Research article Special Issues

Initial coefficient bounds for certain new subclasses of bi-univalent functions with bounded boundary rotation

  • In the current article, we introduced new subclasses of bi-univalent functions associated with bounded boundary rotation. For these new classes, the authors first obtained two initial coefficient bounds. They also verified the special cases where the familiar Brannan and Clunie's conjecture were satisfied. Furthermore, the famous Fekete-Szegö inequality was obtained for the newly defined subclasses of bi-univalent functions, and some of the results improved the earlier results available in the literature.

    Citation: Prathviraj Sharma, Srikandan Sivasubramanian, Nak Eun Cho. Initial coefficient bounds for certain new subclasses of bi-univalent functions with bounded boundary rotation[J]. AIMS Mathematics, 2023, 8(12): 29535-29554. doi: 10.3934/math.20231512

    Related Papers:

    [1] Wei Lu, Yuangang Li, Yixiu Kong, Liangli Yang . Generalized triangular Pythagorean fuzzy weighted Bonferroni operators and their application in multi-attribute decision-making. AIMS Mathematics, 2023, 8(12): 28376-28397. doi: 10.3934/math.20231452
    [2] Attaullah, Shahzaib Ashraf, Noor Rehman, Asghar Khan, Muhammad Naeem, Choonkil Park . Improved VIKOR methodology based on q-rung orthopair hesitant fuzzy rough aggregation information: application in multi expert decision making. AIMS Mathematics, 2022, 7(5): 9524-9548. doi: 10.3934/math.2022530
    [3] Li Li, Mengjing Hao . Interval-valued Pythagorean fuzzy entropy and its application to multi-criterion group decision-making. AIMS Mathematics, 2024, 9(5): 12511-12528. doi: 10.3934/math.2024612
    [4] Nasser Aedh Alreshidi, Muhammad Rahim, Fazli Amin, Abdulaziz Alenazi . Trapezoidal type-2 Pythagorean fuzzy TODIM approach for sensible decision-making with unknown weights in the presence of hesitancy. AIMS Mathematics, 2023, 8(12): 30462-30486. doi: 10.3934/math.20231556
    [5] Shichao Li, Zeeshan Ali, Peide Liu . Prioritized Hamy mean operators based on Dombi t-norm and t-conorm for the complex interval-valued Atanassov-Intuitionistic fuzzy sets and their applications in strategic decision-making problems. AIMS Mathematics, 2025, 10(3): 6589-6635. doi: 10.3934/math.2025302
    [6] Nadia Khan, Sehrish Ayaz, Imran Siddique, Hijaz Ahmad, Sameh Askar, Rana Muhammad Zulqarnain . Sustainable practices to reduce environmental impact of industry using interaction aggregation operators under interval-valued Pythagorean fuzzy hypersoft set. AIMS Mathematics, 2023, 8(6): 14644-14683. doi: 10.3934/math.2023750
    [7] Saleem Abdullah, Muhammad Qiyas, Muhammad Naeem, Mamona, Yi Liu . Pythagorean Cubic fuzzy Hamacher aggregation operators and their application in green supply selection problem. AIMS Mathematics, 2022, 7(3): 4735-4766. doi: 10.3934/math.2022263
    [8] Muhammad Riaz, Khadija Akmal, Yahya Almalki, S. A. Alblowi . Cubic m-polar fuzzy topology with multi-criteria group decision-making. AIMS Mathematics, 2022, 7(7): 13019-13052. doi: 10.3934/math.2022721
    [9] Esmail Hassan Abdullatif Al-Sabri, Muhammad Rahim, Fazli Amin, Rashad Ismail, Salma Khan, Agaeb Mahal Alanzi, Hamiden Abd El-Wahed Khalifa . Multi-criteria decision-making based on Pythagorean cubic fuzzy Einstein aggregation operators for investment management. AIMS Mathematics, 2023, 8(7): 16961-16988. doi: 10.3934/math.2023866
    [10] Hifza, Muhammad Gulistan, Zahid Khan, Mohammed M. Al-Shamiri, Muhammad Azhar, Asad Ali, Joseph David Madasi . A new fuzzy decision support system approach; analysis and applications. AIMS Mathematics, 2022, 7(8): 14785-14825. doi: 10.3934/math.2022812
  • In the current article, we introduced new subclasses of bi-univalent functions associated with bounded boundary rotation. For these new classes, the authors first obtained two initial coefficient bounds. They also verified the special cases where the familiar Brannan and Clunie's conjecture were satisfied. Furthermore, the famous Fekete-Szegö inequality was obtained for the newly defined subclasses of bi-univalent functions, and some of the results improved the earlier results available in the literature.



    In the 1960's, B. O'Neill [1] introduced the notion of Riemannian submersion as a tool to study the geometry of a manifold in terms of the simpler components, namely, fibers and base space. A. L. Besse considered warped product Riemannian submersion [2]. Further, I. K. Erken and C. Murathan [3] studied warped product Riemannian submersion and obtained fundamental geometric properties.

    J. F. Nash [4] started the study of warped product manifolds and proved that every warped product manifold can be embedded as a Riemannian submanifold in some Euclidean spaces. In 1969, B. O'Neill and R. L. Bishop [5] studied the warped product manifold as a fruitful generalization of the Riemannian product manifold.

    Warped product manifolds play key roles in mathematical physics [6]. H. M. Tastan and S. B. Aydin [7,8] introduced the concept of a warped-twisted product as an extension of the twisted product and consequently, the warped product. The warped-twisted product, denoted as M=f2M1×f1M2, refers to the product manifold M1×M2 endowed with the metric tensor g, which is defined by

    g=(f2φ2)2φ1(g1)+f21φ2(g2), (1.1)

    where, φi:M1×M2Mi is the natural projections, for i{1,2}. The function f2C(M2) is named a warping function, and the function f1C(M1×M2) is named a twisting function of M=f2M1×f1M2. If the function f1 solely depends on the points of M2 in this instance, the resulting warped-twisted product can be classified as a base conformal warped product [8]. A warped-twisted product is considered non-trivial if it does not fall into any categories of a doubly warped product, a warped product, or a base conformal warped product. For more details about the concerned studies, we refer the papers [9,10,11,12,13,14,15,16,17,18,19,20,21,22].

    The following is our definition of warped-twisted product submersions:

    Definition 1.1. Suppose that M=f2M1×f1M2 and =ρ21×ρ12 are warped-twisted product manifolds and φi:Mii,i{1,2}, are Riemannian submersion between the manifolds Mi and Ni. Then the map

    φ=φ1×φ2:M=f2M1×f1M2=ρ21×ρ12 (1.2)

    given by φ(x1,x2)=(φ1(x1),φ2(x2)) is a Riemannian submersion, which is called warped-twisted product submersion.

    Our primary objective of this paper is to investigate the fundamental geometric properties associated with warped-twisted product submersions. The notion of warped product generalizes usual products, which is further generalized by the twisted product and doubly warped product. Non-trivial wraped-twisted product is neither twisted nor base conformal nor direct product. The definition of warped product submersion and its geometrical properties was discussed by Murathan, C. in his article "Riemannian warped product submersions". These results, which are presented in that article, serve as our motivation.

    The paper is organized in the following way. In Section 2, we recall definitions and some fundamental results of Riemannian submersions and warped-twisted product manifolds which are useful for this paper. In Section 3, we defined Riemannian warped-twisted product submersion and discuss some geometrical properties for this submersion. In Section 4, we obtain the Ricci tensors for Riemannian warped-twisted product submerion and discuss Einstein's condition on vertical and horizontal distributions of total manifold.

    In this section, we recall some definitions, results and notations that are necessary for the paper.

    Let (M,gM) and (,g) be two Riemannian manifolds with dimM=m and dim=n, where m>n. A smooth map φ:(M,gM)(,g) is said to be Riemannian submersion if the following axioms are satisfied:

    1) φ(derivative map of φ) is onto,

    2) φ preserves the length of horizontal vectors, i.e.,

    g(φX,φY)=gM(X,Y).

    For each p2, ϕ1(p2) is a submanifold of dimension (mn) called fibers. If the fibers are orthogonal then a vector field on M is referred to as horizontal and it is referred to as vertical if the fibers are tangent. Let φ:(M,gM)(,g) be a smooth map. Then Γ(TM) has the following decomposition:

    TM=(kerφ)(kerφ).

    B. O'Neill [1] first introduced the fundamental tensors of submersions, and are defined by

    T(E,F)=TEF=HVEVF+VVEHF, (2.1)
    A(E,F)=AEF=HHEVF+VHEHF, (2.2)

    where E and F are vector fields on M; H and V are the projection morphism on the distribution (kerφ) and (kerφ), respectively. We observe that the tensor fields T and A satisfy

    1) TUV=TVU,U,VΓ(kerφ),

    2) AXY=AYX,X,YΓ(kerφ).

    Equations (2.1) and (2.2) give the following lemma.

    Lemma 2.1. [1]. Let X,YΓ(kerφ) and U,VΓ(kerφ); then we have

    UV=TUV+ˆUV, (2.3)
    UX=HUX+TUX, (2.4)
    XU=AXU+VXU, (2.5)
    XY=HXY+AXY, (2.6)

    where is the Levi-Civita connection of (M,gM) and ˆUV=VUV.

    It is noted that if the tensor field A (respectively T) vanishes, then the horizontal distribution H (respectively, vertical distribution V or fiber) is integrable. Also, any fiber of Riemannian submersion ϕ is totally umbilical if and only if

    TVW=g(V,W)H,

    where H is the mean curvature vector field of the fiber given by

    N=sH,

    such that

    N=si=1TUiUi, (2.7)

    and {U1,U2,,Us} denotes the orthonormal basis of vertical distribution and s denotes the dimension of any fiber. It is easy to see that any fiber of Riemannian submersion ϕ is minimal if and only if the horizontal vector field N vanishes.

    Let (Mi,gMi) be two Riemannian manifolds of dimensions m1 and m2, respectively and f1 and f2 be two positive differentiable functions on M1 and M1×M2, respectively. Let φi:M1×M2Mi be the natural projections from product manifold M1×M2 to Mi, i{1,2}. Then the warped-twisted product manifold M=f2M1×f1M2 is a product manifold M=M1×M2 endowed with the metric gM such that

    gM(X,Y)=(f2φ2)2gM1(φ1(X),φ1(Y))+f12gM2(φ2(X),φ2(Y))

    for any X,YM.

    For any X on M1, the lift of X to f2M1×f1M2 is the vector field ˜X whose value at each (p,q) is the lift Xp to (p,q). Thus the lift of X is the unique vector field on f2M1×f1M2, that is, φ1-related to X and φ2-related to the zero vector field on M2.

    Let and i be the Levi-Civita connections of f2M1×f1M2 and Mi, respectively for i{1,2}. The lifts of vector fields on Mi is denoted by L(Mi).

    Then, the covariant derivative formulae for a warped-twisted product manifold are given as [8]:

    XY=1XYg(X,Y)(ln(f2φ2)), (2.8)
    XV=VX=V(ln(f2φ2))X+X(ln(f1))V, (2.9)
    UV=2UV+U(lnf1)V+V(lnf1)Ug(U,V)(ln(f1)) (2.10)

    for X,YL(M1) and U,VL(M2). Now, for any smooth function ψ on a warped-twisted product (f2M1×f1M2,gM), we have

    hψ(X,U)=X(lnf1)U(ψ)X(ψ)U(lnf2)

    for any XL(M1) and UL(M2), where the definition of the Hessian tensor being used. Now let S and Si be the Ricci tensors of (M,g) and (Mi,gi), respectively. Then we have the following relations:

    Lemma 2.2. [23] Let M=f2M1×f1M2 be a warped-twisted product manifold. Then we have

    S(X,Y)=S1(X,Y)+hlnf2(X,Y)m2{hlnf11(X,Y)+X(lnf1)Y(lnf1)}g(X,Y){Δlnf2+g(lnf2,lnf2)},S(X,U)=(1m2)XU(lnf1)+(m1+m22)X(lnf1)U(lnf2),S(U,V)=S2(U,V)+hlnf1(U,V)+(1m2)hlnf12(U,V)+m2U(lnf1)V(lnf1)g(U,V){Δlnf1+g(lnf1,lnf1)}m1{hlnf22(U,V)+U(lnf2)V(lnf2)U(lnf2)V(lnf1)U(lnf1)V(lnf2)}

    for X,YL(M1) and U,VL(M2), where Δ is Laplacian operator and is gradiant of the function.

    In this section, we define Riemannian warped-twisted product submersion and obtain some fruitful results.

    Proposition 3.1. Let M=f2M1×f1M2 and =ρ21×ρ12 be two warped-twisted product manifolds and let φi:Mii,i{1,2} be Riemannian submersion between the manifolds Mi and i. Then the map

    φ=φ1×φ2:M=f2M1×f1M2=ρ21×ρ12

    given by φ(x1,x2)=(φ1(x1),φ2(x2)) is a Riemannian submersion, which is called warped-twisted product submersion.

    Proof. Making use of Proposition 2 [3], it is easy to show that the map φ is a Riemannian submersion. Now it is enough to show that the map φ is a warped-twisted product. Since

    g(φ(X1,X2),φ(Y1,Y2))=ρ22g1(φ1(X1),φ1(Y1))+ρ21g2(φ2(X2),φ2(Y2))=(f2φ2)2gM1(X1,Y1)+f21gM2(X2,Y2)=gM((X1,Y1),(X2,Y2)).

    It shows that φ preserve the length of the horizontal vector field. Thus φ is a warped-twisted product submersion.

    Next, we obtain fundamental tensors for the Riemannian warped-twisted product submersion in the subsequent lemmas:

    Lemma 3.1. Let M=f2M1×f1M2 and =ρ21×ρ12 are warped-twisted product manifolds and φi:Mii is Riemannian warped-twisted product submersion between the manifolds Mi and i. Then we have

    1) TU1V1=T1U1V1gM(U1,V1)H(ln(f2φ2)),

    2) TU1U2=0,

    3) TU2V2=T2U2V2gM(U2,V2)H(ln(f1))

    for any Ui,ViΓ(Vi), i={1,2}.

    Proof. From Eq (2.3), we get

    U1V1=ˆU1V1+TU1V1. (3.1)

    By using Eq (2.8), we obtain

    U1V1=1U1V1gM(U1,V1)(ln(f2φ2)). (3.2)

    Using Eq (2.3) in Eq (3.2) and combining the result with Eq (3.1), we get result 1).

    By using Eq (2.3), we obtain

    U1V1=ˆU1V1+TU1V1. (3.3)

    Making use of Eq (2.9), we have

    U1U2=U2U1=U2(ln(f2π2))U1+U1(ln(f1))U2. (3.4)

    Combining Eq (3.3) with Eq (3.4), we get

    TU1U2=0.

    From Eq (2.1), we obtain

    TU2V2=H(U2V2). (3.5)

    Using Eq (2.10), we get

    U2V2=2U2V2+U2(lnf1)V2+V2(lnf1)U2gM(U2,V2)(ln(f1)). (3.6)

    From Eq (2.3), we know that

    2U2V2=T2U2V2+V2U2V2. (3.7)

    By using Eqs (3.6) and (3.7) in Eq (3.5), we get the desired result 3).

    Lemma 3.2. Let M=f2M1×f1M2 and =ρ21×ρ12 are warped-twisted product manifolds and φi:Mii is Riemannian warped-twisted product submersion between the manifolds Mi and i. Then we have

    1) HX1Y1=H1X1Y1gM(X1,Y1)H(ln(f2φ2)),

    AX1Y1=A1X1Y1gM(X1,Y1)V(ln(f2φ2)),

    2) HX1X2=HX2X1=X2(ln(f2φ2))X1+X1(ln(f1))X2,

    AX2X1=0=AX1X2,

    3) AX2Y2=A2X2Y2andV(ln(f1))=0,

    HX2Y2=H2X2Y2+X2(lnf1)Y2+Y2(lnf1)X2gM(X2,Y2)H(ln(f1))

    for any Xi,YiΓ(Hi), i={1,2}.

    Proof. From Eq (2.7), we have

    X1Y1=H(X1Y1)+AX1Y1. (3.8)

    From Eqs (2.8) and (3.8), we get

    HX1Y1+AX1Y1=H1X1Y1+A1X1Y1gM(X1,Y1)(ln(f2φ2)). (3.9)

    Separating the horizontal and vertical parts in Eq (3.9), we obtain

    HX1Y1=H1X1Y1gM(X1,Y1)H(ln(f2φ2)),AX1Y1=A1X1Y1gM(X1,Y1)V(ln(f2φ2)).

    From Eq (2.7), we have

    X1X2=HX1X2+AX1X2. (3.10)
    X2X1=HX2X1+AX2X1. (3.11)

    From Eq (2.9), we get

    X1X2=X2X1=X2(ln(f2φ2))X1+X1(ln(f1))X2. (3.12)

    Combining Eqs (3.10)–(3.12), we obtain result 2). We know that

    AX2Y2=VX2Y2.

    Using Eq (2.10) in the above equation, we get

    AX2Y2=V[2X2Y2gM(X2,Y2)(ln(f1))]=V2X2Y2gM(X2,Y2)V(ln(f1))=A2X2Y2gM(X2,Y2)V(ln(f1)), (3.13)

    and

    HX2Y2=H2X2Y2+X2(lnf1)Y2+Y2(lnf1)X2gM(X2,Y2)H(ln(f1)). (3.14)

    Since A and A2 are skew-symmetric tensor fields and gM is a symmetric tensor field, by using Eqs (3.13) and (3.14), we obtain the required result 3).

    Lemma 3.3. Let M=f2M1×f1M2 and =ρ21×ρ12 be warped-twisted product manifolds and let φi:Mii be Riemannian warped-twisted product submersion between the manifolds Mi and i. Then we have

    1) HV1X1=H1V1X1andTV1X1=T1V1X1,

    2) TV1X2=X2(ln(f2φ2))V1=VX2V1and

    AX2V1=V1(ln(f1))X2=HV2X1,

    3) TV2X1=X1(ln(f1))V2=VX1V2and

    AX1V2=V2(ln(f2φ2))X1=HV2X1,

    4) TV2X2=T2V2X2+X2(lnf1)V2and

    HV2X2=H2V2X2+V2(lnf1)X2

    for any ViΓ(Vi), and XiΓ(Hi) where i={1,2}.

    Proof. For V1Γ(Vi) and X1Γ(Hi), by using Eq (2.4), we have

    V1X1=HV1X1+TV1X1. (3.15)

    Making use of Eqs (2.8) and (2.4), we obtain

    V1X1=1V1X1gM(V1,X1)(ln(f2φ2))=H1V1X1+T1V1X1. (3.16)

    Combining Eqs (3.15) and (3.16) and comparing the vertical and the horizontal parts in the resulting expression, we get result 1).

    From Eq (2.9), we have

    V1X2=X2V1=X2(ln(f2φ2))V1+V1(ln(f1))X2. (3.17)

    From Eqs (2.4) and (2.5), we have

    V1X2=HV1X2+TV1X2, (3.18)
    X2V1=AX2V1+VX2V1. (3.19)

    Combining Eqs (3.17)–(3.19) and comparing the vertical and the horizontal parts in the resulting expression, we obtain result 2). On a similar line, we get result 3).

    Further, using Eqs (2.4) and (2.10), we obtain result 4).

    Further, using the above lemmas, we obtained the fundamental geometric properties of Riemannian warped-twisted product submersion in the consequent theorems.

    Theorem 3.1. Let M=f2M1×f1M2 and =ρ21×ρ12 be warped-twisted product manifolds and let φi:Mii be Riemannian warped-twisted product submersion between the manifolds Mi and i with dimM1=m1,dimM2=m2,dim1=n1 and dim2=n2. Then

    (i) φ has totally geodesic fibers if and only if φ1 and φ2 have totally geodesic fibers and f1 and f2 are constants,

    (ii) The fundamental metric tensor T satisfies the following inequality

    T2(n1m1)H(lnf2)2(n2m2)H(lnf1)2

    with the equality holding if and only if φ1 and φ2 have totally geodesic fibers.

    Proof. (ⅰ) Let ekΓ(V1) and k=1,,m1n1 and ecΓ(V2),c=m1n1+ 1,,m1n1+m2n2 be orthonormal vectors of vertical spaces of submersion π. Then using Lemma (3.2), we have

    T2=m1n1k,k1=1gM(T(ek,ek1),T(ek,ek1))+m1n1+m2n2c,d=m1n1+1gM(T(ec,ed),T(ec,ed))=T12+T22+(m1n1)H(lnf2)2+(m2n2)H(lnf1)2.

    (ⅱ) It follows from the above equation.

    Theorem 3.2. Let M=f2M1×f1M2 and =ρ21×ρ12 be warped-twisted product manifolds and let φi:Mii be Riemannian warped-twisted product submersion between the manifolds Mi and i. Then φ has totally umbilical fibers if and only if φ1 and φ2 have totally geodesic fibers and Hφ=H(lnf1)=H(lnf2), where Hφ denotes the mean curvature of φ.

    Proof. From Lemma (3.1) and the fact that φ has totally umbilical fibers, we have

    TU1V1=T1U1V1gM(U1,V1)H(lnf2)=gM(U1,V1)Hφ,TU1U2=0=gM(U1,U2)Hφ=0Hφ,TU2V2=T2U2V2gM(U2,V2)H(lnf1)=gM(U2,V2)Hφ,

    for any Ui,ViΓ(Vi), i{1,2} which gives the following relation

    Hφ=H(lnf1),Hφ=H(lnf2) and T1U1V1=0=T2U2V2. (3.20)

    Converse follow easily.

    Theorem 3.3. Let M=f2M1×f1M2 and =ρ21×ρ12 be warped-twisted product manifolds and let φi:Mii be Riemannian warped-twisted product submersion between the manifolds Mi and i. Then φ has minimal fibers if and only if the mean curvature of φ1 and φ2 is given by H1=m2n2m1n1H(1lnf1) and H2=m1n1m2n2H(lnf2)+H(2lnf1).

    Proof. We suppose that φ has minimal fibers for M. Let ekΓ(V1) and k=1,,m1n1 and ecΓ(V2),c=m1n1+1,m1n1+m2n2 be orthonormal frames of vertical spaces of submersion φ. Then using Eq (2.7) and Lemma (3.1), we have

    H=1m1n1+m2n2(m1n1k=1T(ek,ek)+m1n1+m2n2c=m1n1+1T(ec,ec))=1m1n1+m2n2(m1n1k=1T1(ek,ek)gM(ek,ek)H(lnf2)+m1n1+m2n2c=m1n1+1(T2(ec,ec)gM(ec,ec)H(lnf1))=1m1n1+m2n2((m1n1)(H1H(lnf2))+(m2n2)(H2H(lnf1))).

    Since, H(lnf2)Γ(H2) and H(lnf1)Γ(H1×H2). So, we can write H(lnf1)=H(1lnf1)+H(2lnf1), where H(1lnf1)Γ(H1) and H(2lnf1)Γ(H2). Then, we obtain H1=m2n2m1n1H(1lnf1) and H2=m1n1m2n2H(lnf2)+H(2lnf1). Converse follows quickly from the above relation.

    In this section, we obtain the Ricci tensor for Riemannian warped-twisted product submersion. Further, we discuss Einstein's condition on vertical and horizontal spaces of Mi for Riemannian warped-twisted product submersion.

    Definition 4.1. [2]. A Riemannian manifold (M,gM) is called an Einstein manifold if

    S=λgM, (4.1)

    where λ is a real constant, and S is the Ricci tensor on M.

    Making use of Lemma 2.2, we have

    Lemma 4.1. Let M=f2M1×f1M2 and =ρ21×ρ12 be warped-twisted product manifolds and φi:Mii be Riemannian warped-twisted product submersion between the manifolds Mi and i with dimMi=mi and dimi=ni; i{1,2}. Let S1 and S2 be the lifts of the Ricci curvatures on M1 and M2, respectively. Then for any Xi,YiΓ(Hi) and Ui,ViΓ(Vi), we have following relations:

    1) S(X1,U1)=(1m2)X1U1(lnf1)+(m1+m22)X1(lnf1)U1(lnf2),

    2) S(X1,X2)=(1m2)X1X2(lnf1)+(m1+m22)X1(lnf1)X2(lnf2),

    3) S(X1,Y1)=S1(X1,Y1)+hlnf2(X1,Y1)m2{hlnf11(X1,Y1)+X1(lnf1)Y1(lnf1)} -gM(X1,Y1){Δlnf2+gM(lnf2,lnf2)},

    4) S(X2,Y2)=S2(X2,Y2)+hlnf1(X2,Y2)+(1m2)hlnf12(X2,Y2)+m2X2(lnf1)Y2(lnf1)-gM(X2,Y2){Δlnf1+gM(lnf1,lnf1)}m1{hlnf22(X2,Y2)+X2(lnf2)Y2(lnf2)X2(lnf2)Y2(lnf1)X2(lnf1)Y2(lnf2)},

    5) S(X1,U2)=(1m2)X1U2(lnf1)+(m1+m22)X1(lnf1)U2(lnf2),

    6) S(X2,U1)=(1m2)U1X2(lnf1)+(m1+m22)U1(lnf1)X2(lnf2),

    7) S(X2,U2)=(1m2)X2U2(lnf1)+(m1+m22)X2(lnf1)U2(lnf2),

    8) S(U1,V1)=S1(U1,V1)+hlnf2(U1,V1)m2{hlnf11(U1,V1)+U1(lnf1)V1(lnf1)}-gM(U1,V1){Δlnf2+gM(lnf2,lnf2)},

    9) S(U1,U2)=(1m2)U1U2(lnf1)+(m1+m22)U1(lnf1)U2(lnf2),

    10) S(U2,V2)=S2(U2,V2)+hlnf1(U2,V2)+(1m2)hlnf12(U2,V2)+m2U2(lnf1)V2(lnf1)-gM(U2,V2){Δlnf1+gM(lnf1,lnf1)}m1{hlnf22(U2,V2)+U2(lnf2)V2(lnf2)-U2(lnf2)V2(lnf1)U2(lnf1)V2(lnf2)}.

    Now, we study Einstein conditions for the horizontal and vertical distributions of Riemannian warped-twisted product submersion.

    Theorem 4.1. Let M=f2M1×f1M2 and =ρ21×ρ12 be warped-twisted product manifolds and φi:Mii be Riemannian warped-twisted product submersion between the manifolds Mi and i with dimMi=mi and dimi=ni; i{1,2}. Then

    (i) If the vertical space V1 (or horizontal space H1) of M is Einstein, then vertical space of M1 (resp. horizontal space H1 ) is Einstein assuming that hlnf11,hlnf2 and, U1(lnf1)V1(lnf1) is proportional to constant times the metric gM and Δlnf2+gM(lnf2,lnf2) is constant,

    (ii) If the vertical space V2 (or horizontal space H2) of M is Einstein, then vertical space of M2 (resp. horizontal space H2) is Einstein assuming that hlnf22,hlnf12,hlnf1 U2(lnf1)V2(lnf1) and U2(lnf2)V2(lnf2)U2(lnf2)V2(lnf1)U2(lnf1)V2(lnf2)} are proportional to the metric gM and, {Δlnf1+gM(lnf1,lnf1)} is constant.

    Proof. (i) Suppose, V1 of M is Einstein's manifold. Then, from Eq (4.1) and Lemma (4.1) we have

    S1(U1,V1)=hlnf2(U1,V1)+m2{hlnf11(U1,V1)+U1(lnf1)V1(lnf1)}+gM(U1,V1){λ+Δlnf2+gM(lnf2,lnf2)}.

    Now if hlnf11,hlnf2 and U1(lnf1)V1(lnf1) is proportional to constant times the metric gM and Δlnf2+gM(lnf2,lnf2) are constants, then M1 is Einstein's manifold.

    (ii) For vertical space V2 of M, using definition of Einstein's manifold and relation 10 of Lemma (4.1), we obtain the required result.

    The Einstein equations are of significant importance within the framework of the general theory of relativity, as they form the foundation for the gravitational and cosmological models. The Einstein equation can be expressed as S=λg, which is a non-linear second-order system of differential equations. In the context of this system, the symbol λ is referred to as the Einstein constant, whereas physicists commonly refer to it as the cosmological constant [2]. The geometric properties of warped-twisted products encompass a broader class than both warped products and twisted products. This class exhibits a multitude of applications, not only within the realm of geometry but also in the field of theoretical physics. The exact solution of Einstein's equations exhibits a warped product structure. The technique of Riemannian submersion is commonly employed in the construction of Riemannian manifolds showing positive sectional curvature. Additionally, it is employed in the construction of widely recognized instances of Einstein manifolds. Riemannian submersion finds extensive application within the domains of Kaluza-Klein theory, Yang-Mills theory, superstring and supergravity theory in physics [24,25,26]. The objective of this study is to decompose the Einstein's equation for warped-twisted product into their constituent elements, specifically the base and fiber components.

    In [8,23], H. M. Tastan and S. G. Aydin considered Einstein like warped-twisted product manifolds and warped-twisted product semi-slant submanifolds. On the other hand, I. K. Erken and C. Murathan [3] studied Riemannian warped product submersion. In this paper, we investigated Riemannian warped-twisted product submersions. In light of these studies, it could be interesting to work on warped product semi-slant submersions and warped-twisted product semi-slant submersions.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors are really thankful to the learned reviewers for their careful reading of our manuscript and their many insightful comments and suggestions that have improved the quality of our manuscript. The author Fatemah Mofarreh, expresses her gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

    The authors declare no conflicts of interest.



    [1] R. M. Ali, S. K. Lee, V. Ravichandran, S. Supramanium, Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Appl. Math. Lett., 25 (2012), 344–351. https://doi.org/10.1016/j.aml.2011.09.012 doi: 10.1016/j.aml.2011.09.012
    [2] H. S. Al-Amiri, T. S. Fernando, On close-to-convex functions of complex order, Int. J. Math. Math. Sci., 13 (1990), 840876. https://doi.org/10.1155/S0161171290000473 doi: 10.1155/S0161171290000473
    [3] B. S. T. Alkahtani, P. Goswami, T. Bulboacă, Estimate for initial MacLaurin coefficients of certain subclasses of bi-univalent functions, Miskolc Math. Notes, 17 (2016), 739–748. https://doi.org/10.18514/MMN.2017.1565 doi: 10.18514/MMN.2017.1565
    [4] D. A. Brannan, On functions of bounded boundary rotation-Ⅰ, Proc. Edinburgh Math. Soc., 16 (1969), 339–347. https://doi.org/10.1017/S001309150001302X doi: 10.1017/S001309150001302X
    [5] D. A. Brannan, J. G. Clunie, Aspects of contemporary complex analysis, London: Academic Press, 1980.
    [6] D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, In: Mathematical analysis and its applications, 53–60, Oxford: Pergamon, 1985.
    [7] M. Çağlar, G. Palpandy, E. Deniz, Unpredictability of initial coefficients for m-fold symmetric bi-univalent starlike and convex functions defined by subordinations, Afr. Mat., 29 (2018), 793–802. https://doi.org/10.1007/s13370-018-0578-0 doi: 10.1007/s13370-018-0578-0
    [8] N. E. Cho, E. Analouei Adegani, S. Bulut, A. Motamednezhad, The second Hankel determinant problem for a class of bi-close-to-convex functions, Mathematics, 7 (2019), 986. https://doi.org/10.3390/math7100986 doi: 10.3390/math7100986
    [9] M. Darus, T. N. Shanmugam, S. Sivasubramanian, Fekete–Szegö inequality for a certain class of analytic functions, Mathematica, 49 (2007), 29–34.
    [10] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal., 2 (2013), 49–60. https://doi.org/10.7153/jca-02-05 doi: 10.7153/jca-02-05
    [11] E. Deniz, J. M. Jahangiri, S. K. Kına, S. G. Hamidi, Faber polynomial coefficients for generalized bi-subordinate functions of complex order, J. Math. Inequal., 12 (2018), 645–653. https://doi.org/10.7153/jmi-2018-12-49 doi: 10.7153/jmi-2018-12-49
    [12] P. L. Duren, Univalent functions, New York: Springer, 1983.
    [13] B. A. Frasin, M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett., 24 (2011), 1569–1573. https://doi.org/10.1016/j.aml.2011.03.048 doi: 10.1016/j.aml.2011.03.048
    [14] S. P. Goyal, P. Goswami, Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, J. Egyptian Math. Soc., 20 (2012), 179–182. https://doi.org/10.1016/j.joems.2012.08.020 doi: 10.1016/j.joems.2012.08.020
    [15] T. Hayami, S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J., 22 (2012), 15–26.
    [16] S. Kanas, V. Sivasankari, R. Karthiyayini, S. Sivasubramanian, Second Hankel determinant for a certain subclass of bi-close to convex functions defined by Kaplan, Symmetry, 13 (2021), 567.
    [17] W. Kaplan, Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), 169–185. http://projecteuclid.org/euclid.mmj/1028988895
    [18] S. Kazımoğlu, E. Deniz, Fekete-Szegö problem for generalized bi-subordinate functions of complex order, Hacet. J. Math. Stat., 49 (2020), 1695–1705.
    [19] F. R. Keogh, E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8–12. https://doi.org/10.2307/2035949 doi: 10.2307/2035949
    [20] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc., 101 (1987), 89–95. https://doi.org/10.2307/2046556 doi: 10.2307/2046556
    [21] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18 (1967), 63–68. https://doi.org/10.2307/2035225 doi: 10.2307/2035225
    [22] Y. Li, K. Vijaya, G. Murugusundaramoorthy, H. Tang, On new subclasses of bi-starlike functions with bounded boundary rotation, AIMS Mathematics, 5 (2020) 3346–3356. https://doi.org/10.3934/math.2020215 doi: 10.3934/math.2020215
    [23] A. K. Mishra, M. M. Soren, Coefficient bounds for bi-starlike analytic functions, Bull. Belg. Math. Soc.-Simon Stevin, 21 (2014), 157–167.
    [24] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |ζ|<1, Arch. Rational Mech. Anal., 32 (1969), 100–112. https://doi.org/10.1007/BF00247676 doi: 10.1007/BF00247676
    [25] V. Paatero, Über Gebiete von beschrankter Randdrehung, Ann. Acad. Sci. Fenn. Ser. A., 37 (1933), 9.
    [26] K. S. Padmanabhan, R. Parvatham, Properties of a class of functions with bounded boundary rotation, Ann. Polon. Math., 31 (1975/76), 311–323. https://doi.org/10.4064/ap-31-3-311-323 doi: 10.4064/ap-31-3-311-323
    [27] B. Pinchuk, Functions of bounded boundary rotation, Israel J. Math., 10 (1971), 6–16. https://doi.org/10.1007/BF02771515 doi: 10.1007/BF02771515
    [28] M. O. Reade, On close-to-convex univalent functions, Michigan Math. J., 3 (1955), 59–62.
    [29] M. O. Reade, The coefficients of close-to-convex functions, Duke Math. J., 23 (1956), 459–462.
    [30] M. S. Robertson, On the theory of univalent functions, Ann. Math., 37 (1936), 374–408. https://doi.org/10.2307/1968451 doi: 10.2307/1968451
    [31] S. Sivasubramanian, R. Sivakumar, S. Kanas, S. A. Kim, Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions, Ann. Polon. Math., 113 (2015), 295–304. https://doi.org/10.4064/ap113-3-6 doi: 10.4064/ap113-3-6
    [32] H. M. Srivastava, A. K. Mishra P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett., 23 (2010), 1188–1192. https://doi.org/10.1016/j.aml.2010.05.009 doi: 10.1016/j.aml.2010.05.009
    [33] H. M. Srivastava, D. Bansal, Coefficient estimates for a subclass of analytic and bi-univalent functions, J. Egyptian Math. Soc., 23 (2015), 242–246. https://doi.org/10.1016/j.joems.2014.04.002 doi: 10.1016/j.joems.2014.04.002
    [34] D. K. Thomas, On the coefficients of bounded boundary rotation, Proc. Amer. Math. Soc., 36 (1972), 123–129. https://doi.org/10.1090/S0002-9939-1972-0308384-2 doi: 10.1090/S0002-9939-1972-0308384-2
    [35] Q. H. Xu, Y. C. Gui, H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25 (2012), 990–994. https://doi.org/10.1016/j.aml.2011.11.013 doi: 10.1016/j.aml.2011.11.013
    [36] Q. H. Xu, H. M. Srivastava, H. G. Xiao, A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Appl. Math. Comput., 218 (2012), 11461–11465. https://doi.org/10.1016/j.amc.2012.05.034 doi: 10.1016/j.amc.2012.05.034
    [37] Q. H. Xu, H. M. Srivastava, L. Zhou, A certain subclass of analytic and close-to-convex functions, Appl. Math. Lett., 24 (2011), 396–401. https://doi.org/10.1016/j.aml.2010.10.037 doi: 10.1016/j.aml.2010.10.037
    [38] P. Zaprawa, On the Fekete-Szegö problem for classes of bi-univalent functions, Bull. Belg. Math. Soc.-Simon Stevin, 21 (2014), 169–178.
  • This article has been cited by:

    1. Richa Agarwal, Tanveer Fatima, Sarvesh Kumar Yadav, Shahid Ali, Geometric analysis of Riemannian doubly warped product submersions, 2024, 99, 0031-8949, 075271, 10.1088/1402-4896/ad579a
    2. Foued Aloui, Einstein-like Poisson Warped Product Manifolds and Applications, 2025, 17, 2073-8994, 645, 10.3390/sym17050645
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2022) PDF downloads(98) Cited by(10)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog