The impact of population aging on economic growth is a very important issue in the process of population structure change. This paper first proposes research hypotheses based on a systematic literature review and theoretical analysis of the negative and positive effects of population aging on economic growth. Then, based on the data of 30 provinces in the Chinese Mainland from 2000 to 2019, this paper empirically tests the impact of population aging on economic growth and its impact mechanism using a static panel data model, a dynamic panel data model and a mediating effect model. Our empirical analysis leads to the following conclusions. First, population aging has a significant inhibitory effect on economic growth. Second, the industrial structure upgrading plays a mediating role in the process of population aging affecting economic growth; that is, population aging inhibits economic growth by affecting the overall upgrading of the industrial structure as well as the industrial rationalization and optimization. Last, some policy implications are proposed based on the research conclusions.
Citation: Yue Liu, Liming Chen, Liangting Lv, Pierre Failler. The impact of population aging on economic growth: a case study on China[J]. AIMS Mathematics, 2023, 8(5): 10468-10485. doi: 10.3934/math.2023531
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The impact of population aging on economic growth is a very important issue in the process of population structure change. This paper first proposes research hypotheses based on a systematic literature review and theoretical analysis of the negative and positive effects of population aging on economic growth. Then, based on the data of 30 provinces in the Chinese Mainland from 2000 to 2019, this paper empirically tests the impact of population aging on economic growth and its impact mechanism using a static panel data model, a dynamic panel data model and a mediating effect model. Our empirical analysis leads to the following conclusions. First, population aging has a significant inhibitory effect on economic growth. Second, the industrial structure upgrading plays a mediating role in the process of population aging affecting economic growth; that is, population aging inhibits economic growth by affecting the overall upgrading of the industrial structure as well as the industrial rationalization and optimization. Last, some policy implications are proposed based on the research conclusions.
In 1903, Mittag-Leffler [22] provided the function Eσ(z) defined by
Eσ(z)=∞∑j=0 zjΓ(σj+1),(σ,z∈C,R(σ)>0), |
where Γ is the gamma function and R means the real part.
Wiman [34] introduced the following generalized Mittag-Leffler function
Eσ,μ(z)=∞∑j=0 zjΓ(σj+μ),(σ,μ,z∈C,[R(σ),R(μ)]>0). |
Prabhakar [25] introduced the following function Eρσ,μ(z) in the form
Eρσ,μ(z)=∞∑j=0 (ρ)jΓ(μ+σj).zjj!, (σ,μ,ρ,z∈C,[R(σ),R(μ),R(ρ)]>0). |
Later, Shukla and Prajapati [27] (see also [32]) defined another generalized Mittag-Leffler function
Eρ,kσ,μ(z)=∞∑j=0 (ρ)kjΓ(μ+σj)zjj!,(σ,μ,ρ,z∈C,[R(σ),R(μ),R(ρ)]>0) |
where k∈(0,1)⋃N and (ρ)kj=Γ(ρ+kj)Γ(ρ) is the generalized Pochhammer symbol defined as
kkjk∏m=1(ρ+m−1k)j if k∈N. |
Bansal and Prajapat [5] and Srivastava and Bansal [31] investigated geometric properties of the Mittag-Leffler function Eσ,μ(z), including starlikeness, convexity, and close-to-convexity (see [1,4,6,8,12,13,17,28,29]). In reality, the generalized Mittag-Leffler function Eσ,μ(z) and its extensions are still widely used in geometric function theory and in a variety of applications (see, for details, [2,3,7,16,24]).
Let S(p) be the class of functions of the form
f(z)=zp+∞∑j=p+1ajzj, | (1.1) |
where f is holomorphic and multivalent in the open unit disk O={z:|z|<1}.
Let f and F be two functions in S(p). Then the convolution (or Hadamard product), denoted by f∗F, is defined as
(f∗F)(z)=zp+∞∑j=p+1ajdjzj=(F∗f)(z), |
where f(z) is in (1.1) and F(z)=zp+∞∑j=p+1djzj.
Let f(z) and h(z) be two analytic functions defined in O. The function f(z) is called subordinate to h(z), or h(z) is superordinate to f(z), denoted by f(z)≺h(z) and h(z)≺f(z), respectively, if there is a Schwarz function φ with φ(z)=0,|φ(z)|<1 and f(z)=h(φ(z)). If the function h is univalent in O, then the following equivalence is true if
f(z)≺h(z) (z∈O)⇔f(0)=h(0) and f(O)⊂h(O). |
Definition 1.1. ([18]). Let 0<q<1. Then [j]q! denotes the q-factorial, which is defined as follows:
[j]q!={[j]q[j−1]q…[2]q[1]q, j=1,2,3,…1, j=0 |
where [j]q=1−qj1−q=1+∑j−1m=1 qm and [0]q=0.
Definition 1.2 ([18]). The q-generalized Pochhammer symbol [ρ]j,q, ρ∈C, is given as
[ρ]j,q=[ρ]q[ρ+1]q[ρ+2]q…[ρ+j−1]q, |
and the q-Gamma function is defined as
Γq(ρ+1)=[ρ]qΓq(ρ) and Γq(1)=1. |
It follows that Γq(j+1)=[j]q!.
Lately, many results have been given for some related special functions such as the Wright function [3] and multivalent functions (see [10,23,26]).
Here, we propose a q-extension of specific extensions of the Mittag-Leffler function, motivated by the success of Mittag-Leffler function applications in physics, biology, engineering, and applied sciences. We generalize the Mittag-Leffler function given by Shukla and Prajapati [27] and obtain a new generalized q-Mittag-Leffler function.
Now, we present a new generalized q-Mittag-Leffler function as follows
Eρσ,μ(q;z)=z+∞∑j=2 (ρ)kjΓq(μ+σj)zjj!. | (1.2) |
It is obvious that, when q→1−, the resulting function is the generalized Mittag-Leffler function, which is given by Shukla and Prajapati [27].
Corresponding to the function Eρσ,μ(q;z) in (1.2), we establish the following generalized q-Mittag-Leffler function Eρσ,μ(p,q;z) in multivalent functions S(p), as given below
Eρσ,μ(p,q;z)=zp+∞∑j=p+1 (ρ)k(j−p)Γq(μ+σ(j−p))zj(j−p)!. | (1.3) |
Again, using the new function (1.3), we define the following function:
Gρσ,μ(p,q;z):=zpΓq(μ)Eρσ,μ(p,q;z)=zp+∞∑j=p+1 Γq(μ)(ρ)k(j−p)Γq(μ+σ(j−p))zj(j−p)!. | (1.4) |
Definition 1.3. For f∈S(p), we define the new linear operator Aμ,ρ;kσ;p,qf(z):S(p)→S(p) by
Aμ,ρ;kσ;p,qf(z)=Gρσ,μ(p,q;z)∗f(z)=zp+∞∑j=p+1 χjajzj, | (1.5) |
where χj=Γq(μ)(ρ)kjΓq(μ+σj)j!.
We now define a subclass Qμ,ρ;kσ;q(M,N;τ,p) of the family S(p) using the multivalent linear operator in (1.5) and the subordination concept.
Definition 1.4. Let Aμ,ρ;kσ;p,qf(z) be an operator in (1.5). A function f(z)∈S(p) is said to be in the class Qμ,ρ;kσ;q(M,N;τ,p) if satisfies the following subordination condition:
1p−τ(z(Aμ,ρ;kσ;p,qf(z))′Aμ,ρ;kσ;p,qf(z)−τ)≺1+Mz1+Nz, (z∈O) | (1.6) |
or equivalently
z(Aμ,ρ;kσ;p,qf(z))′Aμ,ρ;kσ;p,qf(z)≺p+(pN+(M−N)(p−τ))z1+Nz, (z∈O) |
and
|z(Aμ,ρ;kσ;p,qf(z))′Aμ,ρ;kσ;p,qf(z)−pNz(Aμ,ρ;kσ;p,qf(z))′Aμ,ρ;kσ;p,qf(z)−[pN+(M−N)(p−τ)]|<1, | (1.7) |
where −1≤M<N≤1, 0≤τ<p, and p∈N.
Remark 1.1. Some well-known special classes of the class Qμ,ρ;kσ;q(M,N;τ,p) can be obtained by choosing the values of the parameters ς,μ,ρ;τ,k,p,q, M, and N.
(1) Q0,0,10,1(M,N;τ,p)=S∗p(M,N;τ,p) was provided by Aouf [2].
(2) Q0,0,10,1(M,N;0,p)=S∗p(M,N;p) was provided by Goel and Sohi [16].
In this work, we introduce a new subclass of multivalent functions Qμ,ρ;kσ;q(M,N;τ,p) defined by the new linear operator Aμ,ρ;kσ;p,qf(z). And we study some geometric properties for the class Qμ,ρ;kσ;q(M,N;τ,p) such as the coefficient estimates, convexity and convex linear combination. Finally, the radius theorems associated with the generalized Srivastava-Attiya integral operator will be investigated.
The first theorem in this section presents the necessary and sufficient condition for the function f(z) in (1.1) belong to the class Qμ,ρ;kσ;q(M,N;τ,p).
Theorem 2.1. A function f(z) is in the class Qμ,ρ;kσ;q(M,N;τ,p) if and only if
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χj|aj|≤(M−N)(p−τ), | (2.1) |
where 1≤M<N≤1, 0≤τ<p, and p∈N.
Proof. Assume that the condition (2.1) is true. Then by (1.7), we have
|z(Aμ,ρ;kσ;p,qf(z))′−pAμ,ρ;kσ;p,qf(z)|−|Nz(Aμ,ρ;kσ;p,qf(z))′−[(M−N)(p−τ)+pN]Aμ,ρ;kσ;p,qf(z)|=|∞∑j=p+1(j−p)χjajzj|−|(M−N)(p−τ)zj−∞∑j=p+1[Nj−((M−N)(p−τ)+pN)]χjajzj|≤−(M−N)(p−τ)+∞∑j=p+1[(1+N)(j−p)+((M−N)(p−τ))]χj|aj|≤0. |
By maximum modulus theorem [11], we get f(z)∈Qμ,ρ;kσ;q(M,N;τ,p).
Conversely, suppose that f(z)∈Qμ,ρ;kσ;q(M,N;τ,p). Then
|z(Aμ,ρ;kσ;p,qf(z))′Aμ,ρ;kσ;p,qf(z)−pNz(Aμ,ρ;kσ;p,qf(z))′Aμ,ρ;kσ;p,qf(z)−[pN+(M−N)(p−τ)]|=|∑∞j=p+1(j−p)χjajzj(M−N)(p−τ)zj−∑∞j=p+1[Nj−((M−N)(p−τ)+pN)]χjajzj|<1. |
Since R(z)≤|z|, we get
R{∑∞j=p+1(j−p)χjajzj(M−N)(p−τ)zj−∑∞j=p+1[Nj−((M−N)(p−τ)+pN)]χjajzj}<1. |
Taking z→1−, we have
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χj|aj|≤(M−N)(p−τ). |
This completes the proof.
Theorem 2.2. Let f1 and f2 be analytic functions in the class Qμ,ρ;kσ;q(M,N;τ,p). Then f1∗f2∈Qμ,ρ;kσ;q(M,N;τ,p), where
τ1=p−(1−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(1−p)+(M−N)(p−τ1))χ1]2−(M−N)2(p−τ)2χ1, | (2.2) |
where χ1=Γq(μ)(ρ)kΓq(μ+ς).
Proof. We will show that τ1 is the largest satisfying
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ1))χj(M−N)(p−τ1)aj,1aj,2≤1. | (2.3) |
Since f1,f2∈Qμ,ρ;kσ;q(M,N;τ,p), by the condition (2.1) and the Cauchy-Schwarz inequality, we get
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)√aj,1aj,2≤1. | (2.4) |
From (2.3) and (2.4), we observe that
√aj,1aj,2≤[((1+N)(j−p)+(M−N)(p−τ))χj](p−τ1)[((1+N)(j−p)+(M−N)(p−τ1))χj](p−τ). |
From (2.4), it is necessary to prove
(M−N)(p−τ)((1+N)(j−p)+(M−N)(p−τ))χj≤[((1+N)(j−p)+(M−N)(p−τ))χj](p−τ1)[((1+N)(j−p)+(M−N)(p−τ1))χj](p−τ). | (2.5) |
Furthermore, from the inequality (2.5) it follows that
τ1≤p−(j−p)(1+N)(M−N)(p−τ)2χj[((1+N)(j−p)+(M−N)(p−τ1))χj]2−(M−N)2(p−τ)2χj. |
Now, set
E(j)=p−(j−p)(1+N)(M−N)(p−τ)2χj[((1+N)(j−p)+(M−N)(p−τ1))χj]2−(M−N)2(p−τ)2χj. |
We observe that the function E(j) is increasing for j∈N. Putting j=1, we have
τ1=E(1)=p−(1−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(1−p)+(M−N)(p−τ1))χ1]2−(M−N)2(p−τ)2χ1. |
This completes the proof.
Theorem 2.3. Let f1 and f2 be analytic functions in the class Qμ,ρ;kσ;q(M,N;τ,p) of forms given in (1.1) with aj,1 and aj,2, respectively. Then
w(z)=zp+∞∑j=p+1(a2j,1+a2j,2)zj∈Qμ,ρ;kσ;q(M,N;τ,p), |
where
η=p−(1−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(1−p)+(M−N)(p−τ1))χ1]2−(M−N)2(p−τ)2χ1. |
Proof. By Theorem 2.1, we have
∞∑j=p+1 [((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)]2a2j,s≤∞∑j=p+1 [((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)aj,s]2≤1, (s=1,2). |
From the above inequality, we obtain
∞∑j=p+1 12[((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)]2(a2j,1+a2j,2)≤1. |
Therefore, the largest η can be obtained such that
((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)≤12[((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)]2. |
That is,
η≤p−2(j−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(j−p)+(M−N)(p−τ1))χ1]2−2(M−N)2(p−τ)2χ1. |
Now, set
E(j)=p−2(j−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(j−p)+(M−N)(p−τ1))χ1]2−2(M−N)2(p−τ)2χ1. |
We observe that the function E(j) is increasing for j∈N. Putting j=1, we have
η=E(1)=p−2(1−p)(1+N)(M−N)(p−τ)2χ1[((1+N)(1−p)+(M−N)(p−τ1))χ1]2−2(M−N)2(p−τ)2χ1. |
This completes the proof.
Theorem 2.4. Let f1,f2∈Qμ,ρ;kσ;q(M,N;τ,p). Then for γ∈[0,1], the function F(z)=(1−γ)f1+γf2 belongs to the class Qμ,ρ;kσ;q(M,N;τ,p).
Proof. Since the functions f1 and f2 belong to the class Qμ,ρ;kσ;q(M,N;τ,p),
F(z)=(1−γ)f1+γf2=zp+∞∑j=p+1ηjzj, |
where ηj=(1−γ)aj,1+γaj,2.
By (2.1), we observe that
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χj[(1−γ)aj,1+γaj,2]=(1−γ)∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χjaj,1+γ∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χjaj,2≤(1−γ)(M−N)(p−τ)+γ(M−N)(p−τ). |
Hence F(z)∈Qμ,ρ;kσ;q(M,N;τ,p).
Theorem 2.5. Let fs(z)=zp+∑∞j=p+1aj,szj be in the class Qμ,ρ;kσ;q(M,N;τ,p) for s=1,2,…,m. Then the function P(z)=∑ms=1ℵsfs, where ∑ms=1ℵs=1, is also in the class Qμ,ρ;kσ;q(M,N;τ,p).
Proof. By Theorem 2.1, we have
∞∑j=p+1 ((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)aj,s≤1. |
Since
P(z)=m∑s=1ℵsfs=m∑s=1ℵs(zp+∞∑j=p+1aj,szj)=zp+∞∑j=p+1(m∑s=1ℵsaj,s)zj, |
∞∑j=p+1((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ)m∑s=1ℵsaj,s≤1. |
Thus P(z)∈Qμ,ρ;kσ;q(M,N;τ,p).
In this section, we investigate radii of multivalent starlikeness, multivalent convexity, and multivalent close-to-convex for the function f(z) in the class Qμ,ρ;kσ;q(M,N;τ,p) with the generalized integral operator of Srivastava-Attiya.
Jung et al. [19] introduced an integral operator with one parameter as follows:
Iδ(f)(z):=2δzΓ(δ)∫z0 (log(zv) )δ−1f(v)dv=z+∞∑j=2 (2j+1)δajzj(δ>0;f∈S). |
In 2007, Srivastava and Attiya [30] investigated a new integral operator, which is called Srivastava-Attiya operator, given by
Ju,mf(z)=z+∞∑j=1(1+uj+u)δajzj. |
Many studies are concerned with the study of the operator of Srivastava-Attiya (see [9,14,15,20]).
Mishra and Gochhayat [21] (also [33]) provided a fractional differintegral operator Jmu,pf(z):S(p)→S(p) which is called a generalized of Srivastava-Attiya integral operator, defined by
Jmu,pf(z)=zp+∞∑j=p+1(p+uj+u)δajzj. | (3.1) |
Theorem 3.1. If f(z)∈Qμ,ρ;kσ;q(M,N;τ,p) and 0≤τ<p, then Jmu,pf(z) in (3.1) is multivalent starlike of order τ in |z|≤r1, where
r1=infj≥p+1{((1+N)(j−p)+(M−N)(p−τ))χj(j+u)δ(M−N)(j−2p+τ)(p+u)δ}. | (3.2) |
Proof. According to the definition of a starlike function in [28], we have
|z(Jmu,pf(z))′Jmu,pf(z)−p|≤p−τ, | (3.3) |
|z(Jmu,pf(z))′Jmu,pf(z)−p|=|∑∞j=p+1(j−p)(p+uj+u)δajzj∑∞j=p+1(p+uj+u)δajzj|≤∑∞j=p+1(j−p)(p+uj+u)δaj|z|j∑∞j=p+1(p+uj+u)δaj|z|j. |
By (3.2), we have
∞∑j=p+1(j−2p+τ)(p+u)δaj|z|j(p−τ)(j+u)δ≤1. |
By (2.1) in Theorem 2.1, it is clear that
(j−2p+τ)(p+u)δ(p−τ)(j+u)δ|z|j≤((1+N)(j−p)+(M−N)(p−τ))χj(M−N)(p−τ). |
Therefore,
|z|≤{((1+N)(j−p)+(M−N)(p−τ))χj(j+u)δ(M−N)(j−2p+τ)(p+u)δ}1j. |
This completes the proof.
Theorem 3.2. If f(z)∈Qμ,ρ;kσ;q(M,N;τ,p) and 0≤τ<p, then Jmu,pf(z) in (3.1) is multivalent convex of order τ in |z|≤r2, where
r2=infj≥p+1{((1+N)(j−p)+(M−N)(p−τ))χjp(j+u)δ(M−N)[j(j−2p+τ)](p+u)δ}. | (3.4) |
Proof. To verify (3.4), it is necessary to prove
|(1+z(Jmu,pf(z))′′(Jmu,pf(z))′)−p|≤p−τ, |
but the result is obtained by repeating the steps in Theorem 3.1.
Corollary 3.1. If f(z)∈Qμ,ρ;kσ;q(M,N;τ,p) and 0≤τ<p, then Jmu,pf(z) in (3.1) is multivalent close-to-convex of order τ in |z|≤r3, where
r3=infj≥1{((1+N)(j−p)+(M−N)(p−τ))χj(j+u)δ(M−N)j(p+u)δ}. | (3.5) |
In this work, we established and investigated a new generalized Mittag-Leffler function, which is a generalization of q-Mittag-Leffler function defined by Shukla and Prajapati [27]. Also, we studied some of the geometric properties of a certain subclass of multivalent functions. In addition, we introduced radius theorem using a generalized Srivastava-Attiya integral operator. Since the Mittag-Leffler function is of importance, it is related to a wide range of problems in mathematical physics, engineering, and the applied sciences. The results obtained in this article may have many other applications in special functions.
The authors express many thanks to the Editor-in-Chief, handling editor, and the reviewers for their outstanding comments that improve our paper.
The authors declare that they have no competing interests concerning the publication of this article.
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