On some ψ Caputo fractional Čebyšev like inequalities for functions of two and three variables

1.   Introduction

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

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.

2.   Preliminaries

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

and

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

and

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

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

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

and

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

and

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

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

We use the following notation:

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

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

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

We use the following notation:

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

3.   Čebyšev inequality involving functions of two variables

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

where

and

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

Similarly we have

Adding the above identities we have

From $(3.6)$ we have

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

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

From the properties of modulus we have

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

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

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

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

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

4.   Čebyšev inequality involving functions of three variables

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

and

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

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

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

Thus we have

Similarly we have

and

Adding the above identities we have

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

Similarly we have

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

From the properties of modulus we have

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

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

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

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.

5.   Conclusions

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.

Conflict of interest

All authors declare no conflict of interest in this paper.