Citation: Edward N Trifonov. Columnar structure of SV40 minichromosome[J]. AIMS Biophysics, 2015, 2(3): 274-283. doi: 10.3934/biophy.2015.3.274
[1] | Saad Ihsan Butt, Artion Kashuri, Muhammad Umar, Adnan Aslam, Wei Gao . Hermite-Jensen-Mercer type inequalities via Ψ-Riemann-Liouville k-fractional integrals. AIMS Mathematics, 2020, 5(5): 5193-5220. doi: 10.3934/math.2020334 |
[2] | Mohammed A. Almalahi, Mohammed S. Abdo, Satish K. Panchal . On the theory of fractional terminal value problem with ψ-Hilfer fractional derivative. AIMS Mathematics, 2020, 5(5): 4889-4908. doi: 10.3934/math.2020312 |
[3] | Naila Mehreen, Matloob Anwar . Some inequalities via Ψ-Riemann-Liouville fractional integrals. AIMS Mathematics, 2019, 4(5): 1403-1415. doi: 10.3934/math.2019.5.1403 |
[4] | Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable fractional integral inequalities for GG- and GA-convex functions. AIMS Mathematics, 2020, 5(5): 5012-5030. doi: 10.3934/math.2020322 |
[5] | Mohammad Esmael Samei, Lotfollah Karimi, Mohammed K. A. Kaabar . To investigate a class of multi-singular pointwise defined fractional $ q $–integro-differential equation with applications. AIMS Mathematics, 2022, 7(5): 7781-7816. doi: 10.3934/math.2022437 |
[6] | Feng Qi, Siddra Habib, Shahid Mubeen, Muhammad Nawaz Naeem . Generalized k-fractional conformable integrals and related inequalities. AIMS Mathematics, 2019, 4(3): 343-358. doi: 10.3934/math.2019.3.343 |
[7] | Gou Hu, Hui Lei, Tingsong Du . Some parameterized integral inequalities for p-convex mappings via the right Katugampola fractional integrals. AIMS Mathematics, 2020, 5(2): 1425-1445. doi: 10.3934/math.2020098 |
[8] | Chunhong Li, Dandan Yang, Chuanzhi Bai . Some Opial type inequalities in (p, q)-calculus. AIMS Mathematics, 2020, 5(6): 5893-5902. doi: 10.3934/math.2020377 |
[9] | Xiuzhi Yang, G. Farid, Waqas Nazeer, Muhammad Yussouf, Yu-Ming Chu, Chunfa Dong . Fractional generalized Hadamard and Fejér-Hadamard inequalities for m-convex functions. AIMS Mathematics, 2020, 5(6): 6325-6340. doi: 10.3934/math.2020407 |
[10] | M. Emin Özdemir, Saad I. Butt, Bahtiyar Bayraktar, Jamshed Nasir . Several integral inequalities for (α, s,m)-convex functions. AIMS Mathematics, 2020, 5(4): 3906-3921. doi: 10.3934/math.2020253 |
P.L Čebyšev in the year 1882 has proved the following interesting inequality:
$ |1b−ab∫af(x)g(x)dx−(1b−ab∫af(x)dx)(1b−ab∫ag(x)dx)|≤112(b−a)2‖f′‖∞‖g′‖∞. $ |
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)nIn−α,ψa+f(x)=1Γ(n−α)(1ψ′(x)ddx)nx∫aψ′(t)(ψ(x)−ψ(t))n−α−1f(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,ψb−f(x)=(−1ψ′(x)ddx)nIn−α,ψb−f(x)=1Γ(n−α)(1ψ′(x)ddx)nx∫aψ′(t)(ψ(t)−ψ(x))n−α−1f(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)x∫ay∫bψ′(s)ψ′(t)(ψ(x)−ψ(s))α1−1(ψ(y)−ψ(t))α2−1f(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)x∫ay∫bz∫cψ′(s)ψ′(t)ψ′(r)×(ψ(x)−ψ(s))α1−1(ψ(y)−ψ(t))α2−1(ψ(z)−ψ(r))α3−1f(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
$ |l∫am∫b[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))l∫am∫b[|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(∂2αf∂ψyα∂ψxα(x,y))=1Γ(α1)Γ(α2)××[x∫ay∫bψ′(t)ψ′(s)(ψ(x)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt−x∫am∫yψ′(t)ψ′(s)(ψ(x)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt−l∫xy∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt+l∫xm∫yψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt]. $ |
Proof. From the given hypotheses for $ (x, y) \in [a, l] \times [b, m] $ we have
$ 1Γ(α1)Γ(α2)x∫ay∫bψ′(t)ψ′(s)×(ψ(x)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt=1Γ(α1)x∫aψ′(s)(ψ(x)−ψ(t))α1−1[∂αf∂ψsα(s,t)|yc]=1Γ(α1)x∫aψ′(s)(ψ(y)−ψ(t))α1−1[∂αf∂ψsα(t,y)−∂αf∂ψsα(t,b)]=f(t,y)|xa−f(t,b)|xa=f(x,y)−f(a,y)−f(x,b)+f(a,b). $ | (3.2) |
Similarly we have
$ 1Γ(α1)Γ(α2)x∫am∫yψ′(t)ψ′(s)×(ψ(x)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt=−f(x,y)−f(a,m)+f(x,m)+f(a,y), $ | (3.3) |
$ 1Γ(α1)Γ(α2)l∫xy∫bψ′(t)ψ′(s)×(ψ(l)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt=−f(x,y)−f(l,b)+f(x,b)+f(l,y), $ | (3.4) |
$ 1Γ(α1)Γ(α2)l∫xm∫yψ′(t)ψ′(s)×(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(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)[x∫ay∫bψ′(t)ψ′(s)(ψ(x)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt−x∫ad∫yψ′(t)ψ′(s)(ψ(x)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt−l∫xy∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(y)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(t,s)dsdt+l∫xm∫yψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1∂2αf∂ψsα∂ψtα(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
$ l∫am∫b[2f(x,y)g(x,y)−g(x,y)G(f(x,y))−f(x,y)G(g(x,y))]dydx=18l∫am∫b[H(∂2αf∂ψyα∂ψxα(x,y))g(x,y)+14f(x,y)H(∂2αg∂ψyα∂ψxα(x,y))]. $ | (3.9) |
From the properties of modulus we have
$ |H(∂2αf∂ψyα∂ψxα(x,y))|≤1Γ(α1)Γ(α2)l∫am∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1|∂2αf∂ψsα∂ψtα(t,s)|dsdt≤(ψ(l)−ψ(a))α1(ψ(m)−ψ(b))α2‖cDα;ψθf‖∞, $ | (3.10) |
$ |H(∂2αg∂ψyα∂ψxα(x,y))|≤1Γ(α1)Γ(α2)l∫am∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1|∂2αg∂ψsα∂ψtα(t,s)|dsdt≤(ψ(l)−ψ(a))α1(ψ(m)−ψ(b))α2‖cDα;ψθg‖∞. $ | (3.11) |
From $ (3.9) $, $ (3.10) $ and $ (3.11) $ we have
$ |l∫am∫b[f(x,y)g(x,y)−12[G(f(x,y))g(x,y)+G(g(x,y))f(x,y)]]dydx|≤18l∫am∫b[|H(∂2αf∂ψyα∂ψxα(x,y))||g(x,y)|+|H(∂2αg∂ψyα∂ψxα(x,y))||f(x,y)|]≤18l∫am∫b{|g(x,y)|[1Γ(α1)Γ(α2)×[l∫am∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1|∂2αf∂ψsα∂ψtα(t,s)|dsdt]+|f(x,y)|×[l∫am∫bψ′(t)ψ′(s)(ψ(l)−ψ(t))α1−1(ψ(m)−ψ(s))α2−1|∂2αg∂ψsα∂ψtα(t,s)|dsdt]}dydx≤18(ψ(l)−ψ(a))α1(ψ(m)−ψ(b))α2×l∫am∫b[|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
$ |l∫am∫b{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))]}dydx≤116{(ψ(l)−ψ(a))α1(ψ(m)−ψ(b))α2}2‖cDα;ψθf‖∞‖cDα;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(∂2αf∂ψyα∂ψxα(x,y))H(∂2αg∂ψyα∂ψxα(x,y)). $ | (3.14) |
Integrating $ (3.14) $ over $ [a, l] \times [b, m] $ and from the properties of modulus we get
$ |l∫am∫b{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|≤116l∫am∫b|H(∂2αf∂ψyα∂ψxα(x,y))||H(∂2αg∂ψyα∂ψxα(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(∂3αp∂ψwα∂ψvα∂ψuα(u,v,w))=1Γ(α1)Γ(α2)Γ(α3)u∫av∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1×(ψ(v)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr−1Γ(α1)Γ(α2)Γ(α3)u∫av∫bn∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1×(ψ(v)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr−1Γ(α1)Γ(α2)Γ(α3)u∫am∫vw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1×(ψ(m)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr−1Γ(α1)Γ(α2)Γ(α3)k∫uv∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1×(ψ(u)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)u∫am∫rn∫wψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1×(ψ(m)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)k∫um∫vw∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1×(ψ(m)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr+1Γ(α1)Γ(α2)Γ(α3)k∫uv∫bn∫wψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1×(ψ(v)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr−1Γ(α1)Γ(α2)Γ(α3)k∫um∫vn∫wψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1×(ψ(m)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αp∂ψtα∂ψsα∂ψrα(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
$ k∫am∫bn∫c[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))]]dwdvdu≤116(ψ(k)−ψ(a))α1(ψ(m)−ψ(b))α2(ψ(n)−ψ(c))α3×k∫am∫bn∫c[|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)u∫av∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr=1Γ(α1)Γ(α2)u∫av∫bψ′(r)ψ′(s)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1∂2αf∂ψsα∂ψrα(r,s,t)|wcdsdr=1Γ(α1)Γ(α2)u∫av∫bψ′(r)ψ′(s)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1∂2αf∂ψsα∂ψrα(r,s,w)dsdr−1Γ(α1)Γ(α2)u∫av∫bψ′(r)ψ′(s)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1∂2αf∂ψsα∂ψrα(r,s,c)dsdr=1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,s,w)|vbdr−1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,s,c)|vbdr=1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,v,w)dr−1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,b,w)dr−1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,v,c)dr+1Γ(α1)u∫aψ′(r)(ψ(u)−ψ(r))α1−1∂αf∂ψrα(r,b,c)dr=f(r,v,w)|ua−f(r,b,w)|ua−f(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)u∫av∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(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)u∫av∫bn∫wψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(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)u∫am∫vw∫cψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(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)k∫uv∫bw∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(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)u∫am∫vn∫wψ′(r)ψ′(s)ψ′(t)(ψ(u)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(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)k∫um∫vw∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1(ψ(w)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(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)k∫uv∫bn∫wψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(v)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(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)k∫um∫vn∫wψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(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
$ k∫am∫bn∫c[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=116k∫am∫bn∫c[g(u,v,w)B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))+f(u,v,w)B(∂3αg∂ψwα∂ψvα∂ψuα(u,v,w))]. $ | (4.14) |
From the properties of modulus we have
$ |B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))|≤k∫am∫bn∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1×(ψ(n)−ψ(t))α3−1∂3αf∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr≤(ψ(k)−ψ(a))α1(ψ(m)−ψ(b))α2(ψ(n)−ψ(c))α3‖CDα;ψΘf‖∞, $ | (4.15) |
$ |B(∂3αg∂ψwα∂ψvα∂ψuα(u,v,w))|≤k∫am∫bn∫cψ′(r)ψ′(s)ψ′(t)(ψ(k)−ψ(r))α1−1(ψ(m)−ψ(s))α2−1×(ψ(n)−ψ(t))α3−1∂3αg∂ψtα∂ψsα∂ψrα(r,s,t)dtdsdr≤(ψ(k)−ψ(a))α1(ψ(m)−ψ(b))α2(ψ(n)−ψ(c))α3‖CDα;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
$ |k∫am∫bn∫c[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))|dwdvdu≤164{(ψ(k)−ψ(a))α1(ψ(m)−ψ(b))α2(ψ(n)−ψ(c))α3}2‖CDα;ψΘf‖∞‖CDα;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(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))B(∂3αg∂ψwα∂ψvα∂ψuα(u,v,w)). $ | (4.18) |
Integrating over $ [a, k] \times [b, m] \times [c, n] $ and from the properties of modulus we have
$ |k∫am∫bn∫c[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))]]|dwdvdu≤164k∫am∫bn∫c|B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w))B(∂3αf∂ψwα∂ψvα∂ψuα(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.
[1] |
Griffith JD (1975) Chromatin structure: deduced from a minichromosome. Science 187: 1202-1203. doi: 10.1126/science.187.4182.1202
![]() |
[2] |
Bellard M, Oudet P, Germond JE, et al. (1976) Subunit structure of simian-virus-40 minichromosome. Eur J Biochem 70: 543-553. doi: 10.1111/j.1432-1033.1976.tb11046.x
![]() |
[3] |
Mengeritsky G, Trifonov EN (1984) Nucleotide sequence-directed mapping of the nucleosomes of SV40 chromatin. Cell Bioph 6: 1-9. doi: 10.1007/BF02788576
![]() |
[4] |
Ambrose C, Lowman H, Rajadhyaksha A, et al. (1990) Location of nucleosomes in simian virus 40 chromatin. J Mol Biol 214: 875-884. doi: 10.1016/0022-2836(90)90342-J
![]() |
[5] | Tatchell K, van Holde KE (1978) Compact oligomers and nucleosome phasing. Proc Natl Acad Sci USA 75: 3583-3587. |
[6] |
Fajkus J, Trifonov EN (2001) Columnar packing of telomeric nucleosomes. Biochem Biophys Res Comm 280: 961-963. doi: 10.1006/bbrc.2000.4208
![]() |
[7] | Salih B, Trifonov EN (2015) Strong nucleosomes of A. thaliana concentrate in centromere regions. J Biomol Str Dyn 33: 10-13. |
[8] |
Salih B, Trifonov EN (2015). Strong nucleosomes reside in meiotic centromeres of C. elegans. J Biomol Str Dyn 33: 365-373. doi: 10.1080/07391102.2013.879263
![]() |
[9] |
Nibhani R, Trifonov EN (2015) Reading sequence-directed computational nucleosome maps. J Biomol Str Dyn 33:1558-1566. doi: 10.1080/07391102.2014.958877
![]() |
[10] |
Gu SG, Goszczynski B, McGhee JD, et al. (2013) Unusual DNA packaging characteristics in endoreduplicated Caenorhabditis elegans oocytes defined by in vivo accessibility to an endogenous nuclease activity. Epigenet Chromatin 6: 37. doi: 10.1186/1756-8935-6-37
![]() |
[11] |
Ponder BAJ, Crawford LV (1977) Arrangement of nucleosomes in nucleoprotein complexes from polyoma-virus and SV40. Cell 11: 35-49. doi: 10.1016/0092-8674(77)90315-4
![]() |
[12] | Zhang T, Talbert PB, Zhang W, et al. (2013) The CentO satellite confers translational and rotational phasing on cenH3 nucleosomes in rice centromeres. Proc Natl Acad Sci U S A 110: E4875-4883. |
[13] | Rattner JB, Hamkalo BA (1978) Higher-order structure in metaphase chromosomes Chromosoma 69: 373-379. |
[14] | Dubochet J, Adrian M, Schultz P, et al. (1986) Cryo-electron microscopy of vitrified SV40 minichromosomes: the liquid drop model. EMBO J 5: 519-528. |
[15] |
Wanner G, Formanek H (2000) A new chromosome model. J Struct Biol 132: 147-161. doi: 10.1006/jsbi.2000.4310
![]() |
[16] |
Karpov VL, Bavykin SG, Preobrazhenskaya OV, et al. (1982) Alignment of nucleosomes along DNA and organization of spacer DNA in drosophila chromatin. Nucl Acids Res 10: 4321-4337. doi: 10.1093/nar/10.14.4321
![]() |
[17] | Widom J (1992) A relationship between the helical twist of DNA and the ordered positioning of nucleosomes in all eukaryotic cells. Proc Natl Acad Sci U S A 89:1095-1099. |
[18] |
Rapoport AE, Frenkel ZM, Trifonov EN (2011) Nucleosome positioning pattern derived from oligonucleotide compositions of genomic sequences. J Biomol Str Dyn 28: 567-574. doi: 10.1080/07391102.2011.10531243
![]() |
[19] |
Frenkel ZM, Bettecken T, Trifonov EN (2011) Nucleosome DNA sequence structure of isochores. BMC Genomics 12: 203. doi: 10.1186/1471-2164-12-203
![]() |
[20] |
Tripathi V, Salih B, Trifonov EN (2015) Universal full-length nucleosome mapping sequence probe. J Biomol Str Dyn 33: 666-673. doi: 10.1080/07391102.2014.891262
![]() |
[21] |
Lowary PT, Widom J (1998) New DNA sequence rules for high affinity binding to histone octamer and sequence directed nucleosome positioning. J Mol Biol 276: 19-42. doi: 10.1006/jmbi.1997.1494
![]() |
[22] |
Trifonov EN, Nibhani R (2015) Review fifteen years of search for strong nucleosomes. Biopolymers, 103: 432-437. doi: 10.1002/bip.22604
![]() |
[23] | Nibhani R, Trifonov EN, TA-periodic (“601”-like) centromeric nucleosomes of A.thaliana. J Biomol Str Dyn [in press]. |
[24] |
Vasudevan D, Chua EYD, Davey CA (2010) Crystal structures of nucleosome core particles containing the ‘601’ strong positioning sequence. J Mol Biol 403: 1-10. doi: 10.1016/j.jmb.2010.08.039
![]() |
[25] | Zhurkin VB (1982) Periodicity in DNA primary structure and specific alignment of nucleosomes. Stud Biophys 87: 151-152. |
[26] |
Zhurkin VB (1983) Specific alignment of nucleosomes on DNA correlates with periodic distribution of purine pyrimidine and pyrimidine purine dimers. FEBS Lett 158: 293-297. doi: 10.1016/0014-5793(83)80598-5
![]() |
[27] |
Zhurkin VB, Lysov YP, Ivanov VI (1979) Anisotropic flexibility of DNA and the nucleosomal structure. Nucl Acids Res 6: 1081-1096. doi: 10.1093/nar/6.3.1081
![]() |
[28] |
Wang D, Ulyanov NB, Zhurkin VB (2010) Sequence-dependent kink-and-slide deformations of nucleosomal DNA facilitated by histone arginines bound in the minor groove. J Biomol Str Dyn 27: 843-859. doi: 10.1080/07391102.2010.10508586
![]() |
[29] | McDowall AW, Smith JM, Dubochet J (1986) Cryo-electron microscopy of vitrified chromosomes in situ. EMBO J 5: 1395-1402. |
[30] | Trifonov EN Nucleosome repeat lengths and columnar chromatin structure. J Biomol Str Dyn [in press]. |
1. | Tamer Nabil, Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative, 2021, 6, 2473-6988, 5088, 10.3934/math.2021301 | |
2. | MAYSAA AL-QURASHI, SAIMA RASHID, YELIZ KARACA, ZAKIA HAMMOUCH, DUMITRU BALEANU, YU-MING CHU, ACHIEVING MORE PRECISE BOUNDS BASED ON DOUBLE AND TRIPLE INTEGRAL AS PROPOSED BY GENERALIZED PROPORTIONAL FRACTIONAL OPERATORS IN THE HILFER SENSE, 2021, 29, 0218-348X, 2140027, 10.1142/S0218348X21400272 |