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Probing the evolutionary history of epigenetic mechanisms: what can we learn from marine diatoms

  • Received: 19 May 2015 Accepted: 13 July 2015 Published: 25 January 2015
  • Recent progress made on epigenetic studies revealed the conservation of epigenetic features in deep diverse branching species including Stramenopiles, plants and animals. This suggests their fundamental role in shaping species genomes across different evolutionary time scales. Diatoms are a highly successful and diverse group of phytoplankton with a fossil record of about 190 million years ago. They are distantly related from other super-groups of Eukaryotes and have retained some of the epigenetic features found in mammals and plants suggesting their ancient origin. Phaeodactylum tricornutum and Thalassiosira pseudonana, pennate and centric diatoms, respectively, emerged as model species to address questions on the evolution of epigenetic phenomena such as what has been lost, retained or has evolved in contemporary species. In the present work, we will discuss how the study of non-model or emerging model organisms, such as diatoms, helps understand the evolutionary history of epigenetic mechanisms with a particular focus on DNA methylation and histone modifications.

    Citation: Achal Rastogi, Xin Lin, Bérangère Lombard, Damarys Loew, Leïla Tirichine. Probing the evolutionary history of epigenetic mechanisms: what can we learn from marine diatoms[J]. AIMS Genetics, 2015, 2(3): 173-191. doi: 10.3934/genet.2015.3.173

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  • Recent progress made on epigenetic studies revealed the conservation of epigenetic features in deep diverse branching species including Stramenopiles, plants and animals. This suggests their fundamental role in shaping species genomes across different evolutionary time scales. Diatoms are a highly successful and diverse group of phytoplankton with a fossil record of about 190 million years ago. They are distantly related from other super-groups of Eukaryotes and have retained some of the epigenetic features found in mammals and plants suggesting their ancient origin. Phaeodactylum tricornutum and Thalassiosira pseudonana, pennate and centric diatoms, respectively, emerged as model species to address questions on the evolution of epigenetic phenomena such as what has been lost, retained or has evolved in contemporary species. In the present work, we will discuss how the study of non-model or emerging model organisms, such as diatoms, helps understand the evolutionary history of epigenetic mechanisms with a particular focus on DNA methylation and histone modifications.


    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\ \ (n\in\mathbb{N}) $ 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, K \in \mathbb{R} $.

    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, K \in \mathbb{R} $.

    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 $ (s-m) $-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: K \rightarrow \mathbb{R} $ exist as

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

    where $ \forall y_{1}, y_{2}\in K $ and $ \delta\in[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 $ y_{1}, y_{2} \in I $, with $ y_{2}\neq y_{1} $, if $ g:I \subseteq \mathbb{R} \rightarrow \mathbb{R} $ is a convex function.

    Definition 2.3. Let $ y_{j}\in K $ for all $ j \in I_{n} $, $ \omega_{j} > 0 $ such that $ \sum_{j \in I_{n}} \omega_{j} = 1 $. Then the Jensen inequality holds

    $ g(jInωjyj)jInωjg(yj), $ (2.3)

    exist if $ g:k \rightarrow \mathbb{R} $ is convex function.

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

    Definition 2.4. Let $ y_{j}\in K $ for all $ j \in I_{n} $, $ \omega_{j} > 0 $ such that $ \sum_{j \in I_{n}} \omega_{j} = 1 $, $ m = \min_{j \in I_{n}}\{y_{j}\} $ and $ n = \max_{j \in I_{n}}\{y_{j}\} $. Then the inequality holds for convex function as

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

    if $ g:k \rightarrow \mathbb{R} $ is convex function.

    Definition 2.5. [55] The inequality holds for exponentially convex function, if a real valued mapping $ g: K \rightarrow \mathbb{R} $ exist as

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

    where $ \forall y_{1}, y_{2}\in K $ and $ \delta\in[0, 1] $ and $ \theta\in \mathbb{R} $.

    Suppose that $ \Omega\subseteq \mathbb{R}^{n} $ is a set. Let $ g:\Omega\rightarrow \mathbb{R} $ continuous function and let $ \xi:\Omega\times\Omega\rightarrow \mathbb{R}^{n} $ be continuous function:

    Definition 2.6. [56] With respect to bifunction $ \xi(., .) $ a set $ \Omega $ is called a invex set, if

    $ y1+δξ(y2,y1), $ (2.6)

    where $ \forall y_{1}, y_{2} \in \Omega, \delta\in[0, 1] $.

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

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

    where $ \forall $ $ y_{1}, y_{2}+\xi(y_{2}, y_{1})\in \Omega, \delta \in [0, 1] $.

    Definition 2.8. A invex set $ \Omega $ with real valued mapping $ g $ and respect to $ \xi(., .) $ 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 $ y_{1}, y_{2}+\xi(y_{2}, y_{1})\in \Omega, \delta \in [0, 1] $ and $ \theta \in \mathbb{R} $.

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

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

    where for all $ y_{1}, y_{2}+\xi(y_{2}, y_{1})\in \Omega, \delta \in [0, 1] $, $ s\in(0, 1] $ and $ \theta \in \mathbb{R} $.

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

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

    where for all $ y_{1}, y_{2}+\xi(y_{2}, y_{1})\in \Omega $, $ \delta, m \in [0, 1] $ and $ \theta \in \mathbb{R} $.

    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 $ \xi_j, \delta_j, \lambda \in\mathbb{C} $, $ (j = 1, \cdots, m) $, $ \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0 $.

    Definition 2.12. [58] Pohhammer symbol is defined for $ \lambda\in \mathbb{C} $ as follows

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

    where $ \Gamma $ 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 $ \xi_j, \delta_j, \lambda, \zeta \in \mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi_{j}) > max\{0: \Re(\sigma)-1\}, \sigma > 0 $. Let $ g \in L \ \ [y_{1}, y_{2}] $ and $ t\in[y_{1}, y_{2}] $. 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 $, $ \sigma = 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 $, $ \delta_{1} = \delta_{1}-1 $ 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 $, $ \delta_{1} = \delta_{1}-1 $, $ \zeta = 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 $, $ \sigma = 1 $, $ \delta_{1} = \delta_{1}-1 $, 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] \rightarrow (0, \infty) $ is convex function such that $ g\in\chi_{c}(m, n) $, $ \forall x, y \in [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 \in [m, n] $, $ t \in [z-1, z] $, $ y_{1} = (z-t)x+(1-z+t)y $ and $ y_{2} = (1-z+t)x+(z-t)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 $ (z-t)^{\delta_j} {\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(z-t)^{\xi_{j}})} $ and integrating with respect to $ t $ from $ [z-1, 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 \in[z-1, 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 $ (z-t)^{\delta_j} {\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(z-t)^{\xi_{j}})} $ and integrating with respect to $ t $ from $ [z-1, 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] \rightarrow (0, \infty) $ is convex function such that $ g\in\chi_{c}(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 $ \forall x, y \in [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 \in[m, n] $, $ t \in [z-1, z] $, $ m+n-y_{1} = (z-t)(m+n-x)+(1-z+t)(m+n-y) $, $ m+n-y_{2} = (1-z+t)(m+n-x)+(z-t)(m+n-y) $, 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 $ (z-t)^{\delta_j} {\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(z-t)^{\xi_{j}})} $ then integrate with respect to $ t $ from $ [z-1, 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 $ (z-t)^{\delta_j} {\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(z-t)^{\xi_{j}})} $ and then integrating with respect to $ t $ from $ [z-1, z] $ we obtain the two inequalities (4.9).

    In this section, we derive some inequalities of $ (s-m) $ preinvex function involving new designed fractional integral operator $ Œ^{(\xi_j, \delta_j)_m}_{\lambda, \sigma, \zeta} g)(z) $ having generalized multi-index Bessel function as its kernel in the form of theorems.

    Theorem 5.1. Suppose a real valued function $ g:[y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow 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]. $

    $ \forall $ $ z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, $ \theta_{1}, \theta_{2} \in R $.

    Proof. Let $ z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, and then for $ t\in [y_{1}, z) $ and $ \delta_{j} > -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 $ y_{1} $ 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\in(z, y_{1}+\xi(y_{2}, y_{1})] $ and $ \mu_{j} > -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 $ y_{1}+\xi(y_{2}, y_{1}) $, 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 $ g\in L_{\infty}[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, 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 $ g\in L_{\infty}[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, 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 $ g\in L_{\infty}[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, 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 $ \xi(y_{2}, y_{1}) = y_{2}-y_{1} $ and $ g\in L_{\infty}[y_{1}, y_{2}] $, 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:[y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow R $ is differentiable and $ |g|^{\prime} $ 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]. $

    $ \forall $ $ z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, $ \theta_{1}, \theta_{2} \in \mathbb{R} $.

    Proof. Let $ z\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, $ t\in[y_{1}, z) $, and applying exponentially (s-m) preinvex of $ |g|^{\prime} $, 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 $ y_{1} $ 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 $ z-t = \alpha $, 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 $ z-\alpha = 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\in[y_{1}, y_{1}+\eta(y_{2}, y_{1})] $ and $ t\in(z, y_{1}+\xi(y_{2}, y_{1})] $, and by exponentially (s-m) preinvex of $ |g^{\prime}| $, 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 $ \xi(y_{2}, y_{1}) = y_{2}-y_{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)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]. $

    $ \forall $ $ t\in[y_{1}, y_{2}] $, $ \theta_{1}, \theta_{2} \in \mathbb{R} $.

    Corollary 5.6. Setting $ \xi(y_{2}, y_{1}) = y_{2}-y_{1} $, 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]. $

    $ \forall $ $ t\in[y_{1}, y_{2}] $, $ \theta_{1}, \theta_{2} \in \mathbb{R} $.

    Definition 5.1. Let $ g: [y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow R $ is a function, and $ g $ is exponentially symmetric about $ \frac{2y_{1}+\xi(y_{2}, y_{1})}{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:[y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow 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 = y_{1}+\delta\xi(y_{2}, y_{1}) $, where $ t\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, and then $ 2y_{1}+\xi(y_{2}, y_{1}) = y_{1}+(1-\delta)\xi(y_{2}, y_{1}) $, 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:[y_{1}, y_{1}+\xi(y_{2}, y_{1})]\rightarrow R $ is exponentially (s-m) preinvex and symmetric about exponentially $ \frac{2y_{1}+\xi(y_{2}, y_{1})}{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\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, 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\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, 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 $ y_{1} $ to $ y_{2} $, 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\in[y_{1}, y_{1}+\xi(y_{2}, y_{1})] $, 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 $ y_{1} $ to $ y_{2} $, 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 $ (z-y_{1})^{\tau_{j}}{\mathrm{J}^{(\mu_j)_m, \lambda}_{(\tau_j)_m, \sigma}(\zeta(z-y_{1})^{\mu_{j}})} $ and integrating with respect to $ t $ from $ y_{1} $ to $ y_{2} $, 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 $ (y_{1}+\xi(y_{2}, y_{1})-z)^{\delta_{j}}{\mathrm{J}^{(\mu_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(y_{1}+\xi(y_{2}, y_{1})-z)^{\mu_{j}})} $ and integrating with respect to variable $ z $ from $ y_{1} $ to $ y_{2} $, 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 $ \xi(y_{2}, y_{1}) = y_{2}-y_{1} $, 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^{(\xi_j, \delta_j)_m}_{\lambda, \sigma}(z) $ in the form of theorems.

    Theorem 6.1. Let $ h $ and $ l $ are integrable functions on $ [y_{1}, \infty) $. Suppose that there exist integrable functions $ \theta_{1}, \theta_{2}, \psi_{1} $ and $ \psi_{2} $ on $ [y_{1}, \infty) $ 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 > y_{1}, y_{1}\geq0 $, $ \xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0 $ and $ (z-b)\in \Omega $, 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\in[y_{1}, z] $, $ z > y_{1} $, 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 $ (y_{1}-b)^{\delta_{j}}\mathrm{J}^{(\xi_j)_m, \lambda}_{(\delta_j)_m, \sigma}(\zeta(y_{1}-b)^{\xi_{j}}) $, $ (z-b) \in \Omega $ and integrating with respect to $ b $ from $ y_{1} $ 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., $ a_{1}+b_{1}\geq2\sqrt{a_{1}b_{1}} $ $ a_{1}, b_{1} \in \Re^{+} $, 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, \infty) $ and satisfying the inequality

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

    For $ z > y_{1}, y_{1}\geq0 $, $ \xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0 $ and $ (z-b)\in \Omega $, 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 $ [y_{1}, \infty) $. Suppose that there exist integrable functions $ \theta_{1}, \theta_{2}, \psi_{1} $ and $ \psi_{2} $ on $ [y_{1}, \infty) $ satisfying $ (R1) $ on $ [y_{1}, \infty) $. Then, for $ z > y_{1}, y_{1}\geq0 $, $ \xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0 $ and $ (z-b), (\tau-z)\in \Omega $, 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 $ \psi_{1}(\alpha)\psi_{2}(\alpha)l^{2}(\alpha) $ 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 $ \alpha $ from $ y_{1} $ to $ z $ and $ z $ to $ y_{2} $ 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 $ [y_{1}, \infty) $. Suppose that there exist integrable functions $ \theta_{1}, \theta_{2}, \psi_{1} $ and $ \psi_{2} $ on $ [y_{1}, \infty) $ satisfying $ (R1) $ on $ [y_{1}, \infty) $. Then, for $ z > y_{1}, y_{1}\geq0 $, $ \xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0 $ and $ (z-b), (\alpha-z)\in \Omega $, 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 $ (z-b), (\alpha-z)\in \Omega $, 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 $ [y_{1}, \infty) $ satisfying $ (R2) $. Then, for $ z > y_{1}, y_{1}\geq0 $, $ \xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0 $ and $ (z-b), (\alpha-z)\in \Omega $, 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^{(\xi_j, \delta_j)_m}_{\lambda, \sigma}(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 $ [y_{1}, \infty) $, and suppose that there exist integrable functions $ \theta_{1}, \theta_{2}, \psi_{1} $ and $ \psi_{2} $ on $ [y_{1}, \infty) $ satisfying $ (R1) $. Then, for $ z > y_{1}, y_{1}\geq0 $, $ \xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0 $ and $ (z-b)(\alpha-z)\in \Omega $ 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, \alpha)\in(y_{1}, z) $ $ (z > y_{1}) $, we defined $ A(b, \alpha) = (h(b)-h(\alpha))(l(b)-l(\alpha)) $ 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 $ \alpha $ from $ y_{1} $ to $ z $ and $ z $ to $ y_{2} $ 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 $ \psi_{1}(z) = \psi_{2}(z) = l(z) = 1 $, we get for any $ \mathrm{J}^{(\xi_j, \delta_j)_m}_{\lambda, \sigma}(z)^{\delta_{j}}\in \Omega $

    $ 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 $ \theta_{1}(z) = \theta_{2}(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 $ [y_{1}, \infty) $, suppose that there exist integrable functions $ \theta_{1}, \theta_{2}, \psi_{1} $ and $ \psi_{2} $ on $ [y_{1}, \infty) $ satisfying $ (R1) $. Then, for $ z > y_{1}, y_{1}\geq0 $, $ \xi_j, \delta_j, \lambda \in\mathbb{C}, (j = 1, \cdots, m), \Re(\lambda) > 0, \Re(\delta_{j}) > -1, \sum^{m}_{j = 1} \Re(\xi)_j > max\{0: \Re(\sigma)-1\}, \sigma > 0 $ and $ (z-b), (\alpha-z)\in \Omega $ 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 $ (s-m) $ 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.

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