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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′∈L∞[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 ψ Caputo and ψ Hilfer fractional derivative of a function with respect to another functions and its applications. The ψ fractional and integral definations are more generalized and it reduces to Riemann Liouville, Hadmard and Erdelyi-Kober fractional definitions for different values of ψ.
Motivated from the above mentioned literature the aim of this paper is to obtain ψ 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, α>0, f is an integrable function defined on I and ψ∈C1(I) an increasing function such that ψ′(x)≠0 for all x∈I then fractional derivative and integral of f is given by
Iα,ψa+f(x)=1Γ(α)x∫aψ′(t)(ψ(x)−ψ(t))α−1f(t)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
Iα,ψb−f(x)=1Γ(α)x∫aψ′(t)(ψ(t)−ψ(x))α−1f(t)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 α>0, n∈N, I is the interval −∞≤a<b≤∞, f,ψ∈Cn(I) two functions such that ψ is increasing and ψ′(x)≠0 for all x∈I. The left ψ-Caputo fractional derivative of f of order α is given by
CDα,ψa+f(x)=In−α,ψa+(1ψ′(x)ddx)nf(x), |
and the right ψ-Caputo fractional derivative of f is given by
CDα,ψb−f(x)=In−α,ψb−(−1ψ′(x)ddx)nf(x). |
For given α∉N
CDα,ψa+f(x)=1Γ(n−α)x∫aψ′(t)(ψ(x)−ψ(t))n−α−1f[n]ψ(t)dt |
and
CDα,ψb−f(x)=1Γ(n−α)x∫aψ′(t)(ψ(t)−ψ(x))n−α−1(−1)nf[n]ψ(t)dt. |
In particular when α∈(0,1) then
CDα,ψa+f(x)=1Γ(1−α)x∫a(ψ(x)−ψ(t))−αf′(t)dt |
and
CDα,ψb−f(x)=1Γ(1−α)x∫a(ψ(t)−ψ(x))−αf′(t)dt. |
In [18] the author has defined the ψ fractional partial integral with respect to another functions as
Definition 2.3 Let θ=(a,b) and α=(α1,α2) where 0≤α1,α2≤1. Also put I=[a,k]×[b,m] where a,b and k,m are positive constants. Also let ψ(.) be an increasing positive monotone function on (a,k]×(b,m] having continuous derivative ψ′(.) on (a,k]×(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 θ=(a,b) and α=(α1,α2) where 0≤α1,α2≤1. Also put I=[a,k]×[b,m] where a,b and a,b are positive constants. Also let ψ(.) be an increasing function on (a,k]×(b,m] and ψ′(.)≠0 on (a,k]×(b,m]. The ψ Caputo fractional partial derivative of functions of two variables of order α is given by
CDα;ψθu(x,y)=I2−α;ψθ(1ψ′(s)ψ′(t)∂2α∂y∂x)u(x,y). |
We use the following notation:
CDα;ψθu(x,y)=∂2αψu∂ψyα∂ψxα(x,y). |
We define the norm for a function of two variables as follows
‖CDα;ψθf‖∞=sup|CDα;ψθf(x,y)|. |
Similarly as in Definition (2.3) and (2.4) we define the ψ fractional partial integral with respect to another functions and ψ Caputo fractional partial derivative of functions of three variables as follows:
Definition 2.5 Let Θ=(a,b,c) and α=(α1,α2,α3) where 0≤α1,α2,α3≤1. Also put I=[a,k]×[b,m]×[c,n] where a,b,c and k,m,n are positive constants. Also let ψ(.) be an increasing positive monotone function on (a,k]×(b,m]×[c,n] having continuous derivative ψ′(.) on (a,k]×(b,m]×(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 θ=(a,b,c) and α=(α1,α2,α3) where 0≤α1,α2,α3≤1. Also put I=[a,k]×[b,m]×[c,n] where a,b,c and k,m,n are positive constants. Also let ψ(.) be an increasing function on (a,k]×(b,m]×(c,n] and ψ′(.)≠0 on (a,k]×(b,m]×(c,n]. The ψ Caputo fractional partial derivative of functions of three variables of order α is given by
CDα;ψΘu(x,y,z)=I3−α;ψΘ(1ψ′(s)ψ′(t)ψ′(r)∂3∂z∂y∂x)u(x,y,z). |
We use the following notation:
CDα;ψΘu(x,y,z)=∂3αψu∂ψzα∂ψyαxα(x,y,z). |
We define the norm for a function of three variables as follows
‖CDα;ψΘf‖∞=sup|CDα;ψΘf(x,y,z)|. |
Now we give the ψ Caputo fractional Čebyšev inequality involving functions of two variables as follows:
Theorem 3.1 Let f,g:[a,l]×[b,m]→R be a continuous function on [a,l]×[b,m] and ∂2αf∂ψyα∂ψxα, ∂2αg∂ψyα∂ψxα exists continuous and bounded on [a,l]×[b,m] and α=(α1,α2). 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)∈[a,l]×[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))=14H(∂2αf∂ψyα∂ψxα(x,y)), | (3.7) |
for (x,y)∈[a,l]×[b,m]. Similarly we have
g(x,y)−G(g(x,y))=14H(∂2αg∂ψyα∂ψxα(x,y)), | (3.8) |
for (x,y)∈[a,l]×[b,m].
Multiplying (3.7) by g(x,y), (3.8) by f(x,y) adding them and Integrating over (x,y)∈[a,l]×[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)),∂2αf∂ψyα∂ψxα,∂2αg∂ψyα∂ψxα 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)∈[a,l]×[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]×[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 ψ 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]×[b,m]×[c,n]→R be a continuous function on [a,l]×[b,m] and ∂3αf∂ψtα∂ψsα∂ψrα, ∂3αg∂ψtα∂ψsα∂ψrα exists and continuous and bounded on [a,k]×[b,m]×[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∈[a,k]×[b,m]×[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(u,v,w)−A(f(u,v,w))=18B(∂3αf∂ψwα∂ψvα∂ψuα(u,v,w)), | (4.12) |
for (u,v,w)∈[a,k]×[b,m]×[c,n].
Similarly we have
g(u,v,w)−A(g(u,v,w))=18B(∂3αg∂ψwα∂ψvα∂ψuα(u,v,w)), | (4.13) |
for (u,v,w)∈[a,k]×[b,m]×[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]×[b,m]×[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, ∂3αf∂ψtα∂ψsα∂ψrα and ∂3αg∂ψtα∂ψsα∂ψrα 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)∈[a,k]×[b,m]×[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]×[b,m]×[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 ψ(x) as x,lnx,xσthen 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 ψ 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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