Research article

Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept

  • Received: 24 August 2020 Accepted: 21 October 2020 Published: 02 November 2020
  • MSC : 30C55, 30C45

  • In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by α[φ(z)φ"(z)+(φ(z))2]+amφm(z)+am1φm1(z)+...+a1φ(z)+a0=0. The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of ez. Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.

    Citation: Rabha W. Ibrahim, Dumitru Baleanu. Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept[J]. AIMS Mathematics, 2021, 6(1): 806-820. doi: 10.3934/math.2021049

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  • In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by α[φ(z)φ"(z)+(φ(z))2]+amφm(z)+am1φm1(z)+...+a1φ(z)+a0=0. The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of ez. Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.


    An algebraic differential equation is a differential equation that can be formulated by consequence with differential algebra. There are different directions to study this class involving complex domains. These studies are considered the second-order homogeneous linear differential equation [1], meropmorphic solution by using Painlevé analysis [2], univalent symmetric solution by applying a special case of Painlevé analysis [3], fractional calculus of CADEs [4,5,6], Nevanlinna method, for normal classes, and algebraic differential equations [7], irregular and regular singular solutions, by utilizing special functions such as Zeta function [8], numerical solution [9] and quantum studies [10].

    The geometric behavior of classes of CADEs is studied in different views. Phong [11] presented a solution of a class of CADEs driven by string theories. The class is also motivating from the view of non-Kahler geometry and the theory of non-linear partial differential equations. Brodsky [12] analyzed a class of CADEs by utilizing the concept of Bourbaki geometric theory with applications in multi-agent system. Seilera and Seib [13] employed the differential geometric theory to recognize the solution of a class of CADEs. Fenyes [14] introduced a complete analysis of solution of a class of CADEs using the quasiconformal geometry. Kravchenko et al. [15] studied the analytic solution by using the geometry behavior of Liouville transformation. More studies of analytic solutions of CADEs can be located in [16,17,18].

    Here, we proceed to study a class of CADEs geometrically. Our tools are based on some concepts from the geometric function theory and univalent function theory. For an analytic function φ which defined in the open unit disk ={zC:|z|<1}, we formulate the following CADE as follows:

    α[φ(z)φ(z)+(φ(z))2]+amφm(z)+am1φm1(z)+...+a1φ(z)+a0=0.

    Our aim is to present sufficient conditions to obtain its analytic solutions. We show that these solutions are subordinated to analytic convex functions in terms of ez. Moreover, we investigate the coefficient bounds of CADE by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.

    A special class of CADEs is studied in [2] taking the structure

    α[φ(z)φ(z)+(φ(z))2]+Λmφ(z)=0,zC, (2.1)

    where α and aıC,ı=0,...,m are constants such that

    Λmφ(z):=amφm(z)+am1φm1(z)+...+a1φ(z)+a0.

    Here, we rearrange (2.1) and investigate the geometric properties by including it in some classes of normalized analytic functions in . Then the solution is majorized by employing special function in . Eq (2.1) implies the homogeneous form when α0

    (zφ(z)φ(z))(zφ(z)φ(z)+zφ(z)φ(z))=0,z. (2.2)

    Fenyes [14] studied a special case of Eq (2.1) as follows: (φ(z))2=q/2, where q indicates the potential energy, by using the Liouville transformation. The same technique is used by Vladislav et al [15] to analyze the equation φ(z)+cφ(z)=0.

    To study Eq (2.2) geometrically, we need the next concepts.

    Definition 2.1. An analytic function φ is subordinated to an analytic function ψ, written φψ, if occurs an analytic function h with |h(z)||z| such that φ=(ψ(h)) (see [19]). The Ma-Minda classes S(ρ) and K(ρ) of starlike and convex functions respectively indicated by (zφ(z)φ(z))ρ(z) and (1+zφ(z)φ(z))ρ(z), where ρ has a positive real part in ,ρ(0)=1,|ρ(0)|>1 and maps onto a starlike-domain with respect to one and symmetric based on the real axis.

    Our study is indicated by using the above inequality to define the following special class.

    Definition 2.2. A function of normalized expansion φ(z)=z+n=2φnzn,z is called in the class M(ρ) if and only if

    P(z):=(zφ(z)φ(z))(zφ(z)φ(z)+zφ(z)φ(z))ρ(z). (2.3)
    (z,ρ(0)=1,|ρ(0)|>1)

    It is clear that P(0)=1. In the sequel, we shall consider a starlike function with positive real part such as ez and a convex function (univalent)

    ρe(z)=zez1=1z2+z212z4720+...

    as well as

    ϱe(z):=1/ρe(z)=1+z2+z26+z324+z4120+...

    is convex univalent in (see [19], P415). Note that the coefficients are converging to the Bernoulli numbers. Moreover, the real part of the function ϱe(z)=(ez1)/z satisfies the inequality

    (eηz1ηz)12,0<η1.793...

    Hence, (eηz1ηz)1ρe(1)=12.

    Our computation is based on analytic technique of Caratheodory functions which are used in [20]. This is the first step. The second step is to majorize Λmφ(z) by a special type of ρ(z),z denoted by Λmφ(z)ρ(z). Note that two functions are under majorization if and only if |λȷ||ρȷ| for all ȷ=1,2,..., where λȷ and ρȷ are the coefficients of Λmφ(z) and ρ(z) respectively. In this case, we illustrate sufficient conditions of the coefficient bounds of Λmφ(z), for different values of m=0,1,.., using a Caratheodory function.

    Majorization-subordination theory creates by Biernacki who exposed in 1936 that if f(z) is subordinate in to F(z) (F(z) is the normalized function in ). In the following works, Goluzin, Tao Shah, Lewandowski and MacGregor studied numerous connected problems, but continuously under the condition that the dominant function F(z) is |z|<0.12. In 1951, Goluzin presented that if f(z) is majorized by a univalent function F(z), then f(z) is majorized by F(z) in |z|<0.12. He conjectured that majorization would continuously arise for |z|<38 and this was shown by Tao Shah in 1958. Later Campbell proved the same result for a parametric class of univalent function (see [21]).

    In this place, we illustrate our computational results.

    Theorem 3.1. Let the function φ achieving the inequality

    1+γ(zP(z)[P(z)]k)z+z2+1,k=0,1,2,

    where P(z)=(zφ(z)φ(z))(zφ(z)φ(z)+zφ(z)φ(z)). Then

    P(z)ρe(z)=zez1,z,

    when γmaxγk,

    minγ0=((e1)(2+2+log(2)log(1+2)))(e2)1.8516..

    and

    maxγ0=(e1)(2+log(2)+log(21))2.106..

    minγ1=(22log(2)+log(1+2))log(e1)1.5..

    and

    maxγ1=((2)log(2)log((2)1))(log(e1)1)2.839..

    minγ2=22+log(1/2+1/2)(e2)1.077..

    and

    maxγ2=e(2+log(2)log(1+2))3.33...

    Proof. Case I: k=01+γ(zP(z))z+z2+1.

    Define a function Γγ:C admitting the structure

    Γγ(z)=1+1γ(z+z2+1log(1+z2+1)1+log(2)).

    Clearly, Γγ(z) is analytic in satisfying Γγ(0)=1 and it is a solution of the differential equation

    1+γ(zΓγ(z))=z+z2+1,z. (3.1)

    Consequently, we have O(z):=γ(zΓγ(z))=z+z2+11 is starlike in . Then for H(z):=O(z)+1, we conclude that

    (zO(z)O(z))=(zH(z)O(z))>0.

    Then Miller-Mocanu Lemma (see [19], P132) implies that

    1+γ(zP(z))1+γzΓγ(z)P(z)Γγ(z).

    To complete this case, we only request to show that Γγ(z)ρe(z). Obviously, the function Γγ(z) is increasing in the interval (1,1) that is achieving the inequality

    Γγ(1)Γγ(1).

    Since the function ρe(z) satisfies the inequality for real ϑ,

    1e1(ρe(z))1cos(ϑ)2+n=1β2ncos(2nϑ)(2n)!ee1

    then the following inequality holds

    1e1Γγ(1)Γγ(1)ee1

    if γ achieves the upper and lower bounds (see Fig 1-first row)

    minγ0=((e1)(2+2+log(2)log(1+2)))(e2)1.8516..
    Figure 1.  The first row represents the min and max of γ0 and the second row indicates γ1, while the third is γ2.

    and

    maxγ0=(e1)(2+log(2)+log(21))2.106...

    This leads to the subordination inequalities

    Γγ(z)zez1P(z)zez1,z.

    Case II: k=11+γ(zP(z)P(z))z+z2+1.

    Define a function Πγ:C formulating the structure

    Πγ(z)=exp(1γ(z+z2+1log(1+z2+1)1+log(2))).

    Clearly, Πγ(z) is analytic in satisfying Πγ(0)=1 and it is a solution of the differential equation

    1+γ(zΠγ(z)Πγ(z))=z+z2+1,z. (3.2)

    By using O(z)=z+z2+11, which is starlike in and H(z)=O(z)+1, we get

    (zO(z)O(z))=(zH(z)O(z))>0,z.

    Then again, according to Miller-Mocanu Lemma, we have

    1+γ(zP(z)P(z))1+γ(zΠγ(z)Πγ(z))P(z)Πγ(z).

    Accordingly, the next inequality carries

    1e1Πγ(1)Πγ(1)ee1

    if γ admits the upper and lower bounds (see Figure 1-second row)

    minγ1=(22log(2)+log(1+2))log(e1)1.5..

    and

    maxγ1=((2)log(2)log((2)1))(log(e1)1)2.839..

    This yields to the subordination inequalities

    Πγ(z)zez1P(z)zez1,z.

    Case III: k=21+γ(zP(z)P2(z))z+z2+1.

    Define a function Θγ:C formulating the structure

    Θγ(z)=(11γ(z+z2+1log(1+z2+1)1+log(2)))1.

    Clearly, Θγ(z) is analytic in satisfying Θγ(0)=1 and it is a solution of the differential equation

    1+γ(zΘγ(z)Θγ(z))=z+z2+1,z. (3.3)

    By using O(z)=z+z2+11, which is starlike in and H(z)=O(z)+1, we get

    (zO(z)O(z))=(zH(z)O(z))>0,z.

    Then again, according to Miller-Mocanu Lemma, we have

    1+γ(zP(z)P2(z))1+γ(zΘγ(z)Θ2γ(z))P(z)Θγ(z).

    Accordingly, we have

    1e1Θγ(1)Θγ(1)ee1

    if γ2 admits the upper and lower bounds (see Figure 1-third row)

    minγ2=22+log(1/2+1/2)(e2)1.077..

    and

    maxγ2=e(2+log(2)log(1+2))3.33...

    This brings the subordination inequalities

    Θγ(z)zez1P(z)zez1,z.

    Next result studies the subordination with respect to the function ϱe(z)=ez1z,z.

    Theorem 3.2. Let the the assumptions of Theorem 3.1 hold. Then

    P(z)ϱe(z)=ez1z,z

    when γmaxγk,

    minγ0=(2+log(2)+log(21))(e2)1.706..

    and

    maxγ0=e(2+2+log(2)log(1+(2)))2.10399...

    minγ1=(2+2+log(2)+log(21))(log(e1)1)1.70..

    and

    maxγ1=(2+log(2)+log(21))/log(e1)2.2...

    minγ2=(e1)(2+2+log(2)log(1+2))1.329..

    and

    maxγ2=((e1)(2+log(2)log(1+2)))(e2)2.932...

    Proof. Consider the convex univalent function ϱe(z)=ez1z. It is clear that ϱ(0)=1 with a positive real part. Moreover it satisfies the inequality

    e1e(ϱe(z))e1,z.

    By the proof of Theorem 3.1, we have the following inequality

    e1eΓγ(1)Γγ(1)e1

    if γ has the upper and lower bounds (see Figure 2-first row)

    minγ0=(2+log(2)+log(21))(e2)1.706..
    Figure 2.  The first row represents the min and max of γ0 and the second row indicates γ1, while the third is γ2.

    and

    maxγ0=e(2+2+log(2)log(1+(2)))2.10399...

    This leads to the subordination inequalities (see Figure 2-second row)

    Γγ(z)ez1zP(z)ez1z,z.

    Similarly, we have

    minγ1=(2+2+log(2)+log(21))(log(e1)1)1.70..

    and

    maxγ1=(2+log(2)+log(21))/log(e1)2.2...

    This yields to the subordination inequalities

    Πγ(z)ez1zP(z)ez1z,z.

    Finally, we have the upper and lower bounds (see Figure 2-third row)

    minγ2=(e1)(2+2+log(2)log(1+2))1.329..

    and

    maxγ2=((e1)(2+log(2)log(1+2)))(e2)2.932...

    This brings the subordination inequalities

    Θγ(z)ez1zP(z)ez1z,z.

    We proceed to include the term Λmφ(z)=amφm(z)+am1φm1(z)+...+a1φ(z)+a0 for some m to study the behavior of solutions of Eq (2.1). Dividing Eq (2.1) by α0, we have

    [φ(z)φ(z)+(φ(z))2]=Λmφ(z)α,zC, (4.1)

    We have the following result

    Theorem 4.1. Consider the CADEs (4.1), with α=1 and a0=1. If φM(ρ) is a convex univalent function in satisfying the condition of Theorem 3.1 then the constant connections aı achieving the following values

    a1=12,a2=712,a3=812,a4=74100,a5=79100. (4.2)

    Proof. From Eq (4.1) together with Theorem 3.1, we have Λmφ(z)ρe(z). Since φ is convex univalent in then it takes the extreme function structure φ(z)=z/(1z)=z+z2+.... Therefore, we have

    Λ0φ(z)=1Λ1φ(z)=1+a1z+a1z2+a1z3+a1z4+a1z5+O(z6)Λ2φ(z)=1+a1z+(a1+a2)z2+(a1+2a2)z3+(a1+3a2)z4+(a1+4a2)z5+O(z6)Λ3φ(z)=1+a1z+(a1+a2)z2+(a1+2a2+a3)z3+(a1+3(a2+a3))z4+(a1+4a2+6a3)z5+O(z6)Λ4φ(z)=1+a1z+(a1+a2)z2+(a1+2a2+a3)z3+(a1+3a2+3a3+a4)z4+(a1+4a2+6a3+4a4)z5+O(z6)Λ5φ(z)=1+a1z+(a1+a2)z2+(a1+2a2+a3)z3+(a1+3a2+3a3+a4)z4+(a1+4a2+6a3+4a4+a5)z5+O(z6)Λ6φ(z)=1+a1z+(a1+a2)z2+(a1+2a2+a3)z3+(a1+3a2+3a3+a4)z4+(a1+4a2+6a3+4a4+a5)z5+O(z6)

    In addition, we have

    ρe(z)=zez1=n=0Bnznn!,

    where Bn is the Bernoulli numbers satisfying the inequality

    |Bn|4πn(nπe)2n,B2n+1=0,
    (B0=1,B1=1/2,B2=1/6,B4=1/30,B6=1/42).

    Compering the coefficients of Λmφ(z) and ρe(z), we have

    a1=B11!=12a2=a1+B22!=712a3=a12a2+B33!=812a4=a13a23a3+B44!=74100a5=a14a26a34a4+B55!=79100.

    Next result indicates the value of constant coefficients of Λmφ when φ is starlike in .

    Theorem 4.2. Consider the CADE (4.1), with α=1 and a0=1. If φM(ρ) is a starlike function in satisfying the condition of Theorem 3.1 then the constant connections aı achieving the following values

    a1=12,a2=1312,a3=2810,a4=795100,a5=24. (4.3)

    Proof. Obviously, from the assumptions, we have Λmφ(z)ρe(z). Since φ is starlike in then it admits the extreme function structure φ(z)=z/(1z)2=z+2z2+.... Therefore, we have

    Λ0φ(z)=1Λ1φ(z)=1+a1z+2a1z2+3a1z3+4a1z4+5a1z5+O(z6)Λ2φ(z)=1+a1z+(2a1+a2)z2+(3a1+4a2)z3+(4a1+10a2)z4+5(a1+4a2)z5+O(z6)Λ3φ(z)=1+a1z+(2a1+a2)z2+(3a1+4a2+a3)z3+(4a1+10a2+6a3)z4+(5a1+20a2+21a3)z5+O(z6)Λ4φ(z)=1+a1z+(2a1+a2)z2+(3a1+4a2+a3)z3+(4a1+10a2+6a3+a4)z4+(5a1+20a2+21a3+8a4)z5+O(z6)Λ5φ(z)=1+a1z+(2a1+a2)z2+(3a1+4a2+a3)z3+(4a1+10a2+6a3+a4)z4+(5a1+20a2+21a3+8a4+a5)z5+O(z6)Λ6φ(z)=1+a1z+(2a1+a2)z2+(3a1+4a2+a3)z3+(4a1+10a2+6a3+a4)z4+(5a1+20a2+21a3+8a4+a5)z5+O(z6)

    Compering the coefficients of Λmφ(z) and ρe(z), we have

    a1=B11!=12a2=2a1+B22!=1312a3=3a14a2+B33!=2810a4=4a110a26a3+B44!=795100a5=5a120a221a38a4+B55!=24.

    Remark 4.3.

    ● Note that Theorems 4.1 and 4.2 show that Λmφ(z) accumulates at m=5, which leads to the expansion structure (see Figure 3)

    Λ5z/(1z)=1z2+z212z4100+O(z6)
    Figure 3.  Λ5z/(1z) and Λ5z/(1z)2 respectively.

    and

    Λ5z/(1z)2=1z2+8z21002z4100+O(z6).

    ● One can generalize Theorems 4.1 and 4.2 in terms of α for all values. In this case, we obtain the constant coefficients Aı=aıα provided α0

    A1=12,A2=712,A3=812,A4=74100,A5=79100

    and

    A1=12,A2=1312,A3=2810,A4=795100,A5=24,

    respectively.

    ● Results in [2] indicate that for n=4, the coefficients satisfy A40 by using Painlevé analysis, which did not apply in case n5. While, the majorization-subordination analysis indicates that for n5, the coefficients are converged by Bernoulli numbers.

    A class of non-linear complex algebraic differential equations (CADEs) is investigated in view of geometric function theory. We defined a class of normalized functions including the structure of CADEs. Based on the subordination inequality, we introduced the values of constant coefficients. As proceeding works in this direction, one can generalize Eq (2.1) in terms of differential operators including fractional differential and convolution operator in the open unit disk. Or can be realized by a quantum calculus.

    The authors would like to thanks the reviewers for the deep comments to improve our work. Also, we express our thanks to editorial office for their advice.

    The authors declare no conflict of interest.



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