Citation: Hongli Niu, Kunliang Xu. A hybrid model combining variational mode decomposition and an attention-GRU network for stock price index forecasting[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7151-7166. doi: 10.3934/mbe.2020367
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Let A denote the class of functions of the form
f(z)=z+a2z2+a3z3+a4z4+⋯, | (1.1) |
which are analytic in the open unit disk D=(z:∣z∣<1) and normalized by f(0)=0 and f′(0)=1. Recall that, S⊂A is the univalent function in D=(z:∣z∣<1) and has the star-like and convex functions as its sub-classes which their geometric condition satisfies Re(zf′(z)f(z))>0 and Re(1+zf′′(z)f′(z))>0. The two well-known sub-classes have been used to define different subclass of analytical functions in different direction with different perspective and their results are too voluminous in literature.
Two functions f and g are said to be subordinate to each other, written as f≺g, if there exists a Schwartz function w(z) such that
f(z)=g(w(z)),zϵD | (1.2) |
where w(0) and ∣w(z)∣<1 for zϵD. Let P denote the class of analytic functions such that p(0)=1 and p(z)≺1+z1−z, zϵD. See [1] for details.
Goodman [2] proposed the concept of conic domain to generalize convex function which generated the first parabolic region as an image domain of analytic function. The same author studied and introduced the class of uniformly convex functions which satisfy
UCV=Re{1+(z−ψ)f″(z)f′(z)}>0,(z,ψ∈A). |
In recent time, Ma and Minda [3] studied the underneath characterization
UCV=Re{1+zf″(z)f′(z)>|zf″(z)f′(z)|},zϵD. | (1.3) |
The characterization studied by [3] gave birth to first parabolic region of the form
Ω={w;Re(w)>∣w−1∣}, | (1.4) |
which was later generalized by Kanas and Wisniowska ([5,6]) to
Ωk={w;Re(w)>k∣w−1∣,k≥0}. | (1.5) |
The Ωk represents the right half plane for k=0, hyperbolic region for 0<k<1, parabolic region for k=1 and elliptic region for k>1 [30].
The generalized conic region (1.5) has been studied by many researchers and their interesting results litter everywhere. Just to mention but a few Malik [7] and Malik et al. [8].
More so, the conic domain Ω was generalized to domain Ω[A,B], −1≤B<A≤1 by Noor and Malik [9] to
Ω[A,B]={u+iv:[(B2−1)(U2+V2)−2(AB−1)u+(A2−1)]2 |
>[−2(B+1)(u2v2)+2(A+B+C)u−2(A+1)]2+4(A−B)2v2} |
and it is called petal type region.
A function p(z) is said to be in the class UP[A,B], if and only if
p(z)≺(A+1)˜p(z)−(A−1)(B+1)˜p(z)−(B−1), | (1.6) |
where ˜p(z)=1+2π2(log1+√z1−√z)2.
Taking A=1 and B=−1 in (1.8), the usual classes of functions studied by Goodman [1] and Kanas ([5,6]) will be obtained.
Furthermore, the classes UCV[A,B] and ST[A,B] are uniformly Janoski convex and Starlike functions satisfies
Re((B−1)(zf′(z))′f′(z)−(A−1)(B+1)(zf′(z))′f′(z)−(A+1))>|(B−1)(zf′(z))′f′(z)−(A−1)(B+1)(zf′(z))′f′(z)−(A+1)−1| | (1.7) |
and
Re((B−1)zf′(z)f′(z)−(A−1)(B+1)zf′(z)f′(z)−(A+1))>|(B−1)zf′(z)f′(z)−(A−1)(B+1)zf′(z)f′(z)−(A+1)−1|, | (1.8) |
or equivalently
(zf′(z))′f′(z)∈UP[A,B] |
and
zf′(z)f′(z)∈UP[A,B]. |
Setting A=1 and B=−1 in (1.7) and (1.8), we obtained the classes of functions investigated by Goodman [2] and Ronning [10].
The relevant connection to Fekete-Szegö problem is a way of maximizing the non-linear functional |a3−λa22| for various subclasses of univalent function theory. To know much of history, we refer the reader to [11,12,13,14] and so on.
The error function was defined because of the normal curve, and shows up anywhere the normal curve appears. Error function occurs in diffusion which is a part of transport phenomena. It is also useful in biology, mass flow, chemistry, physics and thermomechanics. According to the information at hand, Abramowitz [15] expanded the error function into Maclaurin series of the form
Erf(z)=2√π∫z0e−t2dt=2√π∞∑n=0(−1)nz2n+1(2n+1)n! | (1.9) |
The properties and inequalities of error function were studied by [16] and [4] while the zeros of complementary error function of the form
erfc(z)=1−erf(z)=2√π∫∞ze−t2dt, | (1.10) |
was investigated by [17], see for more details in [18,19] and so on. In recent time, [20,21,22] and [23] applied error functions in numerical analysis and their results are flying in the air.
For f given by [15] and g with the form g(z)=z+b2z2+b3z3+⋯ their Hadamard product (convolution) by f∗g and at is defined as:
(f∗g)(z)=z+∞∑n=2anbnzn | (1.11) |
Let Erf be a normalized analytical function which is obtained from (1.9) and given by
Erf=√πz2erf(z)=z+∞∑n=2(−1)n−1zn(2n−1)(n−1)! | (1.12) |
Therefore, applying a notation (1.11) to (1.1) and (1.12) we obtain
ϵ=A∗Erf={F:F(z)=(f∗Erf)(z)=z+∞∑n=2(−1)n−1anzn(2n−1)(n−1)!,f∈A}, | (1.13) |
where Erf is the class that consists of a single function or Erf. See concept in Kanas et al. [18] and Ramachandran et al. [19].
Babalola [24] introduced and studied the class of λ−pseudo starlike function of order β(0≤β≤1) which satisfy the condition
Re(z(f′(z))λf(z))>β, | (1.14) |
where λ≥1∈ℜ(z∈D) and denoted by ∠λ(β). We observed from (1.14) that putting λ=2, the geometric condition gives the product combination of bounded turning point and starlike function which satisfy
Ref′(z)(z(f′(z))f(z))>β |
Olatunji [25] extended the class ∠λ(β) to ∠βλ(s,t,Φ) which the geometric condition satisfy
Re((s−t)z(f′(z))λf(sz)−f(tz))>β, |
where s,t∈C,s≠t,λ≥1∈ℜ,0≤β<1,z∈D and Φ(z) is the modified sigmoid function. The initial coefficient bounds were obtained and the relevant connection to Fekete-Szegö inequalities were generated. The contributions of authors like Altinkaya and Özkan [26] and Murugusundaramoorthy and Janani [27] and Murugusundaramoorthy et al. [28] can not be ignored when we are talking on λ-pseudo starlike functions.
Inspired by earlier work by [18,19,29]. In this work, the authors employed the approach of [13] to study the coefficient inequalities for pseudo certain subclasses of analytical functions related to petal type region defined by error function. The first few coefficient bounds and the relevant connection to Fekete-Szegö inequalities were obtained for the classes of functions defined. Also note that, the results obtained here has not been in literature and varying of parameters involved will give birth to corollaries.
For the purpose of the main results, the following lemmas and definitions are very necessary.
Lemma 1.1. If p(z)=1+p1z+p2z2+⋯ is a function with positive real part in D, then, for any complex μ,
|p2−μp21|≤2max{1,|2μ−1|} |
and the result is sharp for the functions
p0(z)=1+z1−zorp(z)=1+z21−z2(z∈D). |
Lemma 1.2. [29] Let p∈UP[A,B],−1≤B<A≤1 and of the form p(z)=1+∞∑n=1pnzn. Then, for a complex number μ, we have
|p2−μp21|≤4π2(A−B)max(1,|4π2(B+1)−23+4μ(A−Bπ2)|). | (1.15) |
The result is sharp and the equality in (1.15) holds for the functions
p1(z)=2(A+1)π2(log1+√z1−√z)2+22(B+1)π2(log1+√z1−√z)2+2 |
or
p2(z)=2(A+1)π2(log1+z1−z)2+22(B+1)π2(log1+z1−z)2+2. |
Proof. For h∈P and of the form h(z)=1+∞∑n=1cnzn, we consider
h(z)=1+w(z)1−w(z) |
where w(z) is such that w(0)=0 and |w(z)|<1. It follows easily that
w(z)=h(z)−1h(z)+1=12z+(c22−c214)z2+(c32−c2c12+c318)z3+⋯ | (1.16) |
Now, if ˜p(z)=1+R1z+R2z2+⋯, then from (1.16), one may have,
˜p(w(z))=1+R1w(z)+R2(w(z))2+R3(w(z))3⋯ | (1.17) |
where R1=8π2,R2=163π2, and R3=18445π2, see [30]. Substitute R1,R2 and R3 into (1.17) to obtain
˜p(w(z))=1+4c1π2z+4π2(c2−c216)z2+4π2(c3−c1c23+2c3145)z3+⋯ | (1.18) |
Since p∈UP[A,B], so from relations (1.16), (1.17) and (1.18), one may have,
p(z)=(A+1)˜p(w(z))−(A−1)(B+1)˜p(w(z))−(B−1)=2+(A+1)4π2c1z+(A+1)4π2(c2−c216)z2+⋯2+(B+1)4π2c1z+(B+1)4π2(c2−c216)z2+⋯ |
This implies that,
p(z)=1+2(A−B)c1π2z+2(A−B)π2(c2−c216−2(B−1)c21π2)z2+8(A−B)π2[((B+1)2π4+B+16π2190)c21−(B+1π2+112)c1c2+c34]z3+⋯ | (1.19) |
If p(z)=1+∑∞n=1pnzn, then equating coefficients of z and z2, one may have,
p1=2π2(A−B)c1 |
and
p2=2π2(A−B)(c2−c216−2(B−1)c21π2). |
Now for a complex number μ, consider
p2−μp21=2(A−B)π2[c2−c21(16+2(B+1)π2+2μ(A−B)π2)] |
This implies that
|p2−μp21|=2(A−B)π2|c2−c21(16+2(B+1)π2+2μ(A−B)π2)|. |
Using Lemma 1.1, one may have
|p2−μp21|=4(A−B)π2max{1,|2v−1|}, |
where v=16+2(B+1)π2+2μ(A−B)π2, which completes the proof of the Lemma.
Definition 1.3. A function FϵA is said to be in the class UCV[λ,A,B], −1≤B<A≤1, if and only if,
Re((B−1)(z(F′(z)λ))′F′(z)−(A−1)(B+1)(z(F′(z)λ))′F′(z)−(A+1))>|(B−1)(z(F′(z)λ))′F′(z)−(A−1)(B+1)(z(F′(z)λ))′F′(z)−(A+1)−1|, | (1.20) |
where λ≥1ϵR or equivalently (z(F′(z)λ))′F′(z)ϵUP[A,B].
Definition 1.4. A function FϵA is said to be in the class US[λ,A,B], −1≤B<A≤1, if and only if,
Re((B−1)z(F′(z)λ)F(z)−(A−1)(B+1)z(F′(z)λ)F(z)−(A+1))>|(B−1)z(F′(z)λ)F(z)−(A−1)(B+1)z(F′(z)λ)F(z)−(A+1)−1|, | (1.21) |
where λ≥1ϵR or equivalently z(F′(z)λ)F(z)ϵUP[A,B].
Definition 1.5. A function FϵA is said to be in the class UMα[λ,A,B], −1≤B<A≤1, if and only if,
Re((B−1)[(1−α)z(F′(z)λ)F(z)+α(z(F′(z)λ))′F′(z)]−(A−1)(B+1)[(1−α)z(F′(z)λ)F(z)+α(z(F′(z)λ))′F′(z)]−(A+1))>|(B−1)[(1−α)z(F′(z)λ)F(z)+α(z(F′(z)λ))′F′(z)]−(A−1)(B+1)[(1−α)z(F′(z)λ)F(z)+α(z(F′(z)λ))′F′(z)]−(A+1)−1|, |
where α≥0 and λ≥1ϵR or equivalently (1−α)z(F′(z)λ)f(z)+α(z(f′(z)λ))′f′(z)∈UP[A,B].
In this section, we shall state and prove the main results, and several corollaries can easily be deduced under various conditions.
Theorem 2.1. Let F∈US[λ,A,B], −1≤B<A≤1, and of the form (1.13). Then, for a real number μ, we have
|a3−μa22|≤40(A−B)|1−3λ|π2max{1,|4(B+1)π2−13−2(A−B)(1−2λ)2π2(2(2λ2−4λ+1)−9μ(1−3λ)5)|}. |
Proof. If F∈US[λ,A,B], −1≤B<A≤1, the it follows from relations (1.18), (1.19), and (1.20),
z(F′(z)λ)F(z)=(A+1)˜p(w(z))−(A−1)(B+1)˜p(w(z))−(B−1), |
where w(z) is such that w(0)=0 and ∣w(z)∣<1. The right hand side of the above expression get its series form from (1.13) and reduces to
z(F′(z)λ)F(z)=1+2(A−B)c1π2z+2(A−B)π2(c2−c216−2(B−1)c21π2)z2 |
+8(A−B)π2[((B+1)2π4+B+16π2190)c21−(B+1π2+112)c1c2+c34]z3+⋯. | (2.1) |
If F(z)=z+∑∞n=2(−1)n−1anzn(2n−1)(n−1)!, then one may have
z(F′(z)λ)F(z)=1+1−2λ3a2z+(2λ2−4λ+19a22−1−3λ10a3)z2+⋯ | (2.2) |
From (2.1) and (2.2), comparison of coefficient of z and z2 gives,
a2=6(A−B)(1−2λ)π2c1 | (2.3) |
and
2λ2−4λ+19a22−1−3λ10a3=2(A−B)π2(c2−16c21−2(B+1)π2c21). |
This implies, by using (2.3), that
a3=−20(A−B)(1−3λ)π2[c2−16c21−2(B+1)π2c21−2(2λ2−4λ+1)(A−B)(1−2λ)2π2c21]. |
Now, for a real number μ consider
|a3−μa22|= |
|−20(A−B)(1−3λ)π2(c2−16c21−2(B+1)π2c21)+40(A−B)2(2λ2−4λ+1)(1−2λ)2(1−3λ)π4−36μ(A−B)2c21(1−2λ)2π4| |
=20(A−B)(1−3λ)π2|c2−c21(16+2(B+1)π2−2(A−B)(2λ2−4λ+1)(1−2λ)2π2+9μ(A−B)(1−3λ)5(1−2λ)2π2)| |
=20(A−B)(1−3λ)π2|c2−vc21| |
where v=16+2(B+1)π2−(A−B)(1−2λ)2π2(2(2λ2−4λ+1)−9μ(1−3λ)5).
Theorem 2.2. Let F∈UCV[λ,A,B], −1≤B<A≤1, and of the form (1.13). Then, for a real number μ, we have
|a3−μa22|≤40(A−B)3|1+3λ|π2max{1,|4(B+1)π2−13−2(1+3λ)(A−B)(1+2λ)2π2(λ−27μ20)|} |
Proof. If F∈UCV[λ,A,B], −1≤B<A≤1, then it follows from relations (1.18), (1.19), and (1.21),
(zF′(z)λ)′F′(z)=(A+1)˜p(w(z))−(A−1)(B+1)˜p(w(z))−(B+1), |
where w(z) is such that w(0)=0 and ∣w(z)∣<1. The right hand side of the above expression get its series form from (1.13) and reduces to,
(zF′(z)λ)′F′(z)=1+2(A−B)c1π2z+2(A−B)π2(c2−c216−2(B+1)π2c21)z2+8(A−B)π2[(B+1π4+B+16π2+190)c31−(B+1π2+112)c1c2+c34]z3+⋯ | (2.4) |
If F(z)=z+∑(−1)n−1anzn(2n−1)(n−1)!, then we have,
(zF′(z)λ)′F′(z)=1−2(1+2λ)3a2z+(1+3λ)(3a310+2λ9a22)z2+⋯ | (2.5) |
From (2.4) and (2.5), comparison of coefficients of z and z2 gives,
a2=−3(A−B)c1(1+2λ)π2 | (2.6) |
and
(1+3λ)(3a310+2λ9a22)=2(A−B)π2(c2−c216−2(B+1)c21π2) |
This implies, by using (2.6), that
a3=103[2(A−B)(1+3λ)π2(c2−c216−2(B+1)c21π2)+2λ(A−B)2c21(1+2λ)2π4]. |
Now, for a real number μ, consider
|a3−μa22|=|−20(A−B)3(1+3λ)π2(c2−16c1−2(B+1)π2c21)+20(A−B)2c213(1+2λ)π4−9μ(A−B)2c21(1+2λ)2π4| |
=20(A−B)3(1+3λ)π2|c2−c21(16+2(B+1)π2−λ(1+3λ)(A−B)(1+2λ)2π2+27μ(A−B)(1+3λ)20(1+2λ)2π2)| |
=20(A−B)3(1+3λ)π2|c2−vc21|, |
where
v=16+2(B+1)π2−(1+3λ)(A−B)(1+2λ)2π2(λ−27μ20). |
Theorem 2.3. F∈Mα[λ,A,B], −1≤B<A≤1, α≥0 and of the form (1.13). Then, for a real number μ, we have
|a3−μa22|≤40(A−B)π2|3(λ+α+2αλ)+α−1|max{1,|4(B+1)π2−13−4(A−B)[1−2λ−α(3+2λ)]2π2(2λ2(1+2α)+2λ(3α−2)+1−α−9μ(3(λ+α+2αλ)+α−1)10)|}. |
Proof. Let F∈Mα[λ,A,B], −1≤B<A≤1, α≥0 and of the form (1.13). Then, for a real number μ, we have
(1−α)z(F′(z))λF(z)+α(z(F′(z))λ)′F′(z)=(A+1)˜p(w(z))−(A−1)(B+1)˜p(w(z))−(B−1), | (2.7) |
where w(z) is such that w(z0)=0 and |w(z)|<1. The right hand side of the above expression get its series form from (2.7) and reduces to
(1−α)z(F′(z))λF(z)+α(z(F′(z))λ)′F′(z)=1+2(A−B)Gπ2z+2(A−B)π2(c2−c216−2(B+1)π2c21)z2+... | (2.8) |
If F(z)=z+∑∞n=2(−1)n−1anzn(2n−1)(n−1)!, then one may have
(1−α)z(F′(z))λF(z)+α(z(F′(z))λ)′F′(z)=(1−α)[1+1−2λ3a2z+(2λ2−4λ+19a22−1−3λ10a3)z2+...]+α[1−2(1+2λ)3a2z+(1+3λ)(3a310+2λ9a22)z2+...] | (2.9) |
from (2.8) and (2.9), comparison of coefficients of z and z2 gives
a2=6(A−B)c1[1−2λ−α(3+2λ)]π2 | (2.10) |
and
3(λ+α+2αλ)+α−110a3−2λ2(1+2λ)+α−19a22=2(A−B)π2(c2−c216−2(B+1)π2c21) |
This implies, by using (2.10), that
a3=103(λ+α+2αλ)+α−1[2(A−B)π2(c2−c216−2(B+1)π2c21)+4(A−B)2[2λ2(1+2λ)+2λ(3α−2)+1−α][1−2λ−α(3+2λ)]2π4c21] |
Now, for a real number μ, consider
|a3−μa22|=|103(λ+α+2αλ)+α−1[2(A−B)π2(c2−c216−2(B+1)π2c21)+4(A−B)2[2λ2(1+2λ)+2λ(3α−2)+1−α][1−2λ−α(3+2λ)]2π4c21]−36(A−B)2μG2[1−2λ−α(3+2λ)]2π4| |
=|20(A−B)π(3(λ+α+2αλ)+α−1)|c2−c21[16+2(B+1)π2−2(A−B)[2λ2(1+2α)+2λ(3α−2)+1−α](1−2λ−α(3+2λ))2π2+18μ(A−B)[3(λ+α+2αλ)+α−1]10[1−2λ−α(3+2λ)]2π2 |
=20(A−B)π(3(λ+α+2αλ)+α−1)|c2−vc21|, |
where
v=16+2(B+1)π2−2(A−B)[2λ2(1+2α)+2λ(3α−2)+1−α](1−2λ−α(3+2λ))2π2+18μ(A−B)[3(λ+α+2αλ)+α−1]10[1−2λ−α(3+2λ)]2π2. |
The force applied on certain subclasses of analytical functions associated with petal type domain defined by error function has played a vital role in this work. The results obtained are new and varying the parameters involved in the classes of function defined, these will bring new more results that has not been in existence.
The authors would like to thank the referees for their valuable comments and suggestions.
The authors declare that they have no conflict of interests.
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