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Research article Special Issues

Columnar structure of SV40 minichromosome

  • Received: 13 July 2015 Accepted: 20 July 2015 Published: 30 July 2015
  • Like the sequence of the strongest 601 clone nucleosome of Lowary and Widom, the SV40 genome sequence contains tracks of YR dinucleotides separated by small integers of the 10.4n base series (10, 11, 21 and 30 bases). The tracks, however, substantially exceed the nucleosome DNA size and, thus, correspond to more extended structure - columnar chromatin. The micrococcal nuclease digests of the SV40 chromatin do not show uniquely positioned individual nucleosomes. This confirms the columnar structure of the minichromosome, as well as earlier electron microscopy studies.

    Citation: Edward N Trifonov. Columnar structure of SV40 minichromosome[J]. AIMS Biophysics, 2015, 2(3): 274-283. doi: 10.3934/biophy.2015.3.274

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  • Like the sequence of the strongest 601 clone nucleosome of Lowary and Widom, the SV40 genome sequence contains tracks of YR dinucleotides separated by small integers of the 10.4n base series (10, 11, 21 and 30 bases). The tracks, however, substantially exceed the nucleosome DNA size and, thus, correspond to more extended structure - columnar chromatin. The micrococcal nuclease digests of the SV40 chromatin do not show uniquely positioned individual nucleosomes. This confirms the columnar structure of the minichromosome, as well as earlier electron microscopy studies.


    P.L Čebyšev in the year 1882 has proved the following interesting inequality:

    $ |1babaf(x)g(x)dx(1babaf(x)dx)(1babag(x)dx)|112(ba)2fg. $

    where $ f, g $ are absolutely continuous functions defined on $ [a, b] $ and $ f', g' \in L_{\infty}[a, b] $. The left hand side of the above equation is denoted by $ T(f, g) $ is called Cebysev Functional if the integral exists. The applications of above type of inequalities can be found in the field of coding theory, statistics and other branches of mathematics.

    In last few decades many researchers have obtained various extensions and generalizations of above inequalities using various techniques see [1,2]. Study of inequalities have attracted the attention of researchers from various fields due to its wide applications in various fields [3,4].

    During last few years the subject of Fractional Calculus has been developed rapidly due to the applications in various fields of science and engineering. Various new definitions of fractional derivatives and integrals have been obtained by various researchers depending on the applications such as Riemann liouville, Caputo, Saigo, Hilfer, Hadmard, Katugampola and others See [5,6,7,8]. Many results on study of mathematical inequalities using various new fractional definitions such as Conformable and generalized fractional integral were obtained in [9,10]. Recently in [11,12,13,14,15] the authors have obtained the results on Cebysev inequalities using various fractional integral and derivatives definitions.

    In [7] authors have given definations of fractional derivative and integrals of a functions with respect to another functions. Recently in [16,17] authors have studied the $ \psi $ Caputo and $ \psi $ Hilfer fractional derivative of a function with respect to another functions and its applications. The $ \psi $ fractional and integral definations are more generalized and it reduces to Riemann Liouville, Hadmard and Erdelyi-Kober fractional definitions for different values of $ \psi $.

    Motivated from the above mentioned literature the aim of this paper is to obtain $ \psi $ Caputo fractional Čebyšev inequalities involving functions of two and three variables.

    Now in this section we give some basic definitions and properties which are useful in our subsequent discussions. In [7,8] the authors have defined the fractional integrals and fractional derivative of a function with respect to another function as follows.

    Definition 2.1 [7,16]. Let $ I = [a, b] $ be an interval, $ \alpha > 0 $, $ f $ is an integrable function defined on $ I $ and $ \psi \in C^1(I) $ an increasing function such that $ \psi '\left(x \right) \ne 0 $ for all $ x \in I $ then fractional derivative and integral of $ f $ is given by

    $ I_{a + }^{\alpha , \psi } f(x) = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_a^x {\psi '\left( t \right)\left( {\psi \left( x \right) - \psi \left( t \right)} \right)^{\alpha - 1} f\left( t \right)dt} $

    and

    $ Dα,ψa+f(x)=(1ψ(x)ddx)nInα,ψa+f(x)=1Γ(nα)(1ψ(x)ddx)nxaψ(t)(ψ(x)ψ(t))nα1f(t)dt, $

    respectively. Similarly right fractional integral and right fractional derivative are given by

    $ I_{b - }^{\alpha , \psi } f(x) = \frac{1}{{\Gamma \left( \alpha \right)}}\int\limits_a^x {\psi '\left( t \right)\left( {\psi \left( t \right) - \psi \left( x \right)} \right)^{\alpha - 1} f\left( t \right)dt} $

    and

    $ Dα,ψbf(x)=(1ψ(x)ddx)nInα,ψbf(x)=1Γ(nα)(1ψ(x)ddx)nxaψ(t)(ψ(t)ψ(x))nα1f(t)dt. $

    In [16] Almedia has considered a Caputo type fractional derivative with respect to another function.

    Definition 2.2 [16] Let $ \alpha > 0 $, $ n \in \mathbb{N} $, $ I $ is the interval $ -\infty \le a < b \le \infty $, $ f, \psi \in C^n (I) $ two functions such that $ \psi $ is increasing and $ \psi '\left(x \right) \ne 0 $ for all $ x \in I $. The left $ \psi $-Caputo fractional derivative of $ f $ of order $ \alpha $ is given by

    $ {}^CD_{a + }^{\alpha , \psi } f\left( x \right) = I_{a + }^{n - \alpha , \psi } \left( {\frac{1}{{\psi '\left( x \right)}}\frac{d}{{dx}}} \right)^n f\left( x \right), $

    and the right $ \psi $-Caputo fractional derivative of $ f $ is given by

    $ {}^CD_{b - }^{\alpha , \psi } f\left( x \right) = I_{b - }^{n - \alpha , \psi } \left( {-\frac{1}{{\psi '\left( x \right)}}\frac{d}{{dx}}} \right)^n f\left( x \right). $

    For given $ \alpha \notin\mathbb{N} $

    $ {}^CD_{a + }^{\alpha , \psi } f\left( x \right) = \frac{1}{{\Gamma \left( {n - \alpha } \right)}}\int\limits_a^x {\psi '\left( t \right)\left( {\psi \left( x \right) - \psi \left( t \right)} \right)^{n - \alpha - 1} f_\psi ^{\left[ n \right]} \left( t \right)dt} $

    and

    $ {}^CD_{b - }^{\alpha , \psi } f\left( x \right) = \frac{1}{{\Gamma \left( {n - \alpha } \right)}}\int\limits_a^x {\psi '\left( t \right)\left( {\psi \left( t \right) - \psi \left( x \right)} \right)^{n - \alpha - 1} (-1)^n f_\psi ^{\left[ n \right]} \left( t \right)dt}. $

    In particular when $ \alpha \in (0, 1) $ then

    $ {}^CD_{a + }^{\alpha , \psi } f\left( x \right) = \frac{1}{{\Gamma \left( {1 - \alpha } \right)}}\int\limits_a^x {\left( {\psi \left( x \right) - \psi \left( t \right)} \right)^{ - \alpha } f'\left( t \right)dt} $

    and

    $ {}^CD_{b - }^{\alpha , \psi } f\left( x \right) = \frac{1}{{\Gamma \left( {1 - \alpha } \right)}}\int\limits_a^x {\left( {\psi \left( t \right) - \psi \left( x \right)} \right)^{ - \alpha } f'\left( t \right)dt}. $

    In [18] the author has defined the $ \psi $ fractional partial integral with respect to another functions as

    Definition 2.3 Let $ \theta = (a, b) $ and $ \alpha = (\alpha_1, \alpha_2) $ where $ 0 \le \alpha_1, \alpha_2 \le 1 $. Also put $ I = [a, k] \times [b, m] $ where $ a, b $ and $ k, m $ are positive constants. Also let $ \psi(.) $ be an increasing positive monotone function on $ (a, k] \times (b, m] $ having continuous derivative $ \psi'(.) $ on $ (a, k] \times (b, m] $. Then the fractional partial integral is

    $ Iα;ψθu(x,y)=1Γ(α1)Γ(α2)xaybψ(s)ψ(t)(ψ(x)ψ(s))α11(ψ(y)ψ(t))α21f(s,t)dtds. $

    The Caputo fractional partial derivative is defined as follows

    Definition 2.4 Let $ \theta = (a, b) $ and $ \alpha = (\alpha_1, \alpha_2) $ where $ 0 \le \alpha_1, \alpha_2 \le 1 $. Also put $ I = [a, k] \times [b, m] $ where $ a, b $ and $ a, b $ are positive constants. Also let $ \psi(.) $ be an increasing function on $ (a, k] \times (b, m] $ and $ \psi'(.) \neq 0 $ on $ (a, k] \times (b, m] $. The $ \psi $ Caputo fractional partial derivative of functions of two variables of order $ \alpha $ is given by

    $ {}^CD_\theta ^{\alpha ;\psi } u\left( {x, y} \right) = I_\theta ^{2 - \alpha ;\psi } \left( {\frac{1}{{\psi '(s)\psi '(t)}}\frac{{\partial ^2\alpha }}{{\partial y\partial x}}} \right)u\left( {x, y} \right). $

    We use the following notation:

    $ {}^CD_\theta ^{\alpha ;\psi } u\left( {x, y} \right) = \frac{{\partial _\psi ^{2\alpha } u}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}\left( {x, y} \right). $

    We define the norm for a function of two variables as follows

    $ \left\| {{}^CD_\theta ^{\alpha ;\psi } f} \right\|_\infty = \sup \left| {{}^CD_\theta ^{\alpha ;\psi } f\left( {x, y} \right)} \right|. $

    Similarly as in Definition (2.3) and (2.4) we define the $ \psi $ fractional partial integral with respect to another functions and $ \psi $ Caputo fractional partial derivative of functions of three variables as follows:

    Definition 2.5 Let $ \Theta = (a, b, c) $ and $ \alpha = (\alpha_1, \alpha_2, \alpha_3) $ where $ 0 \le \alpha_1, \alpha_2, \alpha_3 \le 1 $. Also put $ I = [a, k] \times [b, m] \times [c, n] $ where $ a, b, c $ and $ k, m, n $ are positive constants. Also let $ \psi(.) $ be an increasing positive monotone function on $ (a, k] \times (b, m] \times [c, n] $ having continuous derivative $ \psi'(.) $ on $ (a, k] \times (b, m] \times (c, n] $.

    Then the fractional partial integral is

    $ Iα;ψΘu(x,y,z)=1Γ(α1)Γ(α2)xaybzcψ(s)ψ(t)ψ(r)×(ψ(x)ψ(s))α11(ψ(y)ψ(t))α21(ψ(z)ψ(r))α31f(s,t,r)drdtds. $

    Definition 2.6 Let $ \theta = (a, b, c) $ and $ \alpha = (\alpha_1, \alpha_2, \alpha_3) $ where $ 0 \le \alpha_1, \alpha_2, \alpha_3\le 1 $. Also put $ I = [a, k] \times [b, m] \times [c, n] $ where $ a, b, c $ and $ k, m, n $ are positive constants. Also let $ \psi(.) $ be an increasing function on $ (a, k] \times (b, m] \times (c, n] $ and $ \psi'(.) \neq 0 $ on $ (a, k] \times (b, m] \times (c, n] $. The $ \psi $ Caputo fractional partial derivative of functions of three variables of order $ \alpha $ is given by

    $ {}^CD_\Theta ^{\alpha ;\psi } u\left( {x, y, z} \right) = I_\Theta ^{3 - \alpha ;\psi } \left( {\frac{1}{{\psi '(s)\psi '(t)\psi '(r)}}\frac{{\partial ^3 }}{{\partial z\partial y\partial x}}} \right)u\left( {x, y, z} \right). $

    We use the following notation:

    $ {}^CD_\Theta ^{\alpha ;\psi } u\left( {x, y, z} \right) = \frac{{\partial _\psi ^{3\alpha } u}}{{\partial _\psi z^\alpha \partial _\psi y^\alpha x^\alpha }}\left( {x, y, z} \right). $

    We define the norm for a function of three variables as follows

    $ \left\| {{}^CD_\Theta ^{\alpha ;\psi } f} \right\|_\infty = \sup \left| {{}^CD_\Theta ^{\alpha ;\psi } f\left( {x, y, z} \right)} \right|. $

    Now we give the $ \psi $ Caputo fractional Čebyšev inequality involving functions of two variables as follows:

    Theorem 3.1 Let $ f, g:[a, l] \times [b, m] \rightarrow R $ be a continuous function on $ [a, l] \times [b, m] $ and $ \frac{{\partial ^{2\alpha } f}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }} $, $ \frac{{\partial ^{2\alpha } g}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }} $ exists continuous and bounded on $ [a, l] \times [b, m] $ and $ \alpha = (\alpha_1, \alpha_2) $. Then

    $ |lamb[f(x,y)g(x,y)12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]dydx]|18(ψ(l)ψ(a))(ψ(m)ψ(b))lamb[|g(x,y)|Dα;ψθf+g(x,y)Dα;ψθg]dydx, $ (3.1)

    where

    $ G(f(x,y))=12[f(a,y)+f(x,m)+f(x,b)+f(l,y)]14[f(a,b)+f(a,m)+f(l,b)+f(l,m)] $

    and

    $ H(2αfψyαψxα(x,y))=1Γ(α1)Γ(α2)××[xaybψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdtxamyψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdtlxybψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt+lxmyψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt]. $

    Proof. From the given hypotheses for $ (x, y) \in [a, l] \times [b, m] $ we have

    $ 1Γ(α1)Γ(α2)xaybψ(t)ψ(s)×(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt=1Γ(α1)xaψ(s)(ψ(x)ψ(t))α11[αfψsα(s,t)|yc]=1Γ(α1)xaψ(s)(ψ(y)ψ(t))α11[αfψsα(t,y)αfψsα(t,b)]=f(t,y)|xaf(t,b)|xa=f(x,y)f(a,y)f(x,b)+f(a,b). $ (3.2)

    Similarly we have

    $ 1Γ(α1)Γ(α2)xamyψ(t)ψ(s)×(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt=f(x,y)f(a,m)+f(x,m)+f(a,y), $ (3.3)
    $ 1Γ(α1)Γ(α2)lxybψ(t)ψ(s)×(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt=f(x,y)f(l,b)+f(x,b)+f(l,y), $ (3.4)
    $ 1Γ(α1)Γ(α2)lxmyψ(t)ψ(s)×(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(s,t)dsdt=f(x,y)+f(l,b)f(x,b)f(l,y). $ (3.5)

    Adding the above identities we have

    $ 4f(x,y)2[f(a,y)+f(x,m)+f(x,b)+f(l,y)]+[f(a,b)+f(a,m)+f(l,b)+f(l,m)]=1Γ(α1)Γ(α2)[xaybψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdtxadyψ(t)ψ(s)(ψ(x)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdtlxybψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(y)ψ(s))α212αfψsαψtα(t,s)dsdt+lxmyψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α212αfψsαψtα(t,s)dsdt]. $ (3.6)

    From $ (3.6) $ we have

    $ f(x, y) - G(f(x, y)) = \frac{1}{4}H\left( {\frac{{\partial ^{2\alpha } f}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}(x, y)} \right), $ (3.7)

    for $ (x, y) \in [a, l] \times [b, m] $. Similarly we have

    $ g(x, y) - G(g(x, y)) = \frac{1}{4}H\left( {\frac{{\partial ^{2\alpha } g}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}(x, y)} \right), $ (3.8)

    for $ (x, y) \in [a, l] \times [b, m] $.

    Multiplying $ (3.7) $ by $ g(x, y) $, $ (3.8) $ by $ f(x, y) $ adding them and Integrating over $ (x, y) \in [a, l] \times [b, m] $ we get

    $ lamb[2f(x,y)g(x,y)g(x,y)G(f(x,y))f(x,y)G(g(x,y))]dydx=18lamb[H(2αfψyαψxα(x,y))g(x,y)+14f(x,y)H(2αgψyαψxα(x,y))]. $ (3.9)

    From the properties of modulus we have

    $ |H(2αfψyαψxα(x,y))|1Γ(α1)Γ(α2)lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αfψsαψtα(t,s)|dsdt(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2cDα;ψθf, $ (3.10)
    $ |H(2αgψyαψxα(x,y))|1Γ(α1)Γ(α2)lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αgψsαψtα(t,s)|dsdt(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2cDα;ψθg. $ (3.11)

    From $ (3.9) $, $ (3.10) $ and $ (3.11) $ we have

    $ |lamb[f(x,y)g(x,y)12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]]dydx|18lamb[|H(2αfψyαψxα(x,y))||g(x,y)|+|H(2αgψyαψxα(x,y))||f(x,y)|]18lamb{|g(x,y)|[1Γ(α1)Γ(α2)×[lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αfψsαψtα(t,s)|dsdt]+|f(x,y)|×[lambψ(t)ψ(s)(ψ(l)ψ(t))α11(ψ(m)ψ(s))α21|2αgψsαψtα(t,s)|dsdt]}dydx18(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2×lamb[|g(x,y)|cDα;ψθf+|f(x,y)|cDα;ψθg]dydx, $ (3.12)

    which is required inequality.

    Theorem 3.2 Let $ f, g, G(f(x, y)), G(g(f(x, y)), {\frac{{\partial ^{2\alpha } f}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}}, {\frac{{\partial ^{2\alpha } g}}{{\partial _\psi y^\alpha \partial _\psi x^\alpha }}} $ be as in Theorem 3.1 then

    $ |lamb{f(x,y)g(x,y)[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)G(f(x,y))G(g(x,y))]}dydx116{(ψ(l)ψ(a))α1(ψ(m)ψ(b))α2}2cDα;ψθfcDα;ψθg, $ (3.13)

    for $ (x, y) \in [a, l] \times [b, m] $.

    Proof. Multiplying left hand side and right hand side of $ (3.7) $ and $ (3.8) $ we have

    $ f(x,y)g(x,y)[f(x,y)G(g(x,y))+g(x,y)G(f(x,y))]=116H(2αfψyαψxα(x,y))H(2αgψyαψxα(x,y)). $ (3.14)

    Integrating $ (3.14) $ over $ [a, l] \times [b, m] $ and from the properties of modulus we get

    $ |lamb{f(x,y)g(x,y)[G(g(x,y))f(x,y)+G(f(x,y))g(x,y)]G(f(x,y))G(g(x,y))}dydx|116lamb|H(2αfψyαψxα(x,y))||H(2αgψyαψxα(x,y))|dydx. $ (3.15)

    Now using $ (3.13), (3.14) $ in $ (3.19) $ we get required inequality $ (3.13) $.

    Now in our result we give the $ \psi $ Caputo fractional Čebyšev inequality involving functions of three variables. We use some notations as follows:

    $ A(p(u,v,w))=18[p(a,b,c)+p(k,m,n)]14[p(u,b,c)+p(u,m,n)+p(u,m,c)+p(u,b,n)]14[p(a,v,c)+p(k,v,n)+p(a,v,n)+p(k,v,c)]14[p(a,b,w)+p(k,m,w)+p(k,b,w)+p(a,m,w)]+12[p(a,v,w)+p(k,v,w)]+12[p(u,b,w)+p(u,m,w)]+12[p(u,v,c)+p(u,v,n)] $ (4.1)

    and

    $ B(3αpψwαψvαψuα(u,v,w))=1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)uavbncψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)uamvwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)kuvbwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(u)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)uamrnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)kumvwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)kuvbnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr1Γ(α1)Γ(α2)Γ(α3)kumvnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11×(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αpψtαψsαψrα(r,s,t)dtdsdr. $ (4.2)

    Now we give our next result as

    Theorem 4.1 Let $ f, g:[a, k] \times [b, m] \times [c, n] \rightarrow R $ be a continuous function on $ [a, l] \times [b, m] $ and $ \frac{{\partial ^{3\alpha } f}}{{\partial _\psi t^\alpha \partial _\psi s^\alpha \partial _\psi r^\alpha }} $, $ \frac{{\partial ^{3\alpha } g}}{{\partial _\psi t^\alpha \partial _\psi s^\alpha \partial _\psi r^\alpha }} $ exists and continuous and bounded on $ [a, k] \times [b, m] \times [c, n] $. Then

    $ kambnc[f(u,v,w)g(u,v,w)12[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))]]dwdvdu116(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3×kambnc[|g(u,v,w)|cDα;ψΘf+|f(u,v,w)|cDα;ψΘg]dwdvdu, $ (4.3)

    where $ A, B $ are as given in $ (4.1), (4.2) $.

    Proof. From the hypotheses we have for $ u, v, w \in [a, k] \times [b, m] \times [c, n] $

    $ 1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr=1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,t)|wcdsdr=1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,w)dsdr1Γ(α1)Γ(α2)uavbψ(r)ψ(s)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α212αfψsαψrα(r,s,c)dsdr=1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,s,w)|vbdr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,s,c)|vbdr=1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,v,w)dr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,b,w)dr1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,v,c)dr+1Γ(α1)uaψ(r)(ψ(u)ψ(r))α11αfψrα(r,b,c)dr=f(r,v,w)|uaf(r,b,w)|uaf(r,v,c)|ua+f(r,b,c)|ua=f(u,v,w)f(a,v,w)f(u,b,w)+f(a,b,w)f(u,v,c)+f(a,v,c)+f(u,b,c)+f(a,b,c). $

    Thus we have

    $ f(u,v,w)=f(a,v,w)+f(u,b,w)f(a,b,w)+f(u,v,c)f(a,v,c)f(u,b,c)f(a,b,c)1Γ(α1)Γ(α2)Γ(α3)uavbwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.4)

    Similarly we have

    $ f(u,v,w)=f(u,v,n)+f(a,v,w)+f(u,b,w)+f(a,b,n)f(a,b,w)f(a,v,n)f(v,b,n)1Γ(α1)Γ(α2)Γ(α3)uavbnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.5)
    $ f(u,v,w)=f(u,m,w)+f(u,v,c)+f(a,m,c)+f(a,v,w)f(u,m,c)f(a,m,w)f(a,v,c)1Γ(α1)Γ(α2)Γ(α3)uamvwcψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.6)
    $ f(u,v,w)=f(k,s,t)+f(k,b,c)+f(u,v,c)+f(u,b,w)f(k,v,c)f(k,b,w)f(u,b,c)1Γ(α1)Γ(α2)Γ(α3)kuvbwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(v)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.7)
    $ f(u,v,w)=f(u,m,w)+f(u,v,n)+f(a,m,n)+f(a,v,w)f(u,m,n)f(a,m,w)f(a,v,n)+1Γ(α1)Γ(α2)Γ(α3)uamvnwψ(r)ψ(s)ψ(t)(ψ(u)ψ(r))α11(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.8)
    $ f(u,v,w)=f(r,m,t)+f(u,v,c)+f(k,s,t)+f(k,m,c)f(k,m,w)f(k,v,c)f(u,m,c)+1Γ(α1)Γ(α2)Γ(α3)kumvwcψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21(ψ(w)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr, $ (4.9)
    $ f(u,v,w)=f(k,v,w)+f(k,b,n)+f(u,v,n)+f(u,b,t)f(k,v,n)f(k,b,w)f(u,b,n)+1Γ(α1)Γ(α2)Γ(α3)kuvbnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(v)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr $ (4.10)

    and

    $ f(u,v,w)=f(k,m,n)+f(k,v,w)+f(u,m,w)+f(u,v,n)f(k,m,w)f(k,v,n)f(u,m,n)+1Γ(α1)Γ(α2)Γ(α3)kumvnwψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr. $ (4.11)

    Adding the above identities we have

    $ f\left( {u, v, w} \right) - A\left( {f\left( {u, v, w} \right)} \right) = \frac{1}{8}B\left( {\frac{{\partial ^{3\alpha } f}}{{\partial _\psi w^\alpha \partial _\psi v^\alpha \partial _\psi u^\alpha }}(u, v, w)} \right), $ (4.12)

    for $ (u, v, w) \in [a, k] \times [b, m] \times [c, n] $.

    Similarly we have

    $ g\left( {u, v, w} \right) - A\left( {g\left( {u, v, w} \right)} \right) = \frac{1}{8}B\left( {\frac{{\partial ^{3\alpha } g}}{{\partial _\psi w^\alpha \partial _\psi v^\alpha \partial _\psi u^\alpha }}(u, v, w)} \right), $ (4.13)

    for $ (u, v, w) \in [a, k] \times [b, m] \times [c, n] $.

    Now multiplying $ (4.12) $ and $ (4.13) $ by $ g(u, v, w) $ and $ f(u, v, w) $ respectively, adding them and Integrating over $ [a, k] \times [b, m] \times [c, n] $ we have

    $ kambnc[f(u,v,w)g(u,v,w)12[g(u,v,w)A(f(u,v,w))g(u,v,w)A(f(u,v,w))]]dwdvdu=116kambnc[g(u,v,w)B(3αfψwαψvαψuα(u,v,w))+f(u,v,w)B(3αgψwαψvαψuα(u,v,w))]. $ (4.14)

    From the properties of modulus we have

    $ |B(3αfψwαψvαψuα(u,v,w))|kambncψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21×(ψ(n)ψ(t))α313αfψtαψsαψrα(r,s,t)dtdsdr(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3CDα;ψΘf, $ (4.15)
    $ |B(3αgψwαψvαψuα(u,v,w))|kambncψ(r)ψ(s)ψ(t)(ψ(k)ψ(r))α11(ψ(m)ψ(s))α21×(ψ(n)ψ(t))α313αgψtαψsαψrα(r,s,t)dtdsdr(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3CDα;ψΘg. $ (4.16)

    Now by substituting the values from equation $ (4.15) $ and $ (4.16) $ in $ (4.14) $ we get the required inequality $ (4.3) $.

    Theorem 4.2 Let $ f, g $, $ \frac{{\partial ^{3\alpha } f}}{{\partial _\psi t^\alpha \partial _\psi s^\alpha \partial _\psi r^\alpha }} $ and $ \frac{{\partial ^{3\alpha } g}}{{\partial _\psi t^\alpha \partial _\psi s^\alpha \partial _\psi r^\alpha }} $ be as in Theorem $ 4.1 $. Then

    $ |kambnc[f(u,v,w)g(u,v,w)[A(f(u,v,w))g(u,v,w)A(g(u,v,w))f(u,v,w)A(f(u,v,w))A(g(u,v,w))|dwdvdu164{(ψ(k)ψ(a))α1(ψ(m)ψ(b))α2(ψ(n)ψ(c))α3}2CDα;ψΘfCDα;ψΘg, $ (4.17)

    for $ (r, s, t) \in [a, k] \times [b, m] \times [c, n] $ and $ A, B $ are as given in $ (4.1), (4.2) $.

    Proof. Multiplying left hand and right hand side of equation $ (4.12) $ and $ (4.13) $ we have

    $ f(u,v,w)g(u,v,w)[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))A(f(u,v,w))A(g(u,v,w))]=164B(3αfψwαψvαψuα(u,v,w))B(3αgψwαψvαψuα(u,v,w)). $ (4.18)

    Integrating over $ [a, k] \times [b, m] \times [c, n] $ and from the properties of modulus we have

    $ |kambnc[f(u,v,w)g(u,v,w)[f(u,v,w)A(g(u,v,w))+g(u,v,w)A(f(u,v,w))A(f(u,v,w))A(g(u,v,w))]]|dwdvdu164kambnc|B(3αfψwαψvαψuα(u,v,w))B(3αfψwαψvαψuα(u,v,w))|dwdvdu. $ (4.19)

    Using $ (4.15) $ and $ (4.16) $ in $ (4.19) $ we get the required inequality $ (4.17) $.

    Remark: If we put different values for $ \psi(x) $ as $ x, ln x, x^{\sigma} $then it reduces to various types of fractional Čebyšev inequalities such as Riemann Liouville fractional, Hadmard Fractional and Erdelyi-Kober fractional inequalities respectively.

    In this paper, we studied Čebyšev like inequalities. We proved some new $ \psi $ Caputo fractional Čebyšev type inequalities involving functions of two and three variables.

    All authors declare no conflict of interest in this paper.

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