dm1−cm2⟶0,where the concerned sequence of functions hmt=hbm+dmt has uniformly converging type on each concerned compact subset of ℂ to a concerned nonconstant meromorphic function Gt. Thus,(9)supcm∈U∬Uhnw1+hwn+1q1−w2q−21−φcmw2sdmw≥∬Ucm,1/ehnw1+hwn+1q1−w2q−21−φcmw2sdmw≥∬U0,rhmnt1+hmtn+1q1−cm+dmt2q−21−φcmcm+dmt2sdm2−qdmt=∬U0,rhmnt1+hmtn+1q1−cm+dmt2dmq−2×cm−cm+dmt1−c¯mcm+dmt2sdmt=∬U0,rhmnt1+hmtn+1q1−cm+dmt2dmq−2×1−11−cm2/dmt−c¯m2sdmt.
Using the uniform convergence techniques, we deduce that(10)∬U0,rhmnt1+hmtn+1qdmt⟶∬U0,rGn#tqdmt,where the last defined integral is positive, since Gt is a concerned nonconstant meromorphic function. Further, by (6), when m⟶∞, we conclude that(11)1−11−cm2/dmt−c¯m2⟶1.
Therefore, we can obtain that(12)∬U0,rhmnt1+hmtn+1q1−am+dmt2dmq−21−11−am2/dmt−a¯m2sdmt⟶∞.
Hence, when2<q<∞, we have that(13)∬Uhnw1+hwn+1q1−w2q−21−φcmw2sdmw⟶∞,
Thus,cm∈U is a bq,n-type sequence for all q, with 2<q<∞. The proof of Theorem 1 is completely finished.Theorem 2.
We can find a concerned non-normal functionh and a concerned sequence am in U which is a bq,n-sequence for all q, with 2<q<∞, whereas am is not a qN,n-sequence.Proof.
Assume that the functionhw=expi/1−w is a non-normal function where i=−1. Considering the concerned sequence cm=m2/1+m2, after simple computation, we deduce that(14)limm⟶∞1−cm2hncm1+hcmn+1=+∞.
Applying Theorem1 for any concerned sequence of specific points am in U, with dam,cm⟶0, we get(15)limm⟶∞∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞,for all q, with 2<q<∞. Let am=m2/1+m2−i/m+m3, and note that dam,cm⟶0. But(16)limm⟶∞1−am2hnam1+hmmn+1=0.
Hence, the concerned sequenceam is our needed sequence of points.Theorem 3.
Leth∈MU and suppose that 2<q1<q<∞ and 0<s1<s<∞. For a concerned sequence of points am in the disc U, when(17)limm⟶∞∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞,(18)limm⟶∞∬Uhnw1+hwn+1q11−w2q1−21−φamw2s1dmw=+∞.Proof.
When condition (17) holds, then for 2<q1<q<∞ and 0<s1<s<∞, using the known inequality of Hölder, we conclude that(19)∬Uhnw1+hwn+1q11−w2q1−21−φamw2s1dmw≤∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmwq1/q×∬U1−φamw2s1−sq1/qq/q−q11−w2−2dmw1−q1/q=∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmwq1/q×∬U1−w2s1q−sq1/q−q1−2dmw1−q1/q.
It is obvious to see thats1q−sq1/q−q1−2=η−2>−1, for η>1, and we obtain(20)∬U1−w2s1q−sq1/q−q1−2dmw=∬U1−w2η−2dmw<C1<∞,where C1>0. Therefore, B#q,q−2,s;n⊂B#q1,q1−2,s1;n.
Thus, the following inequality can be followed:(21)∫∫Uhnw1+hwn+1q11−w2q1−21−φamw2s1dmw≥∫∫Uhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞.
Then, condition (18) must be verified. Thus, the proof is established completely.Remark 1.
Using the specific condition (17), we deduce that the function h not in the classes B#q,q−2,s;n, this because the concerned meromorphic classes B#q,q−2,s;n have a specific nesting property and the meromorphic function h is not belong to the meromorphic classes B#q1,q1−2,s1 when 2<q1<q<∞ and 0<s1<s<∞. Nevertheless, Theorem 3 shows further details on this case which clearing that the similar concerned sequence of points am, for which B#q,q−2,s;n-condition can be excluded, also it excludes the B#q1,q1−2,s1;n−condition.Remark 2.
From the concerned proof of Theorem3, we can clearly show that for a fixed ρ0, 0<ρ0<1 and ρ>0, when(22)limm⟶∞∬Uam,ρ0hnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞.
Thus, we can find a concerned sequence of pointscm in Δρn=w:1−φaw2>ρ, for which(23)limm⟶∞1−cm2hncm1+hcmn+1=+∞.Theorem 4.
Leth∈MU. For a concerned sequence of points am⊂U, when(24)limm⟶∞1−am2hnam1+hamn+1=+∞,for the same concerned sequence am, we have(25)limm⟶∞∬Uam,ρhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞,for all values ofq,swhere2<q<∞and0<s<∞as well asρ, with0<ρ<1.Proof.
Assume that condition (25) holds. Then, we have ρ0, 0<ρ0<1, such that(26)limm⟶∞sup∬Uam,ρ0hnw1+hwn+1q1−w2q−21−φamw2sdmw=K<+∞.
Thus, we can find a concerned subsequenceamk of am, for which(27)∬Uamk,ρ0hnw1+hwn+1q1−w2q−21−φamkw2sdmw≤K+1,this can be verfied for sufficiently large k. Let ρ1, 0<ρ1<ρ0, Uamk,ρ1=w∈U:φamkw<ρ1, which verifies that(28)K+11−ρ2s+q−2<π2.
This implies that(29)∬Uamk,ρ1hnw1+hwn+1qdmw≤K+11−ρ12s+q−2<π2,where 1−φamkw2≥1−ρ12. Applying the theorem of Dufresngy (see [15]), we deduce that(30)1−amk2hnamk1+hamkn+1≤1ρ1,and this is a contradiction of the concerned assumption. Therefore, the concerned proof of Theorem 4 is finished.Theorem 5.
Suppose thath∈MU. For q,s∈0,∞, we can find a concerned sequence of points am⊂U, for which(31)limm⟶∞∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞.Hence, for any concerned sequence of pointscm in U such that dam,cm⟶0, we have(32)limm⟶∞∬Uhnw1+hwn+1q1−w2q−21−φcmw2dmw=+∞.Proof.
LetK1 and K2 be two specific positive constants satisfying K2<K1. Suppose that(33)UK1m=w:1−φamw2>K1andUK2m=w:1−φamw2>K2.
Thus, whenw∈UK1m and w∈U\UK2m, we have that C1−φanw2≤1−φanw2 for some specific constant C>0. Thus, for all m, we conclude that(34)∬U\UM2mhnw1+hwn+1q1−w2q−21−φcmz2sdmw≥Cs∬U\UK2mhnw1+hwn+1q1−w2q−21−φamw2sdmw.
This inequality holds for any concerned sequence of pointscm in U with dam,cm⟶0. When(35)limm⟶∞sup∬U\UK2mhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞.
Therefore, using (10), we obtain(36)limm⟶∞sup∬U\UK2mhnw1+hwn+1q1−w2q−21−φcmw2sdmw=+∞,
If(37)limm⟶∞sup∬UK2mhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞.
Thus, we can consider two cases.Case 1.
In this case, we can find a concerned sequence of pointscm in UK2m, such that dam,cm⟶0, for which(38)limm⟶∞1−cm2hncm1+hcmn+1=+∞,or we can consider the following case.Case 2.
We can findR0, 0<R0<e−K2; also, there exists λ>0, for which(39)1−w2hnw1+hwn+1≤λ,where we consider all w∈Uam,R0. If Case 1 is verified, by Theorem 1, for the aforementioned sequence cm such that dam,cm⟶0, we deduce that(40)limm⟶∞sup∬Uhnw1+hwn+1q1−w2q−21−φcmw2sdmw=+∞.
This is becausedam,cm⟶0. 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 cn such that dam,cm⟶0,(41)limm⟶∞sup∬Uhnw1+hwn+1q1−w2q−21−φcmw2sdmw=+∞.
This is the end of the concerned proof.
Now we are dealing with the following interesting question:
Assume that2<q<∞ for any concerned sequence am and assume also that(42)limm⟶∞∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞.
Is the following equation correct?(43)limm⟶∞sup∬Uhnw1+hwn+1q11−w2q1−21−φamw2sdmw=+∞,for q1 with q<q1.
We give the answer of this important question by introducing Theorem6 with its concerned proof.Definition 4.
Letam be any concerned sequence of points in U; then, am is said to be a mq,n−sequence, when(44)limm⟶∞sup∬Uhnw1+hwn+1q1−φamw2qdmw=+∞.Theorem 6.
Letq∈2,∞ and assume that(45)limm⟶∞∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞.When the concerned sequence of pointsam in U is not a concerned mq,n−sequence, for any q1 with q<q1, we conclude that(46)limm⟶∞∬Uhnw1+hwn+1q11−w2q1−21−φamw2sdmw=+∞.Proof.
As in [1], we can deduce that Nn∩Bq,q−2,;n#. Also, we have(i)
B#q,q−2,s;n⊂Nn for all q, with 2<q<∞ and 0<s<1.(ii)
For all values ofq, q1, 2<q<∞, 0<s<1 with q1+s>1, we have that(47)∪2<q<q1B#q,q−2,s;n⊊B#q1,q1−2,s;n.Therefore, it is obvious to get that(48)limm⟶∞∬Uhnw1+hwn+1q11−w2q1−21−φamw2sdmw=+∞.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 functionh, the concerned families of points am and cm, for which(49)limm⟶∞1−am2hnam1+hamn+1=+∞,n∈ℕ,limm⟶∞∬Uhnw1+hwn+1q1−w2q−21−φamw2sdmw=+∞are introduced and discussed. Several connections between families (sequences) of bq,n and qN,n 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?



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