The goal of this article is to define, explore and analyze two new families of meromorphically harmonic functions by applying the concept of a certain generalized convolution q-operator along with the idea of convolution. We investigate convolution properties and sufficiency criteria for these families of meromorphically harmonic functions. Some of the interesting consequences of our investigation are also included.
Citation: Hari Mohan Srivastava, Muhammad Arif, Mohsan Raza. Convolution properties of meromorphically harmonic functions defined by a generalized convolution q-derivative operator[J]. AIMS Mathematics, 2021, 6(6): 5869-5885. doi: 10.3934/math.2021347
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The goal of this article is to define, explore and analyze two new families of meromorphically harmonic functions by applying the concept of a certain generalized convolution q-operator along with the idea of convolution. We investigate convolution properties and sufficiency criteria for these families of meromorphically harmonic functions. Some of the interesting consequences of our investigation are also included.
A new research area, which was initially developed by Clunie and Sheil-Small [14], is the subject of univalent harmonic functions (see also [31,38]). The significance of such functions is attributed to their usages in the analysis of minimal surfaces as well as in problems relevant to applied mathematics. Hengartner and Schober [21] introduced and analyzed some specific types of harmonic functions which are given in the region ˜U in the complex z-plane C, which is given by
˜U:={z:z∈Cand|z|>1}. |
Hengartner and Schober [21] proved, among other things, that a harmonic complex-valued and sense preserving univalent mapping f, defined in ˜U and such that f(∞)=∞, must satisfy the following representation:
f(z)=L1(z)+¯L2(z)+Qlog|z|, | (1.1) |
with Q∈C and, for 0≦|η2|<|η1|,
L1(z)=η1z+∞∑n=1anz−nandL2(z)=η2¯z+∞∑n=1bn¯z−n. |
In the year 1999, Jahangiri and Silverman [27] gave adequate coefficient criteria for which functions of the type (1.1) are univalent. They also provided necessary and sufficient coefficient criteria within certain constraints for functions to be harmonic and starlike, also see [39]. Later, Jahangiri [25] and Murugusundaramoorthy [33,34] analyzed the families of meromorphically harmonic function in ˜U. The authors in [12,13] used the technique developed by Zou and his co-authors in [47] to examine the nature of meromorphically harmonic starlike functions with respect to symmetrical conjugate points in the punctured unit disk U∗ given by
U∗={z:z∈Cand0<|z|<1}=U∖{0}. |
Particularly, in [13], a sharp approximation of the coefficient and a structural description of these functions were also determined. To understand the basics in a more clear way, let us represent the symbol H by the family of harmonic functions f which have the series form given by
f(z)=λ(z)+¯μ(z)=1z+∞∑n=1(anzn+bn¯zn)(z∈U∗), | (1.2) |
where the functions λ and μ are holomorphic in U∗ and U, respectively, with the following series forms:
λ(z)=1z+∞∑n=1anzn(z∈U∗)andμ(z)=∞∑n=1bnzn(z∈U). | (1.3) |
and
|an|≥1, |bn|≥1, (n=2,3,⋯). |
We denote the family of all complex-valued functions f∈H by MH, which are sense-preserving and univalent in U∗. Indeed, if
μ(z)≡0(z∈U), |
then
f(z)=λ(z)=1z+∞∑n=1anzn(z∈U∗). |
In the view of this point, the function class MH coincides with the class M of normalized holomorphic univalent functions in U∗. The above-mentioned papers have obviously opened up a new door for researchers to investigate further inputs in this area of geometric function theory. In this regard, we consider the collections of meromorphically harmonic-starlike and meromorphically harmonic-convex functions in U∗ given, respectively, by
MS∗H={f:f∈MHandℜ(DHf(z)f(z))<0(z∈U∗)} |
and
MScH={f:f∈MHandℜ(DH(DHf(z))DHf(z))<0(z∈U∗)}, |
where
DHf(z)=zλ′(z)−¯zμ′(z). |
Now we give the definition of weak subordination in U. For some details about subordinations for hamonic mappings, we refer [15,16].
Acomplex valued function f in U is said to be a weakly subordinate to a complex valued function g in U and written as f(z)⪯g(z) or simply as f⪯g if f(∞)=g(∞) and f(U)⊂g(U). If f⪯g and g is univalent in U, then we consider a complex valued function w(z)=g−1(f(z)),z∈U which maps U onto itself with w(∞)=∞. Conversely if w(z)=g−1(f(z)) in U and maps U onto itself with w(∞)=∞, then f⪯g.
This can be written in the following equivalence.
Lemma 1.1. [17] A complex valued function f in U is weakly subordinate to a complex valued function g in U if and only if there exists a complex valued function w which maps U onto itself with w(∞)=∞ such that f(z)=g(w(z)),z∈U.
Many sub-families of meromorphically harmonic functions have also been introduced and investigated by several earlier researchers (see, for example, the works of Bostanci [10], Bostanci and Öztürk [11], Öztürk and Bostanci [35], Wang et al. [46], Al-Dweby and Darus [3], Al-Shaqsi and Darus [4], Ponnusamy and Rajasekaran [37], Ahuja and Jahangiri [2], Al-Zkeri and Al-Oboudi [5], Jahangiri et al. [26] and Stephen et al. [45]).
The theory of the Hadamard product (or convolution) is incredibly essential in the solution of many function-theoretic problems and, as a result of these realities, this technique becomes a significant part of the area of our study. The objective of this section is to examine the properties and consequences of the convolution for the two newly-defined families of meromorphically harmonic functions. For two functions g1, g2∈M with their series expansions as follows:
g1(z)=1z+∞∑n=1anzn andg2(z)=1z+∞∑n=1bnzn, |
the Hadamard product (or convolution), which is denoted by (g1∗g2)(z), is defined by
(g1∗g2)(z)=1z+∞∑n=1anbnzn=(g2∗g1)(z)(z∈U∗). |
The potentially useful properties, which are mentioned below, can only be true if g∈M:
g(z)∗1z(1−z)=g(z)andg(z)∗1−2zz(1−z)2=−zg′(z). | (2.1) |
We now consider a function Nq(with q∈C) of the following form:
Nq(z)=1z−z(1−qz)(1−z)=1z−∞∑n=1[n]q zn(z∈U∗), |
where, as usual, the q-number [ν]q(ν∈C) is given by
[ν]q={1−qν1−q(ν,q∈C)n−1∑k=0qk(ν=n∈N0:=N∪{0}), |
where N denotes the set of natural numbers. Then, clearly, the function Nq is meromorphically starlike for all complex numbers q such that |q|<1. Also, one can easily see that, if q→1−, then the Nq reduces as follows:
limq→1−Nq(z)=1z−z(1−z)2=1z−∞∑n=1nzn(z∈U∗). |
Moreover, the q-derivative (or q-difference) operator Dq of a function f defined on a subset of the complex space C is given by (see [23,24])
(Dqf)(z)={f(z)−f(qz)(1−q)z(z≠0)f′(0)(z=0), | (2.2) |
provided that the first-order derivative of the function f(z) at z=0 exists.
Now, by using the function Nq, we define the generalized convolution q-derivative operator Dq for meromorphically functions f∈M by
Dqf(z)=−1z[f(z)∗(1z−z(1−qz)(1−z))]=−1z[f(z)∗Nq(z)]. | (2.3) |
We observe that, by taking q=1 in the above q-derivative operator Dq, we achieve the ordinary derivative operator ddz. Also, for 0<q<1, we attain Jackson's q-derivative (or q-difference) Dq of the function f which is defined above by (2.2).
The q-derivative (or q-difference) operator Dq has fascinated and inspired many researchers due mainly to its use in various areas of the mathematical and physical sciences. Although the first article in which a link was established between geometric nature of analytic functions associate with the q-derivative operator Dq was initiated in [22] in 1990, yet the usage of q-calculus in geometric function theory as well as a solid and comprehensive foundation was given in 1989 in a book chapter by Srivastava [41]. After this development, many researchers introduced and studied various useful operators in q-calculus together with the applications of the associated convolution concepts. For example, Kanas and Rǎducanu [28] studied the q-derivative operator Dq and examined its behavior in geometric function theory. The operator Dq was generalized for multivalent analytic functions by Arif et al [9]. Analogous to these q-derivative operators, Arif et al. [8] and Khan et al. [30] contributed by introducing the q-integral operators for analytic and multivalent functions. Similarly, in the authors in [6] developed and analyzed some analogues of the q-derivative operators for meromorphically functions. Very recently, a survay-cum-expository review article on the subject of quantum (or q-) calculus and its various applications in geometric function theory was published by Srivastava [40] (see also [1,7,18,19,20,29,32,36,42,43,44]).
We next define an operator for the function f∈MH as follows. Let Dq,τH:MH→MH be a linear operator defined for a function f=λ+¯μ∈MH by
Dq,τHf(z)=zDqλ(z)+τ¯zDqμ(z)(|τ|=1), | (2.4) |
where the convolution q-derivative operator Dq is given by (2.3).
Making use of the operator Dq,τHf(z), we now introduce two families MSH[q,L,M] and MKH[q,L,M] of functions of the Janowski type for |q|≦1 and −1≦M≦L≦1, which are defined below:
MSH[q,L,M]={f:f∈MHand−Dq,τHf(z)f(z)⪯1+Lz1+Mz(z∈U∗)} | (2.5) |
and
MKH[q,L,M]={f:f∈MHand−Dq,τH(Dq,τHf(z))Dq,τHf(z)⪯1+Lz1+Mz(z∈U∗)}, | (2.6) |
respectively. We note the following special cases:
(ⅰ) Taking L=1−2ξ and M=−1 in MSH[q,L,M] and MKH[q,L,M], we have
MSH[q,1−2ξ,−1]=MS H(q,ξ) |
and
MKH[q,1−2ξ,−1]=MK H(q,ξ), |
where
MSH(q,ξ)={f:f∈MHand−ℜ(Dq,τHf(z)f(z))>ξ(0≦ξ<1;z∈U∗)} | (2.7) |
and
MKH(q,ξ)={f:f∈MHand−ℜ(Dq,τH(Dq,τHf(z))Dq,τHf(z))>ξ(0≦ξ<1;z∈U∗)}. | (2.8) |
(ⅱ). By setting L=(1−2ξ)β and M=−β in MSH[q,L,M] and MKH[q,L,M] with 0≦ξ<1 and 0≦β<1, we get the following function classes:
MSH[q,(1−2ξ)β,−β]=MSH(q,ξ,β) |
and
MKH[q,(1−2ξ)β,−β]=MKH(q,ξ,β), |
where
MSH(q,ξ,β)={f:f∈MHand|(Dq,τHf(z)f(z))−1(Dq,τHf(z)f(z))+1−2ξ|<β(z∈U∗)} | (2.9) |
and
MKH(q,ξ,β)={f:f∈MHand|(Dq,τH(Dq,τHf(z))Dq,τHf(z))−1(Dq,τH(Dq,τHf(z))Dq,τ Hf(z))+1−2ξ|<β(z∈U∗)}. | (2.10) |
We observe also that
MSH[L,M]=limq→1−MSH[q,L,M] |
and
MKH[L,M]=limq→1−MKH[q,L,M], |
where
MSH[L,M]={f:f∈MHand−DτHf(z)f(z)⪯1+Lz1+Mz(z∈U∗)} |
and
MKH[L,M]={f:f∈MHand−DτH(DτHf(z))DτHf(z)⪯1+Lz1+Mz(z∈U∗)}. |
From the definitions of the newly-introduced classes, we have
f∈MKH[q,L,M]⇔−zDq,τHf(z)∈MSH[q,L,M]. | (2.11) |
In this paper, we investigate a number of convolution properties and several coefficient estimates for functions in the classes MSH[q,L,M] and MKH[q,L,M], which are associated with the generalized convolution q-derivative operator Dq,τH.
Unless otherwise mentioned, we assume throughout this paper that
−1≦M<L≦1,0<q<1,|τ|=1,ρ≧0andβ<1. |
Our first result in this section, which is asserted by Theorem 3.1 below, provides a necessary and sufficient condition for a given function to be in the class MSH[q,L,M].
Theorem 3.1. A function f defined by (1.2) is in the class MSH[q,L,M] if and only if
[z{f(z)∗(1+ρ(1+(1−q)z)z−qzz(1−qz)(1−z)+τ1+ρ(1+(1−q)¯z)¯z−q¯z¯z(1−q¯z)(1−¯z))}]≠0 | (3.1) |
for all ρ given by
ρ=ρζ:=ζ−1+ML−M(|ζ|=1) |
and also for ρ=0.
Proof. Let f∈MSH[q,L,M] have the series form (1.2). Then, from the concept of weak subordination, a function u exists, with the restrictions that u(∞)=∞ and |u(z)|<1, such that
−Dq,τHf(z)f(z)=1+Lu(z)1+Mu(z), | (3.2) |
which is equivalent to the following assertion:
−Dq,τHf(z)f(z)≠1+Lζ1+Mζ(z∈U;|ζ|=1) | (3.3) |
or
z[−Dq,τHf(z)(1+Mζ)−f(z)(1+Lζ)]≠0. | (3.4) |
Since
Dq,τHf(z)=zDqλ(z)+τ¯zDqμ(z)=(λ(z)∗(1−z)(1−qz)−z2z(1−z)(1−qz))+τ(¯μ(z)∗(1−¯z)(1−q¯z)−¯z2¯z(1−¯z)(1−q¯z)), |
together with following identities:
λ(z)∗1z(1−z)=λ(z) |
and
¯μ(z)∗1¯z(1−¯z)=¯μ(z), |
We find from (3.4) that
0≠z[−∂q,τHf(z)(1+Mζ)−f(z)(1+Lζ)]=z[{(λ(z)∗(1−z)(1−qz)−z2z(1−z)(1−qz))(1+Mζ)−(λ(z)∗1z(1−z))(1+Lζ)}+τ(¯μ(z)∗(1−¯z)(1−q¯z)−¯z2¯z(1−¯z)(1−q¯z))(1+Mζ)−(¯μ(z)∗1¯z(1−¯z))(1+Lζ)]=z[λ(z)∗{((1−z)(1−qz)−z2)(1+Mζ)−(1−qz)(1+Lζ)z(1−z)(1−qz)}+¯μ(z)∗{τ((1−¯z)(1−q¯z)−¯z2)(1+Mζ)−(1−q¯z)(1+Lζ)¯z(1−¯z)(1−q¯z)}]=zζ(M−L)[λ(z)∗{(1−qz)+z(1+(1−q)z)ρz(1−z)(1−qz)}+τ¯μ(z)∗{(1−q¯z)+¯z(1+(1−q)¯z)ρ¯z(1−¯z)(1−q¯z)}]=ζ(M−L)z[f(z)∗{(1−qz)+z(1+(1−q)z)ρz(1−z)(1−qz)+(1−q¯z)+¯z(1+(1−q)¯z)ρ¯z(1−¯z)(1−q¯z)}], |
which leads to (3.1) and the necessary part of the proof of Theorem 3.1 is completed.
Conversely, we suppose that (3.1) holds true for ρ=0. Then it follows that zf(z)≠0 for all z∈U. Consequently, the function Φ(z) given by
Φ(z)=−Dq,τHf(z)f(z) |
is regular at z0=0, with Φ(0)=1. It was shown that the assumption (3.1) is equivalent to (3.3), so we obtain
−Dq,τHf(z)f(z)≠1+Lζ1+Mζ(z∈U;|ζ|=1). | (3.5) |
Let we now put
Ψ(z)=1+Lz1+Mz(z∈U). |
Then the relation (3.5) shows that Φ(U)∩Ψ(∂U)=∅. Thus, clearly, Φ(U) is connected to C∖Ψ(∂U). Hence we have
Φ(0)=Ψ(0), |
which, together with the univalence of Ψ, shows that Φ(z)⪯Ψ(z) in terms of the subordination in (3.2), that is, f∈MSqH[L,M]. This completes the proof of Theorem 3.1.
Upon letting q→1− in Theorem 3.1, we obtain the following result.
Corollary 3.2. A function f defined by (1.2) is in the class MSH[L,M] if and only if
z[f(z)∗(1+(ρ−1)zz(1−z)2+τ1+(ρ−1)¯z¯z(1−¯z)2)]≠0, |
for all ρ given by
ρ=ζ−1+ML−M(|ζ|=1) |
and also for ρ=0.
Putting L=1−2ξ(0≦ξ<1) and M=−1 in Theorem 3.1, we deduce the following result.
Corollary 3.3. A function f defined by (1.2) is in the class MSH(q,ξ) if and only if
z[f(z)∗(1+ϑ(1+(1−q)z)z−qzz(1−qz)(1−z)+τ1+ϑ(1+(1−q)¯z)¯z−q¯z¯z(1−q¯z)(1−¯z))]≠0 |
for all ϑ given by
ϑ=ζ−1−12(1−ξ)(|ζ|=1;0≦ξ<1) |
and also for ϑ=0.
By letting q→1− in Corollary 3.3, we have the following result.
Corollary 3.4. A function f defined by (1.2) is in the class MSH(ξ) if and only if
z[f(z)∗(1+(ϑ−1)zz(1−z)2+τ1+(ϑ−1)¯z¯z(1−¯z)2)]≠0 |
for all ϑ given by
ϑ=ζ−1−12(1−ξ)(|ζ|=1;0≦ξ<1) |
and also for ϑ=0.
Our next result in this section, which is asserted by Theorem 3.5 below, provides a necessary and sufficient condition for a given function to be in the class MKH[q,L,M].
Theorem 3.5. A function f defined by (1.2) is in the class MKH[q,L,M] if and only if
z[f(z)∗((1−q2z)(1−(q+1)z)−ρq(2−q2+qz)z2qz(1−z)(1−qz)(1−q2z) |
+τ(1−q2¯z)(1−(q+1)¯z)−ρq(2−q2+q¯z)¯z2q¯z(1−¯z)(1−q¯z)(1−q2¯z))]≠0 |
for all ρ given by
ρ=ζ−1+ML−M(|ζ|=1) |
and also for ρ=0.
Proof. First of all, we suppose that the function f defined by (1.2) is in the class MKH[q,L,M] if it satisfies the condition (2.6) or, equivalently,
−Dq,τH(Dq,τHf(z))Dq,τHf(z)≠1+Lζ1+Mζ. |
By setting
χ(z)=1+ρ(1+(1−q)z)z−qzz(1−qz)(1−z), |
we note that
−zDqχ(z)=(1−q2z)(1−(q+1)z)−ρq(2−q2+qz)z2qz(1−z)(1−qz)(1−q2z). |
We also recall the following identity:
[−zDqλ(z)]∗χ(z)=λ(z)∗[−zDqχ(z)] |
and the fact that
f(z)∈MKH[q,L,M]⟺−zDq,τHf(z)∈MSH[q,L,M]. |
Hence, clearly, the result asserted by Theorem 3.1, would follow by using the above relations in conjunction with Theorem 3.1.
If we let q→1− in Theorem 3.5, we obtain the following result.
Corollary 3.6. A function f defined by (1.2) is in the class MKH[L,M] if and only if
z[f(z)∗(1−(1+ρ(1+z)z)zz(1−z)3+τ1−(1+ρ(1+¯z)¯z)¯z¯z(1−¯z)3)]≠0 |
for all ρ given by
ρ=ζ−1+ML−M(|ζ|=1) |
and also for ρ=0.
Putting L=1−2ξ(0≦ξ<1) and M=−1 in Theorem 3.5, we obtain the following corollary.
Corollary 3.7. A function f defined by (1.2) is in the class MKH(q,ξ)(0≦ξ<1) if and only if
z[f(z)∗((1−q2z)(1−(q+1)z)−ϑq(2−q2+qz)z2qz(1−z)(1−qz)(1−q2z) |
+τ(1−q2¯z)(1−(q+1)¯z)−ρq(2−q2+q¯z)¯z2q¯z(1−¯z)(1−q¯z)(1−q2¯z))]≠0 |
for all ϑ given by
ϑ=ζ−1−12(1−ξ)(|ζ|=1;0≦ξ<1) |
and also for ϑ=0.
Letting q→1− in Corollary 3.7, we obtain the following result.
Corollary 3.8. A function f defined by (1.2) is in the class MKH(ξ)(0≦ξ<1) if and only if
z[f(z)∗(1−((1+ϑ(1+z)z))zz(1−z)3+τ1−((1+ϑ(1+¯z)¯z))¯z¯z(1−¯z)3)]≠0 |
for all ϑ given by
ϑ=ζ−1−12(1−ξ)(|ζ|=1;0≦ξ<1) |
and also for ϑ=0.
Theorem 3.9 below provides a necessary and sufficient condition for a given function to be in the class MSH[q,L,M].
Theorem 3.9. A function f defined by (1.2) is in the class MSH[q,L,M] if and only if
1−∞∑n=1[n]q(ζ−1+M)+ζ−1+LM−L(anzn+1+τ¯bnzn+1)≠0(z∈U). | (3.6) |
Proof. From Theorem 3.1, we know that f∈MSH[q,L,M] if and only if
z[f(z)∗(1+ρ((1−qz)z+z2)−qzz(1−qz)(1−z)+τ1+ρ((1−q¯z)¯z+¯z2)−q¯z¯z(1−q¯z)(1−¯z))]≠0 | (3.7) |
for all ρ given by
ρ=ρζ:=ζ−1+ML−M(|ζ|=1) |
and also for ρ=0.
The left-hand side of (3.7) can be written as follows:
z[λ(z)∗1−qz+ρ((1−qz)z+z2)z(1−qz)(1−z)+τ¯μ(z)∗1−q¯z+ρ((1−q¯z)¯z+¯z2)¯z(1−q¯z)(1−¯z)]=z[λ(z)∗(1z(1−z)+ρ(z(1−qz)(1−z)+1(1−z)))+τ¯μ(z)∗(1¯z(1−¯z)+ρ(¯z(1−q¯z)(1−¯z)+1(1−¯z)))]=z[λ(z)+ρ(zDqλ(z)+λ(z))+τ(¯μ(z)+ρ(¯zDqμ¯(z)+¯μ(z)))]=z[1z+∞∑n=1(1+ρ([n]q+1))anzn+τ∞∑n=1(1+ρ([n]q+1))¯bnzn]=1+∞∑n=1(1+ρ([n]q+1))anzn+1+τ∞∑n=1(1+ρ([n]q+1))¯bnzn+1. |
This evidently completes our proof of the result asserted by Theorem 3.9.
Upon letting q→1− in Theorem 3.9, we obtain the following result.
Corollary 3.10. A function f defined by (1.2) is in the class MSH[L,M] if and only if
1−∞∑n=1n(ζ−1+M)+ζ−1+LM−L(anzn+1+τ¯bnzn+1)≠0(z∈U). |
Putting L=1−2ξ(0≦ξ<1) and M=−1 in Theorem 3.9, we deduce the following result.
Corollary 3.11. A function f defined by (1.2) is in the class MSH(q,ξ) if and only if
1+∞∑n=1[n]q(ζ−1−1)+ζ−1+1−2ξ2(1−ξ)(anzn+1+τ¯bnzn+1)≠0(z∈U). |
Taking q→1− in Corollary 3.11, we obtain the following result.
Corollary 3.12. A function f defined by (1.2) is in the class MSH(ξ) if and only if
1+∞∑n=1n(ζ−1−1)+ζ−1+1−2ξ2(1−ξ)(anzn+1+τ¯bnzn+1)≠0(z∈U). |
Theorem 3.13. A function f defined by (1.2) is in the class MKH[q,L,M] if and only if
1−∞∑n=1[n]q[n]q(ζ−1+M)+ζ−1+LM−L(anzn+1+τ¯bnzn+1)≠0(z∈U). |
Taking q→1− in Theorem 3.13, we obtain the following result.
Corollary 3.14. A function f defined by (1.2), is in the class MKH[L,M] if and only if
1−∞∑n=1nn(ζ−1+M)+ζ−1+LM−L(anzn+1+τ¯bnzn+1)≠0(z∈U). |
Putting L=1−2ξ,(0≦ξ<1) and M=−1 in Theorem 3.13, we obtain the following result.
Corollary 3.15. A function f defined by (1.2) is in the class MKH(q,ξ) if and only if
1+∞∑n=1[n]q[n]q(ζ−1−1)+ζ−1+1−2ξ2(1−ξ)(anzn+1+τ¯bnzn+1)≠0(z∈U). |
Letting q→1− in Corollary 3.15, we obtain the following result.
Corollary 3.16. A function f defined by (1.2) is in the class MKH(ξ) if and only if
1+∞∑n=1nn(ζ−1−1)+ζ−1+1−2ξ2(1−ξ)(anzn+1+τ¯bnzn+1)≠0(z∈U). |
We now determine the coefficient estimates and inclusion relations for functions belonging to the classes MSH[q,L,M] and MKH[q,L,M].
Theorem 4.1. If a function f defined by (1.2) satisfies the following inequality:
∞∑n=1([n]q(1+|M|)+1+L)(|an|+|bn|)≦L−M, |
then f∈MSH[q,L,M].
Proof. From (3.6), we have
|1−∞∑n=1[n]q(ζ−1+M)+ζ−1+LM−L(anzn+1+τ¯bnzn+1)|>1−∞∑n=1[n]q|(ζ−1+M)+ζ−1+L|L−M(|an|+|bn|)>1−∞∑n=1[n]q(1+|M|)+1+LL−M(|an|+|bn|)≧0, |
which completes the proof of Theorem 4.1.
By letting q→1− in Theorem 4.1, we obtain the following result.
Corollary 4.2. If a function f defined by (1.2) satisfies the following inequality:
∞∑n=1(n(1+|M|)+1+L)(|an|+|bn|)≦L−M, |
then f∈MSH[L,M].
Putting L=1−2ξ(0≦ξ<1) and M=−1 in Theorem 4.1, we obtain the following result.
Corollary 4.3. If the function f defined by (1.2) satisfies the following inequality:
∞∑n=1([n]q+1−ξ)(|an|+|bn|)≦1−ξ, |
then f∈MSH(q,ξ).
Letting q→1− in Corollary 4.3, we obtain the following result.
Corollary 4.4. If a function f defined by (1.2) satisfies the following inequality:
∞∑n=1(n+1−ξ)(|an|+|bn|)≦1−ξ, |
then f∈MSH(ξ).
Similarly, we can prove the next result (Theorem 4.5 below).
Theorem 4.5. If the function f defined by (1.2) satisfies the following inequality:
∞∑n=1[n]q([n]q(1+|M|)+1+L)(|an|+|bn|)≦L−M, |
then f∈MKH[q,L,M].
Upon letting q→1− in Theorem 4.5, we obtain the following result.
Corollary 4.6. If the function f defined by (1.2) satisfies the following inequality:
∞∑n=1n(n(1+|M|)+1+L)(|an|+|bn|)≦L−M, |
then f∈MKH[L,M].
Putting L=1−2ξ(0≦ξ<1) and M=−1 in Theorem 4.5, we obtain the following corollary.
Corollary 4.7. If the function f defined by (1.2) satisfies the following inequality:
∞∑n=1[n]q([n]q+1−ξ)(|an|+|bn|)≦1−ξ, |
then f∈MKH(q,ξ).
If we let q→1− in Corollary 4.7, we obtain the following result.
Corollary 4.8. If the function f defined by (1.2) satisfies the following inequality:
∞∑n=1n(n+1−ξ)(|an|+|bn|)≦1−ξ, |
then f∈MKH(ξ).
In this paper, we have introduced the generalized convolution q-derivative operator Dq, which is defined by
Dqf(z)=−1z[f(z)∗(1z−z(1−qz)(1−z))], |
where q∈C and |q|≦1. By letting q→1−, this generalized convolution q-operator Dq takes the form of the ordinary derivative. Also, for 0<q<1, we obtain the q-analog of derivative operator. By applying this operator, we have defined a corresponding operator for meromorphically harmonic functions and introduced some subclasses of meromorphically harmonic starlike and meromorphically harmonic convex functions and have studied a number of properties and results for functions belonging to each of these function classes.
Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas (see, for example, [41,pp. 351–352] and [40,p. 328]). Moreover, in this recently-published survey-cum-expository review article by Srivastava [40], the so-called (p,q)-calculus was exposed to be a rather trivial and inconsequential variation of the classical q-calculus, the additional parameter p being redundant (see, for details, [40,p. 340]). This observation by Srivastava [40] will indeed apply also to any attempt to produce the rather straightforward (p,q)-variations of the results which we have presented in this paper.
The authors declare no conflicts of interest.
[1] | S. Agrawal, S. K. Sahoo, A generalization of starlike functions of order alpha, Hokkaido Math. J., 46 (2017), 15–27. |
[2] | O. P. Ahuja, J. M. Jahangiri, Certain meromorphic harmonic functions, Bull. Malays. Math. Sci. Soc., 25 (2002), 1–10. |
[3] | H. Al-Dweby, M. Darus, On harmonic meromorphic functions associated with basic hypergeometric functions, Sci. World J., 2013 (2013), 1–8. |
[4] | K. Al-Shaqsi, M. Darus, On meromorphic harmonic functions with respect to symmetric points, J. Inequal. Appl., 2008 (2008), 1–11. |
[5] | H. A. Al-Zkeri, F. M. Al-Oboudi, On a class of harmonic starlike multivalent meromorphic functions, Int. J. Open Probl. Complex Anal., 3 (2011), 68–81. |
[6] |
M. Arif, B. Ahmad, New subfamily of meromorphic multivalent starlike functions in circular domain involving q-differential operator, Math. Slovaca, 68 (2018), 1049–1056. doi: 10.1515/ms-2017-0166
![]() |
[7] | M. Arif, O. Barkub, H. M. Srivastava, S. Abdullah, S. A. Khan, Some Janowski type harmonic q-starlike functions associated with symmetrical points, Mathematics, 8 (2020), 1–16. |
[8] | M. Arif, M. U. Haq, J. L. Liu, A subfamily of univalent functions associated with q-analogue of Noor integral operator, J. Funct. Spaces, 2018 (2018), 1–8. |
[9] | M. Arif, H. M. Srivastava, S. Umar, Some applications of a q-analogue of the Ruscheweyh type operator for multivalent functions, Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM), 113 (2019), 1211–1221. |
[10] |
H. Bostanci, A new subclass of the meromorphic harmonic γ-starlike functions, Appl. Math. Comput., 218 (2011), 683–688. doi: 10.1016/j.amc.2011.03.149
![]() |
[11] |
H. Bostanci, M. Öztürk, A new subclass of the meromorphic harmonic starlike functions, Appl. Math. Lett., 23 (2010), 1027–1032. doi: 10.1016/j.aml.2010.04.031
![]() |
[12] | H. Bostanci, M. Öztürk, On meromorphic harmonic starlike functions with missing coefficients, Hacet. J. Math. Stat., 38 (2009), 173–183. |
[13] |
H. Bostanci, S. Yalçin, M. Öztürk, On meromorphically harmonic starlike functions with respect to symmetric conjugate points, J. Math. Anal. Appl., 328 (2007), 370–379. doi: 10.1016/j.jmaa.2006.05.044
![]() |
[14] | J. Clunie, T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I. Math., 9 (1984), 3–25. |
[15] | J. Dziok, On Janowski harmonic functions, J. Appl. Anal., 21 (2015), 99–107. |
[16] | J. Dziok, Classes of harmonic functions defined by subordination, Abstr. Appl. Anal., 2015 (2015), 1–9. |
[17] |
J. Dziok, Classes of meromorphic harmonic functions and duality principle, Anal. Math. Phys., 10 (2020), 1–13. doi: 10.1007/s13324-019-00351-5
![]() |
[18] | S. Elhaddad, H. Aldweby, M. Darus, On a subclass of harmonic univalent functions involving a new operator containing q-Mittag-Leffler function, Int. J. Math. Comput. Sci., 14 (2019), 833–847. |
[19] | S. Elhaddad, H. Aldweby, M. Darus, Some properties on a class of harmonic univalent functions defined by q-analogue of Ruscheweyh operator, J. Math. Anal., 9 (2018), 28–35. |
[20] | M. U. Haq, M. Raza, M. Arif, Q. Khan, H. Tang, q-analogue of differential subordinations, Mathematics, 7 (2019), 1–16. |
[21] | W. Hengartner, G. Schober, Univalent harmonic functions, Trans. Am. Math. Soc., 299 (1987), 1–31. |
[22] | M. E. H. Ismail, E. Merkes, D. Styer, A generalization of starlike functions, Complex Var. Theory Appl., 14 (1990), 77–84. |
[23] | F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203. |
[24] |
F. H. Jackson, On q-functions and a certain difference operator, Trans. Roy. Soc. Edin., 46 (1909), 253–281. doi: 10.1017/S0080456800002751
![]() |
[25] | J. M. Jahangiri, Harmonic meromorphic starlike functions, Bull. Korean Math. Soc., 37 (2000), 291–301. |
[26] |
J. M. Jahangiri, Y. C. Kim, H. M. Srivastava, Construction of a certain class of harmonic close-to-convex functions associated with the Alexander integral transform, Integr. Transf. Spec. Funct., 14 (2003), 237–242. doi: 10.1080/1065246031000074380
![]() |
[27] | J. M. Jahangiri, H. Silverman, Meromorphic univalent harmonic functions with negative coefficients, Bull. Korean Math. Soc., 36 (1999), 763–770. |
[28] | S. Kanas, D. Rǎducanu, Some class of analytic functions related to conic domains, Math. Slovaca, 64 (2014), 1183–1196. |
[29] | B. Khan, H. M. Srivastava, N. Khan, M. Darus, M. Tahir, A. Samad, Applications of higher-order derivatives to the subclasses of meromorphic starlike functions, J. Appl. Comput. Mech., 7 (2021), 321–333. |
[30] | Q. Khan, M. Arif, M. Raza, G. Srivastava, H. Tang, S. U. Rehman, Some applications of a new integral operator in q-analog for multivalent functions, Mathematics, 7 (2019), 1–13. |
[31] |
H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Am. Math. Soc., 42 (1936), 689–692. doi: 10.1090/S0002-9904-1936-06397-4
![]() |
[32] | S. Mahmood, G. Srivastava, H. M. Srivastava, E. S. A. Abujarad, M. Arif, F. Ghani, Sufficiency criterion for a subfamily of meromorphic multivalent functions of reciprocal order with respect to symmetric points, Symmetry, 11 (2019), 1–7. |
[33] | G. Murugusundaramoorthy, Harmonic meromorphic convex functions with missing coefficients, J. Indones. Math. Soc., 10 (2004), 15–22. |
[34] |
G. Murugusundaramoorthy, Starlikeness of multivalent meromorphic harmonic functions, Bull. Korean Math. Soc., 40 (2003), 553–564. doi: 10.4134/BKMS.2003.40.4.553
![]() |
[35] |
M. Öztürk, H. Bostanci, Certain subclasses of meromorphic harmonic starlike functions, Integr. Transf. Spec. Funct., 19 (2008), 377–385. doi: 10.1080/10652460801895588
![]() |
[36] |
K. Piejko, J. Sokół, K. Trabka-Wiecław, On q-calculus and starlike functions, Iran. J. Sci. Technol. Trans. A: Sci., 43 (2019), 2879–2883. doi: 10.1007/s40995-019-00758-6
![]() |
[37] | S. Ponnusamy, N. Rajasekaran, New sufficient conditions for starlike and univalent functions, Soochow J. Math., 21 (1995), 193–201. |
[38] |
T. Sheil-Small, Constants for planar harmonic mappings, J. London Math. Soc., s2-42 (1990), 237–248. doi: 10.1112/jlms/s2-42.2.237
![]() |
[39] |
H. Silverman, Harmonic univalent functions with negative coefficients, J. Math. Anal. Appl., 220 (1998), 283–289. doi: 10.1006/jmaa.1997.5882
![]() |
[40] |
H. M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A: Sci.,
44 (2020), 327–344. doi: 10.1007/s40995-019-00815-0
![]() |
[41] | H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, In: Univalent Functions, Fractional Calculus, and Their Applications (H. M. Srivastava, S. Owa, Eds.), Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons: New York, Chichester, Brisbane and Toronto, (1989), 329–354. |
[42] |
H. M. Srivastava, M. K. Aouf, A. O. Mostafa, Some properties of analytic functions associated with fractional q-calculus operators, Miskolc Math. Notes, 20 (2019), 1245–1260. doi: 10.18514/MMN.2019.3046
![]() |
[43] | H. M. Srivastava, B. Khan, N. Khan, Q. Z. Ahmad, Coefficient inequalities for q-starlike functions associated with the Janowski functions, Hokkaido Math. J., 48 (2019), 407–425. |
[44] | H. M. Srivastava, M. Tahir, B. Khan, Q. Z. Ahmad, N. Khan, Some general classes of q-starlike functions associated with the Janowski functions, Symmetry, 11 (2019), 1–14. |
[45] | B. A. Stephen, P. Nirmaladevi, T. V. Sudharsan, K. G. Subramanian, A class of harmonic meromorphic functions with negative coefficients, Chamchuri J. Math., 1 (2009), 87–94. |
[46] | Z. G. Wang, H. Bostanci, Y. Sun, On meromorphically harmonic starlike functions with respect to symmetric and conjugate points, Southeast Asian Bull. Math., 35 (2011), 699–708. |
[47] |
Z. Z. Zou, Z. R. Wu, On meromorphically starlike functions and functions meromorphically starlike with respect to symmetric conjugate points, J. Math. Anal. Appl., 261 (2001), 17–27. doi: 10.1006/jmaa.2001.7441
![]() |
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