Research article Topical Sections

“That was terrifying!”: When 2SLGBTQQ+ individuals and rural women experiencing intimate partner violence are stalked

  • Received: 24 April 2024 Revised: 07 August 2024 Accepted: 26 August 2024 Published: 31 August 2024
  • Background 

    Stalking reflects a lesser-studied form of intimate partner violence (IPV; e.g., physical abuse) that may occur pre- and postseparation between two or more partners, incurring lifelong pervasive health impacts on those involved. Intersectionality theory elucidates how Two-Spirit, lesbian, gay, bisexual, trans, queer, questioning, intersex, and asexual (2SLGBTQQIA+) individuals' and rural women's identities are oppressed by society, thus subjecting them to unique IPV experiences. Therefore, this study aims to explore how stalking manifests among 2SLGBTQQIA+ individuals and women living rurally with lived experiences, both of which are underrepresented groups in current stalking literature.

    Methods 

    We used secondary data from two IPV studies conducted among 2SLGBTQQ+ (no intersex or asexual participants) individuals and rural women (n = 29). We interviewed 2SLGBTQQ+ (n = 18) and rural women (n = 11) who resided in Alberta, Canada and experienced IPV via semi-structured, qualitative approaches. A thematic analysis was guided by intersectionality theory to analyze the data, applying inductive and semantic approaches.

    Findings 

    Of the 29 participants, 15 were stalked by their abusive partners and 9 reported on the negative impacts of being stalked. Rural women and 2SLGBTQQ+ individuals were mainly stalked via physical forms of stalking and cyberstalking, respectively. We describe other forms of stalking and the ineffectiveness of legal systems in those seeking support for stalking. The impacts of stalking (e.g., hypervigilance) were so profound that the feeling of being stalked persisted, which we termed phantom stalking.

    Significance 

    2SLGBTQQ+ individuals predominantly experienced stalking through technology and rural women experienced stalking in more public or physical forms, which stemmed from intersections with community and geographical factors, respectively. We posit the notion of “phantom stalking” and discuss and differentiate it from other psychiatric diagnoses. Additionally, we provide important recommendations related to legislation, education, safety, and research.

    Citation: Stefan Kurbatfinski, Kendra Nixon, Susanne Marshall, Jason Novick, Dawn McBride, Nicole Letourneau. “That was terrifying!”: When 2SLGBTQQ+ individuals and rural women experiencing intimate partner violence are stalked[J]. AIMS Medical Science, 2024, 11(3): 265-291. doi: 10.3934/medsci.2024020

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  • Background 

    Stalking reflects a lesser-studied form of intimate partner violence (IPV; e.g., physical abuse) that may occur pre- and postseparation between two or more partners, incurring lifelong pervasive health impacts on those involved. Intersectionality theory elucidates how Two-Spirit, lesbian, gay, bisexual, trans, queer, questioning, intersex, and asexual (2SLGBTQQIA+) individuals' and rural women's identities are oppressed by society, thus subjecting them to unique IPV experiences. Therefore, this study aims to explore how stalking manifests among 2SLGBTQQIA+ individuals and women living rurally with lived experiences, both of which are underrepresented groups in current stalking literature.

    Methods 

    We used secondary data from two IPV studies conducted among 2SLGBTQQ+ (no intersex or asexual participants) individuals and rural women (n = 29). We interviewed 2SLGBTQQ+ (n = 18) and rural women (n = 11) who resided in Alberta, Canada and experienced IPV via semi-structured, qualitative approaches. A thematic analysis was guided by intersectionality theory to analyze the data, applying inductive and semantic approaches.

    Findings 

    Of the 29 participants, 15 were stalked by their abusive partners and 9 reported on the negative impacts of being stalked. Rural women and 2SLGBTQQ+ individuals were mainly stalked via physical forms of stalking and cyberstalking, respectively. We describe other forms of stalking and the ineffectiveness of legal systems in those seeking support for stalking. The impacts of stalking (e.g., hypervigilance) were so profound that the feeling of being stalked persisted, which we termed phantom stalking.

    Significance 

    2SLGBTQQ+ individuals predominantly experienced stalking through technology and rural women experienced stalking in more public or physical forms, which stemmed from intersections with community and geographical factors, respectively. We posit the notion of “phantom stalking” and discuss and differentiate it from other psychiatric diagnoses. Additionally, we provide important recommendations related to legislation, education, safety, and research.



    Fractional calculus signifies the identity of the distinguished materials in the modern research field due to its integrated applications in diverse regions such as mathematical physics, fluid dynamics, mathematical biology, etc. Convex function, exponentially convex function [1,2,3,4,5], related inequalities like as trapezium inequality, Ostrowski's inequality and Hermite Hadamard inequality, integrals [6,7,8,9,10] having succeed in mathematical analysis, approximation theory due to immense applications [11,12] have great importance in mathematics theory. Many authors established quadrature rules in numerical analysis for approximate definite integrals. Recently, Pólya-Szegö and Chebyshev inequalities occupied immense space in the field analysis. Chebyshev [13] was introduced the well-known inequality called Chebyshev inequality.

    In the literature of convex function, the Jensen inequality has gained much importance which describes a connection between an integral of the convex function and the value of the convex function of an interval [14,15,16]. Pshtiwan and Thabet [17] considered the modified Hermite Hadamard inequality in the context of fractional calculus using the Riemann-Liouville fractional integrals. Arran and Pshtiwan [18] discussed the Hermite Hadamard inequality results with fractional integrals and derivatives using Mittag-Leffler kernel. Pshtiwan and Thabet [19] constructed a connection between the Riemann-Liouville fractional integrals of a function concerning a monotone function with nonsingular kernel and Atangana-Baleanu. Pshtiwan and Brevik [20] obtained an inequality of Hermite Hadamard type for Riemann-Liouville fractional integrals, and proved the application of obtained inequalities on modified Bessel functions and q-digamma function. In [21], Set et al. introduced Grüss type inequalities by employing generalized k-fractional integrals. Recently, Nisar et al. [22] gave some new generalized fractional integral inequalities.

    Very recently, the fractional conformable and proportional fractional integral operators were given in [23,24]. Later on, Huang et al. [25] gave Hermite–Hadamard type inequalities by using fractional conformable integrals (FCI). Qi et al. [26] investigated Čebyšev type inequalities involving FCI. The Chebyshev type inequalities and certain Minkowski's type inequalities are found in [27,28,29]. Nisar et al. [30] have investigated some new inequalities for a class of n  (nN) positive, continuous, and decreasing functions by employing FCI. Rahman et al. [31] introduced Grüss type inequalities for k-fractional conformable integrals.

    Some significant inequalities are given as applications of fractional integrals [32,33,34,35,36,37,38]. Recently, Rahman et al. [39,40] presented fractional integral inequalities involving tempered fractional integrals. Qiang et al. [41] discussed a fractional integral containing the Mittag-Leffler function in inequality theory and contributed Hadamard type inequality, continuity, and boundedness, upper bounds of that integral. Nisar et al. [42] established weighted fractional Pólya-Szegö and Chebyshev type integral inequalities by operating the generalized weighted fractional integral involving kernel function. The dynamical approach of fractional calculus [43,44,45,46,47,48,49] in the field of inequalities.

    Grüss inequality [50] established for two integrable function as follows

    |T(h,l)|(kK)(sS)4, (1.1)

    where the h and l are two integrable functions which are synchronous on [a,b] and satisfy:

    sh(z)K,sl(y1)S, z,y1[a,b] (1.2)

    for some s,k,S,KR.

    Pólya and Szegö [51] proved the inequalities

    bah2(z)dzabl2(z)dz(abh(z)l(z)dz)214(KSks+ksKS)2. (1.3)

    Dragomir and Diamond [52], proves the inequality by using the Pólya-szegö inequality

    |T(h,l)|(Ss)(Kk)4(ba)2skSKbah(z)l(z)dz (1.4)

    where h and l are two integrable functions which are synchronous on [a,b], and

    0<sh(z)S<,0<kl(y1)K<, z,y1[a,b] (1.5)

    for some s,k,S,KR.

    The aim of this paper is to estimate a new version of Pólya-Szegö inequality, Chebyshev integral inequality, and Hermite Hadamard type integral inequality by a fractional integral operator having a nonsingular function (generalized multi-index Bessel function) as a kernel, and these established results have great contribution in the field of inequalities. The Hermite Hadamard type integral inequality provides the upper and lower estimate to find the average integral for the convex function of any defined interval.

    The structure of the paper follows:

    In section 2, we present some well-known definitions and mathematical preliminaries. The new generalized fractional integral with nonsingular function as a kernel is defined in section 3. In section 4, we present Hermite Hadamard type Mercer inequality of new designed fractional integral operator with nonsingular function (generalized multi-index Bessel function) as a kernel. some inequalities of (sm)-preinvex function involving new designed fractional integral operator with nonsingular function (generalized multi-index Bessel function) as a kernel are presented in section 5. Here section 6 and 7, we present Pólya-Szegö and Chebyshev integral inequalities involving generalized fractional integral operator with nonsingular function as a kernel, respectively.

    Definition 2.1. The inequality holds for the convex function if a mapping g:KR exist as

    g(δy1+(1δ)y2)δg(y1)+(1δ)g(y2), (2.1)

    where y1,y2K and δ[0,1].

    Definition 2.2. The inequality derived by Hermite [53] call as Hermite Hadamard inequality

    g(y1+y22)1y2y1y2y1g(t)dtg(y1)+g(y2)2, (2.2)

    where y1,y2I, with y2y1, if g:IRR is a convex function.

    Definition 2.3. Let yjK for all jIn, ωj>0 such that jInωj=1. Then the Jensen inequality holds

    g(jInωjyj)jInωjg(yj), (2.3)

    exist if g:kR is convex function.

    Mercer [54] derived the Mercer inequality by applying the Jensen inequality and properties of convex function.

    Definition 2.4. Let yjK for all jIn, ωj>0 such that jInωj=1, m=minjIn{yj} and n=maxjIn{yj}. Then the inequality holds for convex function as

    g(m+niInωjyj)g(m)+g(n)jInωjg(yj), (2.4)

    if g:kR is convex function.

    Definition 2.5. [55] The inequality holds for exponentially convex function, if a real valued mapping g:KR exist as

    g(δy1+(1δ)y2)δg(y1)eθy1+(1δ)g(y2)eθy2, (2.5)

    where y1,y2K and δ[0,1] and θR.

    Suppose that ΩRn is a set. Let g:ΩR continuous function and let ξ:Ω×ΩRn be continuous function:

    Definition 2.6. [56] With respect to bifunction ξ(.,.) a set Ω is called a invex set, if

    y1+δξ(y2,y1), (2.6)

    where y1,y2Ω,δ[0,1].

    Definition 2.7. [57] A invex set Ω and a mapping g with respect to ξ(.,.) is called a preinvex function, as

    g(y1+δξ(y2,y1))(1δ)g(y1)+δg(y2), (2.7)

    where y1,y2+ξ(y2,y1)Ω,δ[0,1].

    Definition 2.8. A invex set Ω with real valued mapping g and respect to ξ(.,.) is called a exponentially preinvex, if the inequality

    g(y1+δξ(y2,y1))(1δ)g(y1)eθy1+δg(y2)eθy2, (2.8)

    where for all y1,y2+ξ(y2,y1)Ω,δ[0,1] and θR.

    Definition 2.9. A invex set Ω with real valued mapping g and respect to ξ(.,.) is called a exponentially s-preinvex, if

    g(y1+δξ(y2,y1))(1δ)sg(y1)eθy1+δsg(y2)eθy2, (2.9)

    where for all y1,y2+ξ(y2,y1)Ω,δ[0,1], s(0,1] and θR.

    Definition 2.10. A invex set Ω with real valued mapping g and respect to ξ(.,.) is called exponentially (s-m)-preinvex, if

    g(y1+mδξ(y2,y1))(1δ)sg(y1)eθy1+mδsg(y2)eθy2, (2.10)

    where for all y1,y2+ξ(y2,y1)Ω, δ,m[0,1] and θR.

    Definition 2.11. [58] Generalized multi-index Bessel function is defined by Choi et al as follows

    J(ξj)m,λ(δj)m,σ(z)=s=0(λ)σsmj=1Γ(ξjs+δj+1)(z)ss!, (2.11)

    where ξj,δj,λC, (j=1,,m), (λ)>0,(δj)>1,mj=1(ξ)j>max{0:(σ)1},σ>0.

    Definition 2.12. [58] Pohhammer symbol is defined for λC as follows

    (λ)s={λ(λ+1)(λ+s1),sN1,s=0, (2.12)
    =Γ(λ+s)Γ(λ),(λC/Z0) (2.13)

    where Γ being the Gamma function.

    This section presents a generalized fractional integral operator with a nonsingular function (multi-index Bessel function) as a kernel.

    Definition 3.1. Let ξj,δj,λ,ζC,(j=1,,m),(λ)>0,(δj)>1,mj=1(ξj)>max{0:(σ)1},σ>0. Let gL  [y1,y2] and t[y1,y2]. Then the corresponding left sided and right sided generalized integral operators having generalized multi-index Bessel function defined as:

    (Œ(ξj,δj)mλ,σ,ζ;y+1g)(z)=zy1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)g(t)dt, (3.1)

    and

    (Œ(ξj,δj)mλ,σ,ζ;y2g)(z)=y2z(tz)δjJ(ξj)m,λ(δj)m,σ(ζ(tz)ξj)g(t)dt. (3.2)

    Remark 3.1. The special cases of generalized fractional integrals with nonsingular kernel are given below:

    1. If set j=m=1, σ=0 and limits from [0,z] in Eq (3.1), we get a fractional integral defined by Srivastava and Singh in [59] as

    (Œξ1,δ1λ,0,ζ;0+g)(z)=z0(zt)δ1Jξ1δ1(ζ(zt)ξ1)g(t)dt=f(z). (3.3)

    2. If set j=m=1, δ1=δ11 in Eq (3.1), we have a fractional integral defined by Srivastava and Tomovski in [60] as

    (Œξ1,δ11λ,σ,ζ;y+1g)(z)=(Eζ;λ,σy+1;ξ1,δ1g)(z). (3.4)

    3. If set j=m=1, δ1=δ11, ζ=0 in Eq (3.1), we get a Riemann-Liouville fractional integral operator defined in [61] as

    (Œξ1,δ1λ,σ,ζ;y+1g)(z)=(Iδ1y+1g)(z). (3.5)

    4. If set j=m=1, σ=1, δ1=δ11, in Eq (3.1) and Eq (3.2), we get the fractional integral operator defined by Prabhakar in [62] as follows

    (Œξ1,δ11λ,1,ζ;y+1g)(z)=E(ξ1,δ1;λ;ζ)g(z)=g(z) (3.6)
    (Œ(ξ1,δ11)λ,1,ζ;y2g)(z)=E(ξ1,δ1;λ;ζ)g(z). (3.7)

    Lemma 3.1. From generalized fractional integral operator, we have

    (Œ(ξj,δj)mλ,σ,ζ;y+11)(z)=zy1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)dt=zy1(zt)δjs=0(λ)σs(ζ)smj=1Γ(ξjs+δj+1)(zt)ξjss!dt=s=0(λ)σs(ζ)smj=1Γ(ξjs+δj+1)s!zy1(zt)ξjs+δjdt=(zy1)δj+1s=0(λ)σs(ζ)smj=1Γ(ξjs+δj+1)s!(zy1)ξjsξjs+δj+1. (3.8)

    Hence, the Eq (3.8) becomes

    (Œ(ξj,δj+1)mλ,σ,ζ;y+11)(z)=(zy1)δj+1J(ξj)m,λ(δj)m+1,σ(ζ(zy1)ξj), (3.9)

    and similarly we have

    (Œ(ξj,δj+1)mλ,σ,ζ;y21)(z)=(y2z)δj+1J(ξj)m,λ(δj)m+1,σ(ζ(y2z)ξj). (3.10)

    In this section, we derive Hermite Hadamard type Mercer inequality of new designed fractional integral operator in a generalized multi-index Bessel function using a kernel.

    Theorem 4.1. Let g:[m,n](0,) is convex function such that gχc(m,n), x,y[m,n] and the operator defined in Eq (5.2) in the form of left sense operator and Eq (3.2) in the form of right sense operator then we have

    g(m+nx+y2)g(m)+g(n)[J(ξj)m,λ(δj)m+1,σ(ζ)]12(yx)[Œ(ξj,δj)mλ,σ,ζ;x+g(y)+Œ(ξj,δj)mλ,σ,ζ;yg(x)] (4.1)
    g(m)+g(n)g(x)+g(y)2. (4.2)

    Proof. Consider the mercer inequality

    g(m+ny1+y22)g(m)+g(n)g(y1)+g(y2)2,y1,y2[m,n]. (4.3)

    Let x,y[m,n], t[z1,z], y1=(zt)x+(1z+t)y and y2=(1z+t)x+(zt)y then inequality (4.3) becomes

    g(m+ny1+y22)g(m)+g(n)g((zt)x+(1z+t)y)+g(1z+t)x+(zt)y)2. (4.4)

    Multiply both sides of Eq (4.4) by (zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj) and integrating with respect to t from [z1,z], we get

    J(ξj)m,λ(δj)m+1,σ(ζ)g(m+nx+y2)J(ξj)m,λ(δj)m+1,σ(ζ)[g(m)+g(n)]12[zz1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)×[g((zt)y1+(1z+t)y2)+g(1z+t)x+(zt)y2]]dt=J(ξj)m,λ(δj)m+1,σ(ζ)[g(m)+g(n)]12[yx(yuyx)δjJ(ξj)m,λ(δj)m,σ(ζ(yuyx)ξj)×g(u)(yx)du+xy(uxyx)δjJ(ξj)m,λ(δj)m,σ(ζ(uxyx)ξj)g(u)(yx)du]=J(ξj)m,λ(δj)m+1,σ(ζ)[g(m)+g(n)]12(yx)[Œ(ξj,δj)mλ,σ,ζ;x+g(y)+Œ(ξj,δj)mλ,σ,ζ;yg(x)],

    we get the desired inequality, as

    g(m+nx+y2)g(m)+g(n)[J(ξj)m,λ(δj)m+1,σ(ζ)]12(yx)[Œ(ξj,δj)mλ,σ,ζ;x+g(y)+Œ(ξj,δj)mλ,σ,ζ;yg(x)]. (4.5)

    Thus, we get the inequality (4.1). Let t[z1,z]. From the convexity of function g we have

    g(x+y2)=g[(zt)x+(1z+t)y+(1z+t)x+(zt)y]2g((zt)x+(1z+t)y)+g((1z+t)x+(zt)y)2. (4.6)

    Both sides multiply of Eq (4.6) by (zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj) and integrating with respect to t from [z1,z], we obtain

    J(ξj)m,λ(δj)m,σ(ζ)g(x+y2)zz1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)×[g((zt)x+(1z+t)y)+g((1z+t)x+(zt)y)]dt=12(yx)[Œ(ξj,δj)mλ,σ,ζ;x+g(y)+Œ(ξj,δj)mλ,σ,ζ;yg(x)].

    We get the inequality of negative sign

    g(x+y2)[J(ξj)m,λ(δj)m+1,σ(ζ)]12(yx)[Œ(ξj,δj)mλ,σ,ζ;x+g(y)+Œ(ξj,δj)mλ,σ,ζ;yg(x)]. (4.7)

    By adding g(m)+g(n) of both sides of inequality (4.7), we have

    g(m)+g(n)g(x+y2)g(m)+g(n)[J(ξj)m,λ(δj)m+1,σ(ζ)]12(yx)[Œ(ξj,δj)mλ,σ,ζ;x+g(y)+Œ(ξj,δj)mλ,σ,ζ;yg(x)].

    Hence, we get the inequality (4.2).

    Theorem 4.2. Let g:[m,n](0,) is convex function such that gχc(m,n) then we have the following inequalities:

    g(m+nx+y2)[J(ξj)m,λ(δj)m,σ(ζ)]12(yx)[Œ(ξj,δj)mλ,σ,ζ;(m+ny)+g(m+nx)+Œ(ξj,δj)mλ,σ,ζ;(m+nx)g(m+ny)]. (4.8)
    g(m+nx)+g(m+ny)2g(m)+g(n)g(m)+g(n)2. (4.9)

    Where x,y[m,n].

    Proof. We see that from the convexity of g as

    g(m+ny1+y22)=g(m+ny1+m+ny22)12[g(m+ny1)+g(m+ny2)],y1,y2[m,n]. (4.10)

    Let x,y[m,n], t[z1,z], m+ny1=(zt)(m+nx)+(1z+t)(m+ny), m+ny2=(1z+t)(m+nx)+(zt)(m+ny), then inequality (4.10) gives

    g(m+ny1+y22)12g[(zt)(m+nx)+(1z+t)(m+ny)]+12g[(1z+t)(m+nx)+(zt)(m+ny)], (4.11)

    multiply of both sides of inequality (4.11) by (zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj) then integrate with respect to t from [z1,z], we get

    J(ξj)m,λ(δj)m,σ(ζ)g(m+nx+y2)12zz1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)g[(zt)(m+nx)+(1z+t)(m+ny)]dt+12zz1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)g[(1z+t)(m+nx)+(zt)(m+ny)]dt=12(yx)[m+nxm+ny(u(m+ny)yx)δj)J(ξj)m,λ(δj)m,σ(ζ(u(m+ny)yx)ξj)g(u)du+m+nym+nx((m+ny)uyx)δj)J(ξj)m,λ(δj)m,σ(ζ((m+ny)uyx)ξj)g(u)du]=12(yx)[Œ(ξj,δj)mλ,σ,ζ;(m+ny)+g(m+nx)+Œ(ξj,δj)mλ,σ,ζ;(m+nx)g(m+ny)].

    Thus, we get the inequality (4.8)

    g(m+nx+y2)[J(ξj)m,λ(δj)m,σ(ζ)]12(yx)[Œ(ξj,δj)mλ,σ,ζ;(m+ny)+g(m+nx)+Œ(ξj,δj)mλ,σ,ζ;(m+nx)g(m+ny)].

    From the convexity of g, we obtain

    g((zt)(m+nx)+(1z+t)(m+ny))(zt)g(m+nx)+(1z+t)g(m+ny), (4.12)

    and

    g((1z+t)(m+nx)+(zt)(m+ny))(1z+t)g(m+nx)+(zt)g(m+ny). (4.13)

    Adding up the above inequalities and applying Jensen-Mercer inequality, we get

    g((zt)(m+nx)+(1z+t)(m+ny))+g((1z+t)(m+nx)+(zt)(m+ny))g(m+nx)+g(m+ny)2[g(m)+g(n)][g(x)+g(y)]. (4.14)

    Multiply both sides of inequality (4.14) by (zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj) and then integrating with respect to t from [z1,z] we obtain the two inequalities (4.9).

    In this section, we derive some inequalities of (sm) preinvex function involving new designed fractional integral operator Œ(ξj,δj)mλ,σ,ζg)(z) having generalized multi-index Bessel function as its kernel in the form of theorems.

    Theorem 5.1. Suppose a real valued function g:[y1,y1+ξ(y2,y1)]R be exponentially (s-m) preinvex function, then the following fractional inequality holds:

    (Œ(ξj,δj)mλ,σ,ζ;y+1g)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+g)(z)(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[g(y1)eθ1y1+mg(z)eθ1z]+(y1+ξ(y2,y1)z)s+1(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+1)(z)[g(y1+ξ(y2,y1))eθ2(y1+ξ(y2,y1))+mg(z)eθ2z].

    z[y1,y1+ξ(y2,y1)], θ1,θ2R.

    Proof. Let z[y1,y1+ξ(y2,y1)], and then for t[y1,z) and δj>1, we have the subsequent inequality

    (zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)(zy1)δjJ(ξj)m,λ(δj)m,σ(ζ(zy1)ξj). (5.1)

    For g is exponentially (s-m)-preinvex function, we obtain

    g(t)(ztzy1)sg(y1)eθ1y1+m(ty1zy1)sg(z)eθ1z. (5.2)

    Taking product (5.1) and (5.2), and integrating with respect to t from y1 to z, we get

    zy1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)g(t)dtzy1(zy1)δjJ(ξj)m,λ(δj)m,σ(ζ(zy1)ξj)×[(ztzy1)sg(y1)eθ1y1+m(ty1zy1)sg(z)eθ1z]dt, (5.3)

    apply definition (13) in Eq (5.3), we have

    (Œ(ξj,δj)mλ,σ,ζ;y+1g)(z)(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[g(y1)eθ1y1+mg(z)eθ1z]. (5.4)

    Analogously for t(z,y1+ξ(y2,y1)] and μj>1, we have

    (tz)μjJ(ξj)m,λ(μj)m,σ(ζ(tz)ξj)(y1+ξ(y2,y1)z)μjJ(ξj)m,λ(μj)m,σ(ζ(y1+ξ(y2,y1)z)ξj). (5.5)

    Further, the exponentially (s-m) convexity of g, we get

    g(t)(tzy1+ξ(y2,y1)z)sg(y1+ξ(y2,y1))eθ2(y1+ξ(y2,y1))+m(y1+ξ(y2,y1)ty1+ξ(y2,y1)z)sg(z)eθ2z. (5.6)

    Taking product of (5.5) and (5.6) and integrating with respect to t from z to y1+ξ(y2,y1), we have

    y1+ξ(y2,y1)z(tz)μjJ(ξj)m,λ(μj)m,σ(ζ(tz)ξj)g(t)dty1+ξ(y2,y1)z(y1+ξ(y2,y1)z)μjJ(ξj)m,λ(μj)m,σ(ζ(y1+ξ(y2,y1)z)ξj)×[(tzy1+ξ(y2,y1)z)sg(y1+ξ(y2,y1))eθ2(y1+ξ(y2,y1))+m(y1+ξ(y2,y1)ty1+ξ(y2,y1)z)sg(z)eθ2z]dt, (5.7)

    apply the definition (13) in inequality (5.7), we have

    (Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+g)(z)(y1+ξ(y2,y1)z)s+1(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+1)(z)[g(y1+ξ(y2,y1))eθ2(y1+ξ(y2,y1))+mg(z)eθ2z]. (5.8)

    Now, add the inequalities (5.4) and (5.8), we get the result

    (Œ(ξj,δj)mλ,σ,ζ;y+1g)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+g)(z)(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[g(y1)eθ1y1+mg(z)eθ1z]+(y1+ξ(y2,y1)z)s+1(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+1)(z)[g(y1+ξ(y2,y1))eθ2(y1+ξ(y2,y1))+mg(z)eθ2z].

    Corollary 5.1. If gL[y1,y1+ξ(y2,y1)], then under the assumption of theorem (5.1), we have

    (Œ(ξj,δj)mλ,σ,ζ;y+1g)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+g)(z)||g||s+1[(zy1)(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)(1eθ1y1+m1eθ1z)+(y1+η(y2,y1)z)(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+1)(z)(1eθ2(y1+ξ(y2,y1))+m1eθ2z)].

    Corollary 5.2. Setting m=1 and gL[y1,y1+ξ(y2,y1)], then under the assumption of theorem (5.1), we have

    (Œ(ξj,δj)mλ,σ,ζ;y+1g)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+g)(z)||g||s+1[(zy1)(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)(1eθ1y1+m1eθ1z)+(y1+ξ(y2,y1)z)(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+1)(z)(1eθ2(y1+ξ(y2,y1))+1eθ2z)].

    Corollary 5.3. Setting m=s=1 and gL[y1,y1+ξ(y2,y1)], then under the assumption of theorem (5.1), we have

    (Œ(ξj,δj)mλ,σ,ζ;y+1g)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+g)(z)||g||2[(zy1)(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)(1eθ1y1+m1eθ1z)+(y1+ξ(y2,y1)z)(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+1)(z)(1eθ2(y1+ξ(y2,y1))+1eθ2z)].

    Corollary 5.4. Setting ξ(y2,y1)=y2y1 and gL[y1,y2], then under the assumption of theorem (5.1), we have

    (Œ(ξj,δj)mλ,σ,ζ;y+1g)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))+g)(z)||g||s+1[(zy1)(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)(1eθ1y1+m1eθ1z)+(y2z)(Œ(ξj,μj)mλ,σ,ζ;y+21)(z)(1eθ2y2+1eθ2z)].

    Theorem 5.2. Suppose a real value function g:[y1,y1+ξ(y2,y1)]R is differentiable and |g| is exponentially (s-m) preinvex, then the following fractional inequality for (3.1) and (3.2) holds:

    |(Œ(ξj)m,(δj1)mλ,σ,ζ:y+1g)(z)+(Œ(ξj)m,(μj1)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z)[(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)]g(y1)[(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)]g(y1+ξ(y2,y1))|(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[|g(y1)|eθ1y1+m|g(z)|eθ1z]+((y1+ξ(y2,y1)z)s+1(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)[|g(y1+ξ(y2,y1))|eθ1(y1+ξ(y2,y1))+m|g(z)|eθ1z].

    z[y1,y1+ξ(y2,y1)], θ1,θ2R.

    Proof. Let z[y1,y1+ξ(y2,y1)], t[y1,z), and applying exponentially (s-m) preinvex of |g|, we get

    |g(t)|(ztzy1)s|g(y1)|eθ1y1+m(ty1zx1)s|g(z)|eθ1z. (5.9)

    Get the inequality (5.9), we have

    g(t)(ztzy1)s|g(y1)|eθ1y1+m(ty1zy1)s|g(x)|eθ1z. (5.10)

    Subsequently inequality as:

    (zt)δjJ(ξj)m,λ(δj)m,k(ζ(zt)ξj)(zy1)δjJ(ξj)m,λ(δj)m,σ(ζ(zy1)ξj). (5.11)

    Conducting product of inequality (5.10) and (5.11), we have

    (zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)g(t)(zy1)δjJ(ξj)m,λ(δj)m,σ(ζ(zy1)ξj)×[(ztzy1)s|g(y1)|eθ1y1+m(ty1zy1)s|g(x)|eθ1z], (5.12)

    integrating before mention inequality with respect to t from y1 to z, we have

    zy1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)g(t)dtzy1(zy1)δjJ(ξj)m,λ(δj)m,k(ζ(zy1)ξj)[(ztzy1)s|g(y1)|eθ1y1+m(ty1zy1)s|g(z)|eθ1z]dt=(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[|g(y1)|eθ1y1+m|g(z)|eθ1z]. (5.13)

    Now, solving left side of (5.13) by putting zt=α, then we have

    zy1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)g(t)dt=zy10αδjJ(ξj)m,λ(δj)m,σ(ζ(α)ξj)g(zα)dα=(zy1)δjJ(ξj)m,λ(δj)m,σ(ζ(zy1)ξj)g(y1)+zy10αδj1J(ξj)m,λ(δj)m1,σ(ζ(α)ξj)g(zα)dα.

    Now, again subsisting zα=t, we get

    zy1(zt)δjJ(ξj)m,λ(δj)m,σ(ζ(zt)ξj)g(t)dt=zy1(zt)δj1J(ξj)m,λ(δj)m1,σ(ζ(zt)ξj)g(t)dt(zy1)δjJ(ξj)m,λ(δj)m,σ(ζ(zy1)ξj)g(y1)=(Œ(ξj)m,(δj1)mλ,σ,ζ:y+1g)(z)[(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)]g(y1).

    Therefore, the inequality (5.13) have the following form

    (Œ(ξj)m,(δj1)mλ,σ,ζ:y+1g)(x)[(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)]g(y1)(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[|g(y1)|eθ1y1+m|g(z)|eθ1z]. (5.14)

    Also from (5.9), we get

    g(t)(ztzy1)s|g(y1)|eθ1y1m(ty1zy1)s|g(z)|eθ1z. (5.15)

    Adopting the same procedure as we have done for (5.10), we obtain

    (Œ(ξj)m,(δj1)mλ,σ,ζ:y+1g)(z)[(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)]g(y1)(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[|g(y1)|eθ1y1+m|g(z)|eθ1z]. (5.16)

    From (5.14) and (5.16), we get

    |(Œ(ξj)m,(δj1)mλ,σ,ζ:y+1g)(z)[(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)]g(y1)|(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[|g(y1)|eθ1y1+m|g(z)|eθ1z]. (5.17)

    Now, we let z[y1,y1+η(y2,y1)] and t(z,y1+ξ(y2,y1)], and by exponentially (s-m) preinvex of |g|, we get

    |g(t)|(tzy1+ξ(y2,y1)z)s|g(y1+ξ(y2,y1))|eθ2(y1+ξ(y2,y1))+m(y1+ξ(y2,y1)ty1+ξ(y2,y1)z)s|g(z)|eθ2z, (5.18)

    repeat the same procedure from Eq (5.9) to Eq (5.17), we get

    |(Œ(ξj)m,(μj1)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z)[(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)]g(y1+ξ(y2,y1))|((y1+ξ(y2,y1)z)s+1(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)[|g(y1+ξ(y2,y1))|eθ1(y1+ξ(y2,y1))+m|g(z)|eθ1z]. (5.19)

    From inequalities (5.17) and (5.19), we have

    |(Œ(ξj)m,(δj1)mλ,σ,ζ:y+1g)(z)+(Œ(ξj)m,(μj1)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z)[(Œ(ξj,δj)mλ,k,ζ;y+11)(z)]g(y1)[(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)]g(y1+ξ(y2,y1))|(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[|g(y1)|eθ1y1+m|g(z)|eθ1z]+((y1+ξ(y2,y1)z)s+1(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)[|g(y1+ξ(y2,y1))|eθ1(y1+ξ(y2,y1))+m|g(z)|eθ1z].

    Corollary 5.5. Setting ξ(y2,y1)=y2y1, then under the assumption of theorem (5.2), we have

    |(Œ(ξj)m,(δj1)mλ,σ,ζ:y+1g)(z)+(Œ(ξj)m,(μj1)mλ,σ,ζ;y2g)(z)[(Œ(ξj,δj)mλ,k,ζ;y+11)(z)]g(y1)[(Œ(ξj,μj)mλ,σ,ζ;y21)(z)]g(y2)|(zy1)s+1(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[|g(y1)|eθ1y1+m|g(z)|eθ1z]+(y2z)s+1(Œ(ξj,μj)mλ,σ,ζ;y21)(z)[|g(y2)|eθ1(y2)+m|g(z)|eθ1z].

    t[y1,y2], θ1,θ2R.

    Corollary 5.6. Setting ξ(y2,y1)=y2y1, along with m=s=1 then under the assumption of theorem (5.2), we have

    |(Œ(ξj)m,(δj1)mλ,σ,ζ:y+1g)(z)+(Œ(ξj)m,(μj1)mλ,σ,ζ;y2g)(z)[(Œ(ξj,δj)mλ,k,ζ;y+11)(z)]g(y1)[(Œ(ξj,μj)mλ,σ,ζ;y21)(z)]g(y2)|(zy1)2(Œ(ξj,δj)mλ,σ,ζ;y+11)(z)[|g(y1)|eθ1y1+|g(z)|eθ1z]+(y2z)2(Œ(ξj,μj)mλ,σ,ζ;y21)(z)[|g(y2)|eθ1(y2)+|g(z)|eθ1z].

    t[y1,y2], θ1,θ2R.

    Definition 5.1. Let g:[y1,y1+ξ(y2,y1)]R is a function, and g is exponentially symmetric about 2y1+ξ(y2,y1)2 if

    g(z)eθz=g(2y1+ξ(y2,y1)z)eθ(2y1+ξ(y2,y1)z),θR. (5.20)

    Lemma 5.1. Let g:[y1,y1+ξ(y2,y1)]R be exponentially symmetric, then

    g(2y1+ξ(y2,y1)2)(1+m)g(z)2seθz,θR. (5.21)

    Proof. For g is exponentially (s-m) preinvex, therefore

    g(2y1+ξ(y2,y1)2)g(y1+δξ(y2,1))2seθ(y1+δξ(y2,y1))+mg(y1+(1δ)ξ(y2,y1))2seθ(y1+(1δ)ξ(y2,y1)). (5.22)

    Let t=y1+δξ(y2,y1), where t[y1,y1+ξ(y2,y1)], and then 2y1+ξ(y2,y1)=y1+(1δ)ξ(y2,y1), we have

    g(2y1+ξ(y2,y1)2)g(z)2seθz+mg(2y1+ξ(y2,y1)z)2seθ(2y1+ξ(y2,y1)z). (5.23)

    applying that g is exponentially symmetric, we obtain

    g(2y1+ξ(y2,y1)2)(1+m)g(z)2seθz. (5.24)

    Theorem 5.3. Suppose a real valued function g:[y1,y1+ξ(y2,y1)]R is exponentially (s-m) preinvex and symmetric about exponentially 2y1+ξ(y2,y1)2, then the following integral inequality for (3.1) and (3.2) holds:

    2s1+mf(2y1+ξ(y2,y1)2)[eθy1(Œ(μj,τj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(y1)+(Œ(μj,δj)mλ,σ,ζ;y+11)(y1+ξ(y2,y1))](Œ(μj,τj)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z)+(Œ(μj,τj)mλ,σ,ζ;y+1g)(y1+ξ(y2,y1))ξ(y2,y1)s+1(g(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+mg(y1)eθ1y1)×[(Œ(ξj,δj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(y1+ξ(y2,y1))]. (5.25)

    Proof. For z[y1,y1+ξ(y2,y1)], we have

    (zy1)δjJ(ξj)m,λ(δj)m,σ(ζ(zy1)ξj)(ξ(y2,y1))δjJ(ξj)m,λ(δj)m,σ(ζ(ξ(y2,y1))ξj), (5.26)

    the real value function g is exponentially (s-m) preinvex, then for z[y1,y1+ξ(y2,y1)], we get

    g(z)(zy1ξ(y2,y1))sg(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+m((y1+ξ(y2,y1)z)ξ(y2,y1))sg(y1)eθ1y1. (5.27)

    Conducting product of (5.26) and (5.27), and integrating with respect to z from y1 to y2, we get

    y2y1(zy1)δjJ(ξj)m,λ(δj)m,σ(ζ(zy1)ξj)g(z)dzy2y1(ξ(y2,y1))δjJ(ξj)m,λ(δj)m,σ(ζ(ξ(y2,y1))ξj)×[(zy1ξ(y2,y1))sg(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+m((y1+ξ(y2,y1)z)ξ(y2,y1))sg(y1)eθ1y1]dz, (5.28)

    then we have

    (Œ(ξj,δj)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z)(ξ(y2,y1))δjJ(ξj)m,λ(δj)m,σ(ζ(ξ(y2,y1))ξj)ξ(y2,y1)s+1[g(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+mg(y1)eθ1y1]=(Œ(ξj,δj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)ξ(y2,y1)s+1[g(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+mg(y1)eθ1y1]. (5.29)

    Analogously for z[y1,y1+ξ(y2,y1)], we have

    (y1+ξ(y2,y1)z)μjJ(ξj)m,λ(μj)m,σ(ζ(zy1)ξj)(ξ(y2,y1))μjJ(ξj)m,λ(μj)m,σ(ζ(ξ(y2,y1))ξj). (5.30)

    Conducting product of (5.27) and (5.30), and integrating with respect to z from y1 to y2, we have

    y2y1(y1+ξ(y2,y1)z)μjJ(ξj)m,λ(μj)m,σ(ζ(zy1)ξj)g(z)dzy2y1(ξ(y2,y1))μjJ(ξj)m,λ(μj)m,σ(ζ(ξ(y2,y1))ξj)[(zy1ξ(y2,y1))sg(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+m((y1+ξ(y2,y1)z)ξ(y2,y1))sg(y1)eθ1y1]dz=(ξ(y2,y1))μjJ(ξj)m,λ(μj)m,σ(ζ(ξ(y2,y1))ξj)ξ(y2,y1)s+1[g(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+mg(y1)eθ1y1],

    then

    (Œ(ξj,μj)mλ,σ,ζ;y+1g)(z)(Œ(ξj,μj)mλ,σ;(y1+ξ(y2,y1))1)(y1+ξ(y2,y1))ξ(y2,y1)s+1[g(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+mg(y1)eθ1y1]. (5.31)

    Summing (5.29) and (5.31), we obtain

    (Œ(ξj,δj)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z)+(Œ(ξj,μj)mλ,σ,ζ;y+1g)(z)ξ(y2,y1)s+1(g(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+mg(y1)eθ1y1)[(Œ(ξj,δj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(y1+ξ(y2,y1))]. (5.32)

    Take the product of Eq (5.21) with (zy1)τjJ(μj)m,λ(τj)m,σ(ζ(zy1)μj) and integrating with respect to t from y1 to y2, we have

    g(2y1+ξ(y2,y1)2)y2y1(zy1)τjJ(μj)m,λ(τj)m,σ(ζ(zy1)μj)dz(1+m)2sy2y1(zy1)τjJ(μj)m,λ(τj)m,σ(ζ(zy1)μj)g(z)eθzdz (5.33)

    using definition (13), we have

    g(2y1+ξ(y2,y1)2)(Œ(μj,τj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(y1)(1+m)2seθy1(Œ(μj,τj)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z). (5.34)

    Taking product (5.21) with (y1+ξ(y2,y1)z)δjJ(μj)m,λ(δj)m,σ(ζ(y1+ξ(y2,y1)z)μj) and integrating with respect to variable z from y1 to y2, we have

    g(2y1+ξ(y2,y1)2)(Œ(μj,δj)mλ,σ,ζ;y+11)(y1+ξ(y2,y1))(1+m)2seθ1(y1+ξ(y2,y1))(Œ(μj,τj)mλ,σ,ζ;y+1g)(y1+ξ(y2,y1)). (5.35)

    Summing up (5.34) and (5.35), we get

    2s1+mg(2y1+ξ(y2,y1)2)[eθy1(Œ(μj,τj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(y1)+(Œ(μj,δj)mλ,σ,ζ;y+11)(y1+ξ(y2,y1))](Œ(μj,τj)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z)+(Œ(μj,τj)mλ,σ,ζ;y+1g)(y1+ξ(y2,y1)). (5.36)

    Now, combining (5.32) and (5.36), we get inequality

    2s1+mg(2y1+ξ(y2,y1)2)[eθy1(Œ(μj,τj)mλ,σ,ζ;(y1+η(y2,y1))1)(y1)+(Œ(μj,δj)mλ,σ,ζ;y+11)(y1+ξ(y2,y1))](Œ(μj,τj)mλ,σ,ζ;(y1+ξ(y2,y1))g)(z)+(Œ(μj,τj)mλ,σ,ζ;y+1g)(y1+ξ(y2,y1))ξ(y2,y1)s+1(g(y1+ξ(y2,y1))eθ1(y1+ξ(y2,y1))+mg(y1)eθ1y1)×[(Œ(ξj,δj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(z)+(Œ(ξj,μj)mλ,σ,ζ;(y1+ξ(y2,y1))1)(y1+ξ(y2,y1))].

    Corollary 5.7. Setting ξ(y2,y1)=y2y1, then under the assumption of theorem (5.3), we have

    2s1+mg(y1+y22)[eθy1(Œ(μj,τj)mλ,σ,ζ;y21)(y1)+(Œ(μj,δj)mλ,σ,ζ;y+11)(y2)](Œ(μj,τj)mλ,σ,ζ;y2g)(z)+(Œ(μj,τj)mλ,σ,ζ;y+1g)(y2)(y2y1)s+1(g(y2y1)eθ1(y2y1)+mg(y1)eθ1y1)×[(Œ(ξj,δj)mλ,σ,ζ;y21)(z)+(Œ(ξj,μj)mλ,σ,ζ;y21)(y2)]. (5.37)

    In this section, we derive some Pólya-Szegö inequalities for four positive integrable functions having fractional operator Œ(ξj,δj)mλ,σ(z) in the form of theorems.

    Theorem 6.1. Let h and l are integrable functions on [y1,). Suppose that there exist integrable functions θ1,θ2,ψ1 and ψ2 on [y1,) such that:

    (R1) 0<θ1(b)h(b)θ2(b),0<ψ1(b)l(b)ψ2(b) (b[y1,z],z>y1).

    Then, for z>y1,y10, ξj,δj,λC,(j=1,,m),(λ)>0,(δj)>1,mj=1(ξ)j>max{0:(σ)1},σ>0 and (zb)Ω, then the following inequalities hold:

    Œ(ξj,δj)mλ,σ,ζ;y1+[(ψ1ψ2)h2](z)Œ(ξj,δj)mλ,σ,ζ;y+1[(θ1θ2)l2](z)[Œ(ξj,δj)mλ,σ,ζ;y+1[(θ1ψ1+θ2ψ2)hl](z)]214. (6.1)

    Proof. From (R1), for b[y1,z], z>y1, we have

    h(b)l(b)θ2(b)ψ1(b), (6.2)

    the inequality write as

    (θ2(b)ψ1(b)h(b)l(b))0. (6.3)

    Similarly, we get

    θ1(b)ψ2(b)h(b)l(b), (6.4)

    thus

    (h(b)l(bθ1(b)ψ2(b))0. (6.5)

    Multiplying Eq (6.3) and Eq (6.5), it follows

    (θ2(b)ψ1(b)h(b)l(b))(h(b)l(b)θ1(b)ψ2(b))0, (6.6)

    i.e.

    (θ2(b)ψ1(b)+θ1(b)ψ2(b))h(b)l(b)h2(b)l2(b)+θ1(b)θ2(b)ψ1(b)ψ2(b). (6.7)

    The last inequality can be written as

    (θ1(b)ψ1(b)+θ2(b)ψ2(b))h(b)l(b)ψ1(b)ψ2(b)h2(b)+θ1(b)θ2(b)l2(b). (6.8)

    Consequently, multiply both sides of (6.8) by (y1b)δjJ(ξj)m,λ(δj)m,σ(ζ(y1b)ξj), (zb)Ω and integrating with respect to b from y1 to z, we get

    Œ(ξj,δj)mλ,σ,ζ;y1+[(θ1ψ1+θ2ψ2)hl](z)Œ(ξj,δj)mλ,σ,ζ;y1+[ψ1ψ2h2](z)+Œ(ξj,δj)mλ,σ,ζ;y1+[θ1θ2l2](z). (6.9)

    Besides, by AM-GM (arithmetic mean- geometric mean) inequality, i.e., a1+b12a1b1 a1,b1+, we get

    Œ(ξj,δj)mλ,σ,ζ;y1+[(θ1ψ1+θ2ψ2)hl](x)2Œ(ξj,δj)mλ,σ,ζ;y1+[ψ1ψ2h2](z)+Œ(ξj,δj)mλ,σ,ζ;y1+[θ1θ2l2](z), (6.10)

    and it follows straightforward the statement of Eq (6.1).

    Corollary 6.1.. Let h and l be two integrable functions on [0,) and satisfying the inequality

    (R2) 0<sh(b)S,0<kl(b)K(b[y1,τ],z>y1). (6.11)

    For z>y1,y10, ξj,δj,λC,(j=1,,m),(λ)>0,(δj)>1,mj=1(ξ)j>max{0:(σ)1},σ>0 and (zb)Ω, then the following inequalities hold:

    Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[l2](z)(Œ(ξj,δj)mλ,σ,ζ;y1+[hl](z))214(SKsk+skSK)2. (6.12)

    Theorem 6.2. Let h and l are positive integrable functions on [y1,). Suppose that there exist integrable functions θ1,θ2,ψ1 and ψ2 on [y1,) satisfying (R1) on [y1,). Then, for z>y1,y10, ξj,δj,λC,(j=1,,m),(λ)>0,(δj)>1,mj=1(ξ)j>max{0:(σ)1},σ>0 and (zb),(τz)Ω, then the following inequalities hold:

    Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)Œ(ξj,δj)mλ,σ,ζ;y2[ψ1ψ2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[θ1θ2](z)Œ(ξj,δj)mλ,σ,ζ;y2[l2](z)[Œ(ξj,δj)mλ,σ,ζ;y1+[θ1h](z)Œ(ξj,δj)mλ,σ,ζ;y2[ψ1h](z)+Œ(ξj,δj)mλ,σ,ζ;y1+[θ2h](z)Œ(ξj,δj)mλ,σ,ζ;y2[ψ2l](z)]214. (6.13)

    Proof. By condition (R1), it is clear that

    (θ2(b)ψ1(α)h(b)l(α))0, (6.14)

    and

    (h(b)l(α)θ1(b)ψ2(α))0, (6.15)

    these inequalities implies that

    (θ1(b)ψ2(α)+θ2(b)ψ1(α))h(b)l(α)h2(b)l2(α)+θ1(b)θ2(b)ψ1(α)ψ2(α). (6.16)

    The Eq (6.16), multiply by ψ1(α)ψ2(α)l2(α) of both sides, we have

    θ1(b)h(b)ψ1(α)l(α)+θ2(b)h(b)ψ2(α)l(α)ψ1(α)ψ2(α)h2(b)+θ1(b)θ2(b)l2(α). (6.17)

    Hence, the Eq (6.17) multiply both sides by

    (zb)δjJ(ξj)m,λ(δj)m,σ(ζ(zb)ξj),(αz)δjJ(ξj)m,λ(δj)m,σ(ζ(αz)ξj). (6.18)

    And integrating double with respect to b and α from y1 to z and z to y2 respectively, we have

    Œ(ξj,δj)mλ,σ,ζ;y1+[θ1h](z)Œ(ξj,δj)mλ,σ,ζ;y2[ψ1l](z)+Œ(ξj,δj)mλ,σ,ζ;y1+[θ2h](z)Œ(ξj,δj)mλ,σ,ζ;y2[ψ2l](z)Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)Œ(ξj,δj)mλ,σ,ζ;y2[ψ1ψ2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[θ1θ2](z)Œ(ξj,δj)mλ,σ,ζ;y2[l2](z). (6.19)

    At last, we come to Eq (6.13) by using the arithmetic and geometric mean inequality to the upper inequality.

    Theorem 6.3. Let h and l are integrable functions on [y1,). Suppose that there exist integrable functions θ1,θ2,ψ1 and ψ2 on [y1,) satisfying (R1) on [y1,). Then, for z>y1,y10, ξj,δj,λC,(j=1,,m),(λ)>0,(δj)>1,mj=1(ξ)j>max{0:(σ)1},σ>0 and (zb),(αz)Ω, then the following inequalities hold:

    Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)Œ(ξj,δj)mλ,σ,ζ;y2[l2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[(θ2hl)/ψ1](z)Œ(ξj,δj)mλ,σ,ζ;y2[(ψ2hl)/θ1]. (6.20)

    Proof. We have for any (zb),(αz)Ω, from Eq (6.2), thus

    zy1(zb)δjJ(ξj,δj)mλ,σ(ζ(zb)ξj)h2(b)dby1z(αz)ξjJ(ξj,δj)mλ,σ(ζ(αz)ξj)θ2(α)ψ1(α)h(α)l(α)dα,

    which implies

    Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[(θ2hl)/ψ1](z). (6.21)

    and analogously, by Eq (6.4), we get

    Œ(ξj,δj)mλ,σ,ζ;y2[l2](x)Œ(ξj,δj)mλ,σ,ζ;y2[(ψ2hl)/θ1](z), (6.22)

    hence, by multiplying Eq (6.21) and Eq (6.22), follow Eq (6.20).

    Corollary 6.2. Let h and l be integrable functions on [y1,) satisfying (R2). Then, for z>y1,y10, ξj,δj,λC,(j=1,,m),(λ)>0,(δj)>1,mj=1(ξ)j>max{0:(σ)1},σ>0 and (zb),(αz)Ω, we obtain

    Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)Œ(ξj,δj)mλ,σ,ζ;y2[l2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[hl](z)Œ(ξj,δj)mλ,σ,ζ;y2[hl](z)SKsk. (6.23)

    In this section, Chebyshev type integral inequalities established involving the fractional operator Œ(ξj,δj)mλ,σ(z) and using the Pólya-Szegö fractional integral inequalities of theorem (6.1) in the form of theorem, and then discuss its corollary.

    Theorem 7.1. Let h and l be integrable functions on [y1,), and suppose that there exist integrable functions θ1,θ2,ψ1 and ψ2 on [y1,) satisfying (R1). Then, for z>y1,y10, ξj,δj,λC,(j=1,,m),(λ)>0,(δj)>1,mj=1(ξ)j>max{0:(σ)1},σ>0 and (zb)(αz)Ω the following inequality hold:

    |Œ(ξj,δj)mλ,σ,ζ;y1+[hl](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+Œ(νj,μj)mλ,σ,ζ;y2[hl](z)Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)Œ(ξj,δj)mλ,σ,ζ;y1+[h](z)Œ(νj,μj)mλ,σ,ζ;y2[l](z)Œ(ξj,δj)mλ,σ,ζ;y1+[l](z)Œ(νj,μj)mλ,σ,ζ;y2[h](z)|2[Gy1,y2(h,θ1,θ2)Gy1,y2(l,ψ1,ψ2)]12. (7.1)

    where

    Gy1,y2(b,y,x)(z)=18[Œ(ξj,δj)mλ,σ,ζ;y1+[(y+x)b](z)]2Œ(ξj,δj)mλ,σ,ζ;y1+[yx](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+18[Œ(νj,μj)mλ,σ,ζ;y2[(y+x)b](z)]2Œ(μj,νj)mλ,σ,ζ;y2[yx](z)Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)Œ(ξj,δj)mλ,σ,ζ;y1+[b](z)Œ(νj,μj)mλ,σ,ζ;y2[b](z).

    Proof. For (b,α)(y1,z) (z>y1), we defined A(b,α)=(h(b)h(α))(l(b)l(α)) which is the same

    A(b,α)=h(b)l(b)+h(α)l(α)h(b)l(α)h(α)l(b). (7.2)

    Further, the Eq (7.2), multiply both sides by

    (zb)ξjJ(ξj,δj)mλ,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj), (7.3)

    and integrating double with respect to b and α from y1 to z and z to y2 respectively, we get

    zy1y2z(zb)ξjJ(ξj,δj)mλ,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)A(b,α)dbdα=zy1(zb)ξjJ(ξj,δj)mλ,σ(ζ(zb)δj)h(b)l(b)dby2z(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)dα+zy1(zb)ξjJ(ξj,δj)mλ,σ(ζ(zb)δj)dby2z(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)h(α)l(α)dαy1z(zb)ξjJ(ξj,δj)mλ,σ(ζ(zb)δj)h(b)dby2z(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)h(α)dαy1z(zb)ξjJ(ξj,δj)mλ,σ(ζ(zb)δj)l(b)dby2z(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)h(α)dα=Œ(ξj,δj)mλ,σ,ζ;y1+[hl](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)Œ(νj,μj)mλ,σ,ζ;y2[hl](z)Œ(ξj,δj)mλ,σ,ζ;y1+[h](z)Œ(νj,μj)mλ,σ,ζ;y2[l](z)Œ(ξj,δj)mλ,σ,ζ;y1+[l](z)Œ(νj,μj)mλ,σ,ζ;y2[h](z). (7.4)

    Now, applying Cauchy-Schwartz inequality for integrals, we get

    |zy1y2z(zb)ξjJ(ξj)m,λ(δj)m,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)A(b,α)dbdα|(zy1y2z(zb)ξjJ(ξj)m,λ(δj)m,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)α[h(b)]2dbdα+zy1y2z(zb)ξjJ(ξj)m,λ(δj)m,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)[h(α)]2dbdα2zy1y2z(zb)ξjJ(ξj)m,λ(δj)m,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)h(b)h(α)dbdα)1/2×(zy1y2z(zb)ξjJ(ξj)m,λ(δj)m,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)α[l(b)]2dbdα+zy1y2z(zb)ξjJ(ξj)m,λ(δj)m,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)[l(α)]2dbdα2zy1y2z(zb)ξjJ(ξj)m,λ(δj)m,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)l(b)l(α)dbdα)1/2, (7.5)

    it follow as

    |zy1y2z(zb)ξjJ(ξj)m,λ(δj)m,σ(ζ(zb)δj)(αz)νjJ(μj)m,λ(νj)m,σ(ζ(αz)μj)A(b,α)dbdα|2{1/2Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+1/2Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)Œ(νj,μj)mλ,σ,ζ;y2[h2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[h](z)Œ(νj,μj)mλ,σ,ζ;y2[h](z)}1/2×{1/2Œ(ξj,δj)mλ,σ,ζ;y1+[l2](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+1/2Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)Œ(νj,μj)mλ,σ,ζ;y2[l2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[l](z)Œ(νj,μj)mλ,σ,ζ;y2[l](z)}1/2. (7.6)

    By applying lemma (6.1) for ψ1(z)=ψ2(z)=l(z)=1, we get for any J(ξj,δj)mλ,σ(z)δjΩ

    Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)14[Œ(ξj,δj)mλ,σ,ζ;y+1[(θ1+θ2)h](z)]2Œ(ξj,δj)mλ,σ,ζ;y+1[(θ1θ2)](z), (7.7)

    this implies

    1/2Œ(ξj,δj)mλ,σ,ζ;y1+[h2](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+1/2Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)Œ(νj,μj)mλ,σ,ζ;y2[h2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[h](z)Œ(νj,μj)mλ,σ,ζ;y2[h](z)18[Œ(ξj,δj)mλ,σ,ζ;y+1[(θ1+θ2)h](z)]2Œ(ξj,δj)mλ,σ,ζ;y+1[(θ1θ2)](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+18Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)[Œ(νj,μj)mλ,σ,ζ;y+1[(θ1+θ2)h](z)]2Œ(νj,μj)mλ,σ,ζ;y+1[(θ1θ2)](z)Œ(ξj,δj)mλ,σ,ζ;y1+[h](z)Œ(νj,μj)mλ,σ,ζ;y2[h](z)=Gy1,y2(h,θ1,θ2). (7.8)

    Analogously, it is clear when θ1(z)=θ2(z)=h(z)=1, according to Lemma (6.1), we get

    1/2Œ(ξj,δj)mλ,σ,ζ;y1+[l2](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+1/2Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)Œ(νj,μj)mλ,σ,ζ;y2[l2](z)Œ(ξj,δj)mλ,σ,ζ;y1+[l](z)Œ(νj,μj)mλ,σ,ζ;y2[l](x)18[Œ(ξj,δj)mλ,σ,ζ;y+1[(ψ1+ψ2)l](z)]2Œ(ξj,δj)mλ,σ,ζ;y+1[(ψ1ψ2)](z)Œ(νj,μj)mλ,σ,ζ;y2[1](z)+18Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)[Œ(νj,μj)mλ,σ,ζ;y+1[(ψ1+ψ2)l](z)]2Œ(νj,μj)mλ,σ,ζ;y+1[(ψ1ψ2)](z)Œ(ξj,δj)mλ,σ,ζ;y1+[l](z)Œ(νj,μj)mλ,σ,ζ;y2[l](z)=Gy1,y2(l,ψ1,ψ2). (7.9)

    Thus, by resulting Eqs (7.4), (7.6), (7.8) and (7.9), we get the desired inequality (7.1).

    Corollary 7.1. Let h and l be integrable functions on [y1,), suppose that there exist integrable functions θ1,θ2,ψ1 and ψ2 on [y1,) satisfying (R1). Then, for z>y1,y10, ξj,δj,λC,(j=1,,m),(λ)>0,(δj)>1,mj=1(ξ)j>max{0:(σ)1},σ>0 and (zb),(αz)Ω the following inequalities hold:

    |Œ(ξj,δj)mλ,σ,ζ;y1+[hl](z)Œ(ξj,δj)mλ,σ,ζ;y1+[1](z)Œ(ξj,δj)mλ,σ,ζ;y1+[h](z)Œ(ξj,δj)mλ,σ,ζ;y1+[l](z)|[Gy1,y2(h,θ1,θ2)Gy1,y1(l,θ1,θ2)]12,

    where

    Gy1,y1(b,y,x)(z)=14[Œ(ξj,δj)mλ,σ,ζ;y1+[(y+x)b](z)]2Œ(ξj,δj)mλ,σ,ζ;y1+[yx](z)Œ(ξj,δj)mλ,σ,ζ;y1+[1](Œ(ξj,δj)mλ,σ,ζ;y1+[b](z))2.

    This article analyzed the generalized fractional integral operator having nonsingular function (generalized multi-index Bessel function) as kernel and developed a new version of inequalities. We estimate some inequalities (Hermite Hadamard type Mercer inequality, exponentially (sm) preinvex inequality, Pólya-Szegö type integral inequality and the Chebyshev type inequality) with the generalized fractional integral operator in which nonsingular function as the kernel. Introducing the new version of inequalities of newly constricted operators have strengthened the idea and results.

    The authors declare that they have no competing interest.


    Acknowledgments



    We thank: participants for sharing their stories and experiences to further research; RESOLVE staff for their support in making this work a reality; our Albertan community agency partners for assisting in recruitment, including FearIsNotLove and Wheatland Crisis Society; Olivia Giacobbo (OG) for her assistance in conducting interviews; Keira Griggs for her support in finalizing the manuscript; Drs. Scott Patten and Jennifer Woo for their medical expertise to help describe the hypervigilance and anxiety phenomenon of phantom stalking; and the Prairieaction Foundation for funding of this research.

    Ethics approval of research and informed consent



    Prior to participating in the interviews and studies, participants provided written informed consent via signatures. Ethical approval for both studies was obtained from the University of Calgary Conjoint Health Research Ethics Board (REB20-1461_REN3, REB20-1241_REN6).

    Conflict of interest



    The authors declare no conflict of interest.

    [1] Krug EG, Mercy JA, Dahlberg LL, et al. (2002) The world report on violence and health. Lancet 360: 1083-1088. https://doi.org/10.1016/S0140-6736(02)11133-0
    [2] Cotter A (2021) Intimate partner violence in Canada, 2018: an overview. Juristat: Canadian Centre for Justice Statistics : 1-23. Available from: https://www150.statcan.gc.ca/n1/pub/85-002-x/2021001/article/00003-eng.htm.
    [3] Coleman FL (1997) Stalking behavior and the cycle of domestic violence. J Interpers Violence 12: 420-432. https://doi.org/10.1177/088626097012003007
    [4] Mechanic MB, Weaver TL, Resick PA (2000) Intimate partner violence and stalking behavior: exploration of patterns and correlates in a sample of acutely battered women. Violence Vict 15: 55-72.
    [5] Government of Canada, Criminal Code (R.S.C., 1985, c. C-46). Ontario Government of Canada, 2023. Available from: https://laws-lois.justice.gc.ca/eng/acts/c-46/section-264.html#:~:text=264%20(1)%20No%20person%20shall,or%20the%20safety%20of%20anyone
    [6] Braveman P, Gottlieb L (2014) The social determinants of health: it's time to consider the causes of the causes. Public Health Rep 129: 19-31. https://doi.org/10.1177/00333549141291s206
    [7] Bauer GR, Churchill SM, Mahendran M, et al. (2021) Intersectionality in quantitative research: a systematic review of its emergence and applications of theory and methods. SSM Popul Health 14: 100798. https://doi.org/10.1016/j.ssmph.2021.100798
    [8] Kattari SK, Walls NE, Speer SR (2017) Differences in experiences of discrimination in accessing social services among transgender/gender nonconforming individuals by (dis)ability. J Soc Work Disabil Rehabil 16: 116-140. https://doi.org/10.1080/1536710x.2017.1299661
    [9] Kattari SK, Walls NE, Whitfield DL, et al. (2017) Racial and ethnic differences in experiences of discrimination in accessing social services among transgender/gender-nonconforming people. J Ethn Cult Diversit 26: 217-235. https://doi.org/10.1080/15313204.2016.1242102
    [10] Statistics Canada, Section 3: Police-reported intimate partner violence in Canada, 2019. Ontario Statistics Canada, 2021. Available from: https://www150.statcan.gc.ca/n1/pub/85-002-x/2021001/article/00001/03-eng.htm
    [11] Faller YN, Wuerch MA, Hampton MR, et al. (2021) A web of disheartenment with hope on the horizon: intimate partner violence in rural and northern communities. J Interpers Violence 36: 4058-4083. https://doi.org/10.1177/0886260518789141
    [12] Peek-Asa C, Wallis A, Harland K, et al. (2011) Rural disparity in domestic violence prevalence and access to resources. J Womens Health 20: 1743-1749. https://doi.org/10.1089/jwh.2011.2891
    [13] Cao L, Wang SYK (2020) Correlates of stalking victimization in Canada: a model of social support and comorbidity. Int J Law Crime Justice 63: 100437. https://doi.org/10.1016/j.ijlcj.2020.100437
    [14] Brownridge DA (2008) Understanding the elevated risk of partner violence against aboriginal women: a comparison of two nationally representative surveys of Canada. J Fam Viol 23: 353-367. https://doi.org/10.1007/s10896-008-9160-0
    [15] Panchuk K, Hart C, Lewchuk DR (2022) Women+ and intimate partner violence in rural, remote and northern communities. Rural and Northern Social Work Practice: Canadian Perspectives .
    [16] Statistics Canada, Population growth in Canada's rural areas, 2016 to 2021. Ontario Statistics Canada, 2022. Available from: https://www12.statcan.gc.ca/census-recensement/2021/as-sa/98-200-x/2021002/98-200-x2021002-eng.cfm
    [17] Jaffray B (2021) Intimate partner violence: experiences of sexual minority women in Canada, 2018. Available from: https://www150.statcan.gc.ca/n1/pub/85-002-x/2021001/article/00005-eng.htm
    [18] Jaffray B (2021) Intimate partner violence: experiences of sexual minority men in Canada, 2018. Juristat . Available from: https://www150.statcan.gc.ca/n1/pub/85-002-x/2021001/article/00004-eng.htm.
    [19] Langenderfer-Magruder L, Walls NE, Whitfield DL, et al. (2020) Stalking victimization in LGBTQ adults: a brief report. J Interpers Violence 35: 1442-1453. https://doi.org/10.1177/0886260517696871
    [20] McCart MR, Smith DW, Sawyer GK (2010) Help seeking among victims of crime: a review of the empirical literature. J Trauma Stress 23: 198-206. https://doi.org/10.1002/jts.20509
    [21] Rollè L, Giardina G, Caldarera AM, et al. (2018) When intimate partner violence meets same sex couples: a review of same sex intimate partner violence. Front Psychol 9: 1506. https://doi.org/10.3389/fpsyg.2018.01506
    [22] Daley A, Brotman S, MacDonnell JA, et al. (2020) A framework for enhancing access to equitable home care for 2SLGBTQ+ communities. Int J Environ Res Public Health 17: 7533. https://doi.org/10.3390/ijerph17207533
    [23] Henriquez NR, Ahmad N (2021) “The message is you don't exist”: exploring lived experiences of rural lesbian, gay, bisexual, transgender, queer/questioning (LGBTQ) people utilizing health care services. SAGE Open Nurs 7. https://doi.org/10.1177/23779608211051174
    [24] Abboud S, Veldhuis C, Ballout S, et al. (2022) Sexual and gender minority health in the Middle East and North Africa region: a scoping review. Int J Nurs Stud Adv 4: 100085. https://doi.org/10.1016/j.ijnsa.2022.100085
    [25] Javed S, Chattu VK (2021) Patriarchy at the helm of gender-based violence during COVID-19. AIMS Public Health 8: 32-35. https://doi.org/10.3934/publichealth.2021003
    [26] Kardashevskaya M, Arisman K, Novick J, et al. (2022) Responding to women who experience intimate partner violence in rural municipalities across the prairies: final report. Available from: https://www.umanitoba.ca/sites/resolve/files/2022-09/Rural%20IPV%20Final%20Report.pdf
    [27] Haller A, White S, Bresch L, et al. (2022) Examining the nature & context of intimate partner violence in 2SLGBTQ+ communities: final report. Available from: https://www.umanitoba.ca/sites/resolve/files/2022-09/2SLGBTQ%20Final%20Report.pdf
    [28] Statistics Canada, Illustrated Glossary: Rural area (RA). Ontario Statistics Canada, 2022. Available from: https://www150.statcan.gc.ca/n1/pub/92-195-x/2021001/geo/ra-rr/ra-rr-eng.htm
    [29] Woulfe JM, Goodman LA (2021) Identity abuse as a tactic of violence in LGBTQ communities: Initial validation of the identity abuse measure. J Interpers Violence 36: 2656-2676. https://doi.org/10.1177/0886260518760018
    [30] Chowdhury R (2023) The role of religion in domestic violence and abuse in UK muslim communities. Oxf J Law Relig 12: 178-198. https://doi.org/10.1093/ojlr/rwad008
    [31] McLemore A (2021) Stalking by way of the courts: Tennessee's abusive civil action law and why all states should adopt a similar approach to abusive Litigation in the family law context. UCLA Womens Law J 28: 333. https://doi.org/10.5070/L328155792
    [32] Statistics Canada, Census profile, 2021 census of population: profiles of a community or region: 98-316-X2021001. Ontario Statistics Canada, 2023. Available from: https://www150.statcan.gc.ca/n1/en/catalogue/98-316-X2021001
    [33] Sutton D (2023) Gender-related homicide of women and girls in Canada. Available from: https://www150.statcan.gc.ca/n1/pub/85-002-x/2023001/article/00003-eng.htm
    [34] Otter.ai (2019, August). Available from: https://otter.ai./
    [35] Braun V, Clarke V (2006) Using thematic analysis in psychology. Qual Res Psychol 3: 77-101. https://doi.org/10.1191/1478088706qp063oa
    [36] Kurbatfinski S, Whitehead J, Hodge L, et al. (2023) 2SLGBTQQIA+ experiences of intimate partner abuse and help-seeking: an intersectional scoping review.
    [37] Capaldi DM, Knoble NB, Shortt JW, et al. (2012) A systematic review of risk factors for intimate partner violence. Partner Abuse 3: 231-280. https://doi.org/10.1891/1946-6560.3.2.231
    [38] Gilchrist G, Potts LC, Connolly DJ, et al. (2023) Experience and perpetration of intimate partner violence and abuse by gender of respondent and their current partner before and during COVID-19 restrictions in 2020: a cross-sectional study in 13 countries. BMC Public Health 23: 316. https://doi.org/10.1186/s12889-022-14635-2
    [39] Whitton SW, Lawlace M, Dyar C, et al. (2021) Exploring mechanisms of racial disparities in intimate partner violence among sexual and gender minorities assigned female at birth. Cultur Divers Ethnic Minor Psychol 27: 602-612. https://doi.org/10.1037/cdp0000463
    [40] Reyns BW, Scherer H (2018) Stalking victimization among college students: the role of disability within a lifestyle-routine activity framework. Crime Delinquency 64: 650-673. https://doi.org/10.1177/0011128717714794
    [41] Breiding MJ, Smith SG, Basile KC, et al. (2014) Prevalence and characteristics of sexual violence, stalking, and intimate partner violence victimization—national intimate partner and sexual violence survey, United States, 2011. MMWR Surveill Summ 63: 1-18.
    [42] Reyns BW, Scherer H (2019) Disability type and risk of sexual and stalking victimization in a national sample: a lifestyle–routine activity approach. Crim Justice Behav 46: 628-647. https://doi.org/10.1177/0093854818809148
    [43] Statistics Canada, Family matters: to have kids or not to have kids: that is the question! Ontario Statistics Canada, 2023. Available from: https://www150.statcan.gc.ca/n1/pub/11-627-m/11-627-m2023006-eng.htm
    [44] Nikupeteri A, Katz E, Laitinen M (2021) Coercive control and technology-facilitated parental stalking in children's and young people's lives. J Gen Based Violence 5: 395-412. https://doi.org/10.1332/239868021X16285243258834
    [45] Humphreys C, Diemer K, Bornemisza A, et al. (2019) More present than absent: men who use domestic violence and their fathering. Child Fam Soc Work 24: 321-329. https://doi.org/10.1111/cfs.12617
    [46] Bristow SM, Jackson D, Power T, et al. (2022) Rural mothers' feelings of isolation when caring for a child chronic health condition: a phenomenological study. J Child Health Care 26: 185-198. https://doi.org/10.1177/13674935211007324
    [47] Logan TK, Stevenson E, Evans L, et al. (2004) Rural and urban women's perceptions of barriers to health, mental health, and criminal justice services: implications for victim services. Violence Vict 19: 37-62. https://doi.org/10.1891/vivi.19.1.37.33234
    [48] Anderson KM, Renner LM, Bloom TS (2014) Rural women's strategic responses to intimate partner violence. Health Care Women Int 35: 423-441. https://doi.org/10.1080/07399332.2013.815757
    [49] Riddell T, Ford-Gilboe M, Leipert B (2009) Strategies used by rural women to stop, avoid, or escape from intimate partner violence. Health Care Women Int 30: 134-159. https://doi.org/10.1080/07399330802523774
    [50] Bocij P, McFarlane L (2003) Cyberstalking: the technology of hate. Police J 76: 204-221. https://doi.org/10.1350/pojo.76.3.204.19442
    [51] Fraser C, Olsen E, Lee K, et al. (2010) The new age of stalking: technological implications for stalking. Juv Fam Court J 61: 39-55. https://doi.org/10.1111/j.1755-6988.2010.01051.x
    [52] Castro Á, Barrada JR (2020) Dating apps and their sociodemographic and psychosocial correlates: a systematic review. Int J Environ Res Public Health 17: 6500. https://doi.org/10.3390/ijerph17186500
    [53] Blackwell C, Birnholtz J, Abbott C (2015) Seeing and being seen: co-situation and impression formation using Grindr, a location-aware gay dating app. New Media Soc 17: 1117-1136. https://doi.org/10.1177/1461444814521595
    [54] Kaur P, Dhir A, Tandon A, et al. (2021) A systematic literature review on cyberstalking. An analysis of past achievements and future promises. Technol Forecast Soc 163: 120426. https://doi.org/10.1016/j.techfore.2020.120426
    [55] Ogbe E, Harmon S, Van den Bergh R, et al. (2020) A systematic review of intimate partner violence interventions focused on improving social support and/ mental health outcomes of survivors. PLoS One 15: e0235177. https://doi.org/10.1371/journal.pone.0235177
    [56] Benitez CT, McNiel DE, Binder RL (2010) Do protection orders protect?. J Am Acad Psychiatry Law 38: 376-385.
    [57] Holt VL, Kernic MA, Wolf ME, et al. (2003) Do protection orders affect the likelihood of future partner violence and injury?. Am J Prev Med 24: 16-21. https://doi.org/10.1016/s0749-3797(02)00576-7
    [58] Holt VL, Kernic MA, Lumley T, et al. (2002) Civil protection orders and risk of subsequent police-reported violence. JAMA 288: 589-594. https://doi.org/10.1001/jama.288.5.589
    [59] McFarlane J, Malecha A, Gist J, et al. (2004) Protection orders and intimate partner violence: an 18-month study of 150 black, Hispanic, and white women. Am J Public Health 94: 613-618. https://doi.org/10.2105/ajph.94.4.613
    [60] Logan TK, Walker R (2010) Civil protective order effectiveness: justice or just a piece of paper?. Violence Vict 25: 332-348. https://doi.org/10.1891/0886-6708.25.3.332
    [61] Carlson MJ, Harris SD, Holden GW (1999) Protective orders and domestic violence: risk factors for re-abuse. J Fam Violence 14: 205-226. https://doi.org/10.1023/A:1022032904116
    [62] Koshan J (2023) Preventive justice? Domestic violence protection orders and their intersections with family and other laws and legal systems. Can J Fam L 35: 241. https://doi.org/10.2139/ssrn.4372318
    [63] American Psychiatric Association (2013) Diagnostic and Statistical Manual of Mental Disorders (DSM-5). 5 Eds., Washington: American Psychiatric Association. https://doi.org/10.1176/appi.books.9780890425596
    [64] Zimmermann K, Carnahan LR, Paulsey E, et al. (2016) Health care eligibility and availability and health care reform: Are we addressing rural women's barriers to accessing care?. J Health Care Poor Underserved 27: 204-219. https://doi.org/10.1353/hpu.2016.0177
    [65] Sankar P, Mora S, Merz JF, et al. (2003) Patient perspectives of medical confidentiality: a review of the literature. J Gen Intern Med 18: 659-669. https://doi.org/10.1046/j.1525-1497.2003.20823.x
    [66] Sabri B, Tharmarajah S, Njie-Carr VPS, et al. (2022) Safety planning with marginalized survivors of intimate partner violence: challenges of conducting safety planning intervention research with Marginalized women. Trauma Violence Abuse 23: 1728-1751. https://doi.org/10.1177/15248380211013136
    [67] Natarajan M (2016) Police response to domestic violence: a case study of TecSOS mobile phone use in the London metropolitan police service. Polic J Policy Pract 10: 378-390. https://doi.org/10.1093/police/paw022
    [68] Hearn J, Hall M, Lewis R, et al. (2023) The spread of digital intimate partner violence: ethical challenges for business, workplaces, employers and management. J Bus Ethics 187: 695-711. https://doi.org/10.1007/s10551-023-05463-4
    [69] Waite S, Pajovic V, Denier N (2010) Lesbian, gay and bisexual earnings in the Canadian labor market: new evidence from the Canadian community health survey. Res Soc Strat Mobil 67: 100484. https://doi.org/10.1016/j.rssm.2020.100484
    [70] Coy M, Scott E, Tweedale R, et al. (2015) “It's like going through the abuse again”: domestic violence and women and children's (un) safety in private law contact proceedings. J Soc Welf Fam Law 37: 53-69. https://doi.org/10.1080/09649069.2015.1004863
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