Theory and Modern Applications

# Bivariate Montgomery identity for alpha diamond integrals

## Abstract

In the paper, some variants of Montgomery identity with the help of delta and nabla integrals are established which are useful to produce Montgomery identity involving alpha diamond integrals for function of two variables. The aforementioned identity is discussed in discrete, continuous, quantum calculus as well and employed to obtain Ostrowski type inequality for monotonically increasing function with respect to both parameters.

## Introduction

Mitrinovic, Pecaric, and Fink proved Montgomery identities for functions defined on the real line  in the following form:

Let $$f: [ {a,b} ] \to \mathbb{R}$$ be a differentiable function and $$f': [ {a,b} ] \to \mathbb{R}$$ be integrable on $$[ {a,b} ]\in \mathbb{R}$$, then

$$f ( x ) = \frac{1}{{b - a}} \int _{a}^{b} {f ( t )} \,dt + \int _{a}^{b} {p ( {x,t} )} f' ( t )\,dt,$$
(1.1)

where $$p ( {x,t} )$$ is called Peano kernel, defined in . Some applications of Montgomery identity in the form of inequalities can be found in [13, 15, 16].

Dragomir et al. extended it for a function of two variables on the real line  in the following form:

\begin{aligned}[b] f ( {x,y} ) & = \frac{1}{{b - a}} \int _{a}^{b} {f ( {t,y} )\,dt + } \frac{1}{{d - c}} \int _{c}^{d} {f ( {x,s} )\,ds} \\ &\quad {} - \frac{1}{{ ( {b - a} ) ( {d - c} )}} \int _{a}^{b} { \int _{c}^{d} {f ( {t,s} )\,ds\,dt + } } \int _{a}^{b} { \int _{c}^{d} {p ( {x,t} )q ( {y,s} ) \frac{{{\partial ^{2}}f ( {t,s} )}}{ {\partial t\partial s}}\,ds\,dt} }, \end{aligned}
(1.2)

where $$f:I = [ {a,b} ] \times [ {c,d} ] \to \mathbb{R}$$ is differentiable, the derivative $${\frac{{{\partial ^{2}}f ( {t,s} )}}{{\partial t\partial s}}}$$ is integrable on I, $$p(x,t)$$ and $$q(y,s)$$ are Peano kernels, defined in .

In 1988, Hilger, a German mathematician, presented time scales theory which deals with both discrete and continuous cases simultaneously. Delta and nabla calculus were first two approaches in the theory of time scales. For introduction to the time scales calculus, the readers are referred to [9, 10], and some related inequalities can be seen in [1, 3, 5, 6, 8, 11, 14].

Bohner and Matthews proved the following Montgomery identity for functions of one variable by using delta integrals .

Let $$a,b,s,t \in \mathbb{T}$$; $$a < b$$ and $$f: [ {a,b} ] \to \mathbb{R}$$ be differentiable, then

$$f ( t ) = \frac{1}{{b - a}} \int _{a}^{b} {{f^{ \sigma }} ( s )} \Delta s + \frac{1}{{b - a}} \int _{a}^{b} {p ( {t,s} ){f^{\Delta }} ( s )} \Delta s,$$
(1.3)

where

$$p ( {t,s} ) = \textstyle\begin{cases} s - a, & a \le s < t, \\ s - b, & t \le s < b. \end{cases}$$

### Remark 1.1

For nabla integrals, (1.3) can be written as follows:

Let $$a,b,s,t \in \mathbb{T}$$; $$a < b$$ and $$f: [ {a,b} ] \to \mathbb{R}$$ be differentiable, then

$$f ( t ) = \frac{1}{{b - a}} \int _{a}^{b} {{f^{ \rho }} ( s )} \nabla s + \frac{1}{{b - a}} \int _{a}^{b} {p ( {t,s} ){f^{\nabla }} ( s )} \nabla s,$$
(1.4)

where

$$p ( {t,s} ) = \textstyle\begin{cases} s - a, & a < s \le t, \\ s - b, & t < s \le b. \end{cases}$$

Özkan and Yildrim gave the representation of functions depending on two variables in the form of delta integrals . In 2006, Sheng et al. introduced the combined dynamic derivative, also called alpha diamond dynamic derivative $$(\alpha \in [0,1])$$, as a linear convex combination of the delta and nabla dynamic derivatives on time scales . The aim of the paper is to extend Montgomery identity by using alpha diamond integrals  and to establish respective Ostrowski type inequality for alpha diamond integrals.

### Definition 1.2

Let $$\mathbb{T}$$ be a time scale and $$f ( t )$$ be differentiable on $$t \in \mathbb{T}$$ in delta and nabla senses. For $$t \in \mathbb{T}$$, we define the alpha diamond derivative $${f^{{\diamondsuit _{ \alpha }}}} ( t )$$ by

$${f^{{\diamondsuit _{\alpha }}}} ( t ) = \alpha {f^{\Delta }} ( t ) + ( {1 - \alpha } ){f^{\nabla }} ( t ) \quad \forall \alpha \in [ {0,1} ] .$$
(1.5)

Note that the alpha diamond derivative reduces to the standard delta derivative for $$\alpha = 1$$ and nabla derivative for $$\alpha = 0$$.

Properties of alpha diamond derivative

### Theorem 1.3

Let f and g be alpha diamond differentiable functions, then:

The sum $$f + g :{\mathbb{T}} \to {\mathbb{R}}$$ is alpha diamond differentiable at $$t \in {\mathbb{T,}}$$ satisfying

$${ ( {f + g} )^{{{\diamondsuit _{\alpha }}}}} ( t ) = {f^{{\diamondsuit _{\alpha }}}} ( t ) + {g^{ {\diamondsuit _{\alpha }}}} ( t ).$$

The product $$f g :{\mathbb{T}} \to {\mathbb{R}}$$ is alpha diamond differentiable at $$t \in {\mathbb{T}}$$, satisfying

$${ ( {f g} )^{{{\diamondsuit _{\alpha }}}}} ( t ) = {f^{{\diamondsuit _{\alpha }}}} ( t ).g ( t ) + \alpha {f^{\sigma }} ( t ){g^{\Delta }} ( t ) + ( {1 - \alpha } ){f^{\rho }} ( t ) {g^{\nabla }} ( t ).$$

If $$g ( t ){g^{\sigma }} ( t ) {g^{\rho }} ( t ) \ne 0, \frac{f}{g} :{\mathbb{T}} \to {\mathbb{R}}$$ is alpha diamond differentiable at $$t \in {\mathbb{T,}}$$ then

$${ \biggl( {\frac{f}{g}} \biggr)^{{\diamondsuit _{\alpha }}}} ( t ) = \frac{{{f^{{\diamondsuit _{\alpha }}}} ( t ) {g^{\sigma }} ( t ){g^{\rho }} ( t ) - \alpha {f^{\sigma }} ( t ){g^{\rho }} ( t ){g^{ \Delta }} ( t ) - ( {1 - \alpha } ){f^{ \rho }} ( t ){g^{\sigma }} ( t ){g^{\nabla }} ( t )}}{{g ( t ){g^{\sigma }} ( t ) {g^{\rho }} ( t )}}.$$

### Definition 1.4

Let $${a_{1}}, {a_{2}} \in \mathbb{T}$$ and $$f: {\mathbb{T}} \to {\mathbb{R}}$$, then for $$\alpha \in [ {0 , 1}]$$, the alpha diamond integral of f is defined by

$$\int _{{a_{1}}}^{{a_{2}}} {f ( t )} {\diamondsuit _{\alpha }} ( t ) = \alpha \int _{{a_{1}}}^{{a_{2}}} {f ( t )} \Delta t + ( {1 - \alpha } ) \int _{ {a_{1}}}^{{a_{2}}} {f ( t )} \nabla t,$$

provided delta and nabla integrals of f exist on $$\mathbb{T}$$.

Properties of alpha diamond integration

### Theorem 1.5

Let $$f, g:{\mathbb{T}} \to {\mathbb{R}}$$ be alpha diamond integrable on $$[ {a_{1},a_{2}} ]_{\mathbb{T}}$$. Let $$a_{3} \in { [ {a_{1},a_{2}} ]_{\mathbb{T}}}$$ with $$a_{1} < a_{3} < a_{2}$$ and $$\lambda \in {\mathbb{R}}$$, then

\begin{aligned} & \int _{{a_{1}}}^{{a_{1}}} {f}(t) {\diamondsuit _{\alpha }}t = 0; \\ & \int _{{a_{1}}}^{{a_{2}}} {{f}(t) {\diamondsuit _{\alpha }}t = \int _{{a_{1}}}^{{a_{3}}} {{f}(t) {\diamondsuit _{\alpha }}t + \int _{{a_{3}}}^{{a_{2}}} {{f}(t) {\diamondsuit _{\alpha }} t;} } } \\ & \int _{{a_{1}}}^{{a_{2}}} {f(t)} {\diamondsuit _{\alpha }}t = - \int _{{a_{2}}}^{{a_{1}}} {f(t){\diamondsuit _{\alpha }}t;} \\ & \int _{{a_{1}}}^{{a_{2}}} {({f} + {g}) (t) { \diamondsuit _{ \alpha }}t = \int _{{a_{1}}}^{{a_{2}}} {{f}(t) {\diamondsuit _{ \alpha }}(t) + \int _{{a_{1}}}^{{a_{2}}} {{g}(t) { \diamondsuit _{\alpha }}t;} } } \\ & \int _{{a_{1}}}^{{a_{2}}} \lambda {f}(t) { \diamondsuit _{\alpha }}t = \lambda \int _{{a_{1}}}^{{a_{2}}} {{f}(t) {\diamondsuit _{\alpha }}t.} \end{aligned}

The following result can be found in  and is used in the proof of next results.

#### Fubini’s theorem on time scales

Let $${\mathbb{T}_{1}}$$, $${\mathbb{T}_{2}}$$ be two time scales. Suppose that $$S:{\mathbb{T}_{1}} \times {\mathbb{T}_{2}} \to \mathbb{R}$$ is integrable with respect to both time scales. Define $$\phi ( {{y_{2}}} ) = \int _{{\mathbb{T}_{1}}} {S ( {{y_{1}}, {y_{2}}} )} \Delta {y_{1}}$$ for a.e. $${y_{2}} \in {\mathbb{T} _{2}}$$ and $$\psi ( {{y_{1}}} ) = \int _{{\mathbb{T}_{2}}} {S ( {{y_{1}}, {y_{2}}} )} \Delta {y_{2}}$$ for a.e. $${y_{1}} \in {\mathbb{T}_{1}}$$. Then ϕ and ψ are both differentiable in time scales settings and

$$\int _{{\mathbb{T}_{1}}} {\Delta {y_{1}} \int _{{\mathbb{T}_{2}}} {S ( {{y_{1}}, {y_{2}}} )} \Delta {y_{2}} = } \int _{{\mathbb{T}_{2}}} {\Delta {y_{2}} \int _{{\mathbb{T}_{1}}} {S ( {{y_{1}}, {y_{2}}} )} \Delta {y_{1}}}.$$

## Montgomery identities for function of two variables on time scales

### Lemma 2.1

()

Let $$m_{1},n_{1} \in {\mathbb{T}_{1}}$$, $$m_{2},n_{2} \in {\mathbb{T}_{2}}$$, and $${f} \in CC_{\mathrm{rd}}^{1} ( {{{ [ {{m_{1}},{n_{1}}} ]}_{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{{\mathbb{T}_{2}}}}, \mathbb{R}} )$$, then $$\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ] _{{\mathbb{T}_{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{ {\mathbb{T}_{2}}}}$$, we have the representation

\begin{aligned}[b] f(x,y) & = \frac{1}{k} \biggl[ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ), {\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{\Delta _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} } \frac{{\partial f ( {{\sigma _{1}} ( s ),t} )}}{{{\Delta _{2}}t}}{\Delta _{2}}t{\Delta _{1}}s \\ & \quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {p ( {x,s} )} } \frac{{\partial f ( {s, {\sigma _{2}} ( t )} )}}{{{\Delta _{1}}s}}{\Delta _{2}}t{\Delta _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} ) \frac{ {{\partial ^{2}}f ( {s,t} )}}{{{\Delta _{1}}s{\Delta _{2}}t}}} } {\Delta _{2}}t{\Delta _{1}}s \biggr], \end{aligned}
(2.1)

where $${p}: [ {{m_{1}},{n_{1}}} ] \times [ {{m_{1}}, {n_{1}}} ] \to \mathbb{R}$$ and $${q}: [ {{m_{2}},{n_{2}}} ] \times [ {{m_{2}},{n_{2}}} ] \to \mathbb{R}$$ are defined as follows:

$${p}(x,s) = \textstyle\begin{cases} s - {m_{1}},& \textit{if }s \in [{m_{1}},x], \\ s - {n_{1}},& \textit{if }s \in (x,{n_{1}}], \end{cases}\displaystyle \qquad {q}(y,t) = \textstyle\begin{cases} t - {m_{2}}, & \textit{if }t \in [{m_{2}},y], \\ t - {n_{2}}, & \textit{if }t \in (y,{n_{2}}], \end{cases}$$

and $$k = (n_{1}-m_{1})(n_{2}-m_{2})$$.

### Note

Throughout the paper, $$p(x,s)$$, $$q(y,t)$$, and k are as defined in Lemma 2.1.

### Lemma 2.2

Let $$m_{1},n_{1} \in {\mathbb{T}_{1}}$$, $$m_{2},n_{2} \in {\mathbb{T} _{2}}$$, and $${f} \in C{C^{1}} ( {{{ [ {{m_{1}},{n_{1}}} ]} _{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{ {\mathbb{T}_{2}}}},\mathbb{R}} )$$, then $$\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T}_{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{{\mathbb{T}_{2}}}}$$, we have the representation

\begin{aligned}[b] f ( {x,y} ) &= \frac{1}{k} \biggl[ { \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{\Delta _{1}}s} \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} ){f^{{\Delta _{1}}}} \bigl( {s,{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t { \Delta _{1}}s \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} ){f^{{\nabla _{2}}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \Delta _{1}}s \\ &\quad {}+ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} ){f^{{\nabla _{2}}{\Delta _{1}}}} ( {s,t} )} } {\nabla _{2}}t{\Delta _{1}}s} \biggr]. \end{aligned}
(2.2)

### Proof

Apply (1.3) to $$f(\cdot ,y)$$ for fixed $$y \in { [ {{m_{2}}, {n_{2}}} ]_{{\mathbb{T}_{2}}}}$$ to get

$$f ( {x,y} ) = \frac{1}{{n_{1} - m_{1}}} \int _{{m _{1}}}^{{n_{1}}}{f \bigl( {{\sigma _{1}} ( s ),y} \bigr) {\Delta _{1}}s + \frac{1}{{n_{1} - m_{1}}} \int _{{m_{1}}}^{{n _{1}}}{p ( {x,s} ){f^{{\Delta _{1}}}} ( {s,y} ) {\Delta _{1}}s} }.$$
(2.3)

Apply (1.4) to $$f(s,\cdot )$$ for fixed $$s \in { [ {{m_{1}}, {n_{1}}} ]_{{\mathbb{T}_{1}}}}$$ to obtain

$$f ( {s,y} ) = \frac{1}{{n_{2} - m_{2}}} \int _{{m _{2}}}^{{n_{2}}}{f \bigl( {s,{\rho _{2}} ( t )} \bigr)} {\nabla _{2}}t + \frac{1}{{n_{2} - m_{2}}} \int _{{m_{2}}}^{{n _{2}}}{q ( {y,t} ){f^{{\nabla _{2}}}} ( {s,t} ) {\nabla _{2}}t}.$$
(2.4)

Again apply (1.4) to $${f^{{\Delta _{1}}}} ( {s,\cdot } )$$ for $$s \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T}_{1}}}}$$ to get

$${f^{{\Delta _{1}}}} ( {s,y} ) = \frac{1}{{n_{2} - m_{2}}} \int _{{m_{2}}}^{{n_{2}}}{{f^{{\Delta _{1}}}} \bigl( {s,{ \rho _{2}} ( t )} \bigr)} {\nabla _{2}}t + \frac{1}{{n_{2} - m _{2}}} \int _{{m_{2}}}^{{n_{2}}}{q ( {y,t} )} {f^{ {\nabla _{2}}{\Delta _{1}}}} ( {s,t} ){\nabla _{2}}t.$$
(2.5)

Substitute (2.4) and (2.5) in (2.3), then use Fubini’s theorem to obtain

\begin{aligned} {f} ( {x,y} ) & = \frac{1}{{{n_{1}} - {m_{1}}}} \int _{{m_{1}}}^{{n_{1}}} \biggl[ { \frac{1}{{{n_{2}} - {m_{2}}}} \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl( {{\sigma _{1}} ( s ), {\rho _{2}} ( t )} \bigr)} {\nabla _{2}}t} \\ &\quad {}+ \frac{1}{{{n_{2}} - {m_{2}}}} \int _{{m_{2}}}^{{n _{2}}} {{q} ( {y,t} )f^{{\nabla _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} {\nabla _{2}}t \biggr]{\Delta _{1}}s \\ &\quad {} + \frac{1}{{{n_{1}} - {m_{1}}}} \int _{{m_{1}}}^{{n_{1}}} {p} ( {x,s} ) \biggl[ { \frac{1}{{{n_{2}} - {m_{2}}}} \int _{{m_{2}}}^{{n_{2}}} {f^{{\Delta _{1}}} \bigl( {s,{ \rho _{2}} ( t )} \bigr){\nabla _{2}}t} } \\ &\quad {} + \frac{1}{ {{n_{2}} - {m_{2}}}} \int _{{m_{2}}}^{{n_{2}}} {{q} ( {y,t} )} f^{{\nabla _{2}}{\Delta _{1}}} ( {s,t} ){\nabla _{2}}t \biggr]{\Delta _{1}}s, \end{aligned}

which gives the required result. □

### Lemma 2.3

Let $$m_{1},n_{1} \in {\mathbb{T}_{1}}$$, $$m_{2},n_{2} \in {\mathbb{T} _{2}}$$, and $${f} \in C{C^{1}} ( {{{ [ {{m_{1}},{n_{1}}} ]} _{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{ {\mathbb{T}_{2}}}}, \mathbb{R}} )$$, then $$\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T} _{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{{\mathbb{T}_{2}}}}$$, we have the representation

\begin{aligned}[b] f(x,y) & = \frac{{{1}}}{{{k}}} \biggl[ \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{{f}} \bigl( {{\rho _{1}} ( s ),{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{\nabla _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{{p}} ( {{{x,s}}} ) {{{f}}^{{\nabla _{1}}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr)} } { \Delta _{2}}t{\nabla _{1}}s \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {{{q}} ( {{{y,t}}} ){{{f}} ^{{\Delta _{2}}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{\nabla _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{{p}} ( {{{x,s}}} ) {{q}} ( {{{y,t}}} ){{{f}}^{{\nabla _{1}} {\Delta _{2}}}} ( {s,t} )} } { \Delta _{2}}t{ \nabla _{1}}s \biggr]. \end{aligned}
(2.6)

### Proof

It can be proved accordingly, by shuffling the roles of delta and nabla integrals in Lemma 2.2. □

### Lemma 2.4

Let $$m_{1},n_{1} \in {\mathbb{T}_{1}}$$, $$m_{2},n_{2} \in {\mathbb{T} _{2}}$$, and $${f}, {g} \in CC_{\mathrm{ld}}^{1} ( {{{ [ {{m_{1}}, {n_{1}}} ]}_{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}}, {n_{2}}} ]}_{{\mathbb{T}_{2}}}},\mathbb{R}} )$$, then $$\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ] _{{\mathbb{T}_{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{ {\mathbb{T}_{2}}}}$$, we have the representation

\begin{aligned} f(x,y) & = \frac{1}{k} \biggl[ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\rho _{1}} ( s ), {\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{\nabla _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} } \frac{{\partial f ( {{\rho _{1}} ( s ),t} )}}{{{\nabla _{2}}t}}{\nabla _{2}}t{\nabla _{1}}s \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {p ( {x,s} )} } \frac{{\partial f ( {s, {\rho _{2}} ( t )} )}}{{{\nabla _{1}}s}}{\nabla _{2}}t {\nabla _{1}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {p ( {x,s} )q ( {y,t} ) \frac{{{\partial ^{2}}f ( {s,t} )}}{{{\nabla _{1}}s{\nabla _{2}}t}}} } {\nabla _{2}}t {\nabla _{1}}s \biggr]. \end{aligned}
(2.7)

### Proof

It can be proved with the help of (1.4) and nabla derivatives and integrals with respect to both variables, instead of (1.3), accordingly as Lemma 2.2. □

### Theorem 2.5

Let $$m_{1},n_{1} \in {\mathbb{T}_{1}}$$, $$m_{2},n_{2} \in {\mathbb{T} _{2}}$$, and $${f} \in C{C^{1}} ( {{{ [ {{m_{1}},{n_{1}}} ]} _{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{ {\mathbb{T}_{2}}}}, \mathbb{R}} )$$, then $$\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T} _{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{{\mathbb{T}_{2}}}}$$, we have

\begin{aligned}[b] k{f}(x,y) &= \biggl[ {\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl({\sigma _{1}}(s), {\sigma _{2}}(t) \bigr){\Delta _{2}}t {\Delta _{1}}s} } \\ &\quad {}+ {\alpha _{1}}(1 - {\alpha _{2}}) \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl({\sigma _{1}}(s), {\rho _{2}}(t) \bigr){\nabla _{2}}t {\Delta _{1}}s} } \\ &\quad {} + {\alpha _{2}}(1 - {\alpha _{1}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl({\rho _{1}}(s),{\sigma _{2}}(t) \bigr) {\Delta _{2}}t {\nabla _{1}}s} } \\ &\quad {}+ (1 - {\alpha _{1}}) (1 - {\alpha _{2}}) { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl({\rho _{1}}(s),{\rho _{2}}(t) \bigr){\nabla _{2}}t {\nabla _{1}}s} } } \biggr] \\ &\quad {} + \biggl[ {{\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s)f^{{\diamondsuit _{{\alpha _{1}}}}} \bigl(s, {\sigma _{2}}(t) \bigr){\Delta _{2}}t { \diamondsuit _{{\alpha _{1}}}}s} } } \\ &\quad {}+ (1 - {\alpha _{2}}) \int _{{m_{1}}}^{{n _{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s)f^{{\diamondsuit _{{\alpha _{1}}}}} \bigl(s, {\rho _{2}}(t) \bigr){\nabla _{2}}t { \diamondsuit _{{\alpha _{1}}}}s} } \biggr] \\ &\quad {} + \biggl[ {\alpha _{1}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q}(y,t)f^{{\diamondsuit _{\alpha 2}}} \bigl( {\sigma _{1}}(s),t \bigr){\Delta _{2}}t{ \diamondsuit _{{\alpha _{1}}}}s} } \\ &\quad {}+ (1 - {\alpha _{1}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q}(y,t)f^{{\diamondsuit _{{\alpha _{2}}}}} \bigl({\rho _{1}}(s),t \bigr){\nabla _{2}}t { \diamondsuit _{{\alpha _{1}}}}s} } \biggr] \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s){q}(y,t)f^{{\diamondsuit _{{\alpha _{1}}}}{\diamondsuit _{\alpha _{2}}}}(s,t) {\diamondsuit _{{\alpha _{2}}}}t {\diamondsuit _{{\alpha _{1}}}}s}. \end{aligned}
(2.8)

### Proof

Multiply (2.1) by $${\alpha _{1}}{\alpha _{2}}$$, (2.2) by $${\alpha _{1}} ( {1 - {\alpha _{2}}} )$$, (2.6) by $${\alpha _{2}} ( {1 - {\alpha _{1}}} )$$, and (2.7) by $$( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} )$$, then add the resultants to obtain

\begin{aligned} k{f}(x,y) =& {\alpha _{1}} {\alpha _{2}} \biggl[ \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{\Delta _{1}}s \\ &{}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} )f^{{\Delta _{1}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t { \Delta _{1}}s \\ &{} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )f^{{\Delta _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{ \Delta _{1}}s \\ &{}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} )f^{{\Delta _{1}}{\Delta _{2}}} ( {s,t} )} } {\Delta _{2}}t{\Delta _{1}}s \biggr] \\ &{} + {\alpha _{1}} ( {1 - {\alpha _{2}}} ) \biggl[ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl( {{\sigma _{1}} ( s ),{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{\Delta _{1}}s} \\ &{}+ \int _{{m _{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} )f ^{{\Delta _{1}}} \bigl( {s,{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{ \Delta _{1}}s \\ &{} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )f^{{\nabla _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \Delta _{1}}s \\ &{}+ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} )f^{{\Delta _{1}}{\nabla _{2}}} ( {s,t} )} } {\nabla _{2}}t{\Delta _{1}}s} \biggr] \\ &{} + {\alpha _{2}} ( {1 - {\alpha _{1}}} ) \biggl[ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{f} \bigl( {{\rho _{1}} ( s ),{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{\nabla _{1}}} s\biggr] \\ &{}+ \int _{{m _{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )f ^{{\nabla _{1}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr)} } {\Delta _{2}}t{ \nabla _{1}}s \\ &{} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )f^{{\Delta _{2}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{ \nabla _{1}}s \\ &{}+ \int _{ {m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} )f^{{\nabla _{1}}{\Delta _{2}}} ( {s,t} )} } {\Delta _{2}}t{\nabla _{1}}s \\ &{} + ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) \biggl[ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{ {n_{2}}} {{f} \bigl( {{\rho _{1}} ( s ),{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{\nabla _{1}}s} \\ &{}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} )f^{{\nabla _{1}}} \bigl( {s,{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{ \nabla _{1}}s \\ &{} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )f^{{\nabla _{2}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \nabla _{1}}s \\ &{}+ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} )f^{{\nabla _{1}}{\nabla _{2}}} ( {s,t} )} } {\nabla _{2}}t{\nabla _{1}}s} \biggr] \\ = & \biggl[ {{\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl({\sigma _{1}}(s),{\sigma _{2}}(t) \bigr) {\Delta _{2}}t {\Delta _{1}}s} } } \\ &{}+{\alpha _{1}}(1 - {\alpha _{2}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {f \bigl({\sigma _{1}}(s),{\rho _{2}}(t) \bigr){\nabla _{2}}t {\Delta _{1}}s} } \\ &{}+{\alpha _{2}}(1 - {\alpha _{1}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl({\rho _{1}}(s),{\sigma _{2}}(t) \bigr) {\Delta _{2}}t {\nabla _{1}}s} } \\ &{}+ (1 - { \alpha _{1}}) (1 - {\alpha _{2}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {f \bigl({\rho _{1}}(s),{\rho _{2}}(t) \bigr){\nabla _{2}}t {\nabla _{1}}s} } \biggr] \\ &{} + \biggl[ {{\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s)f^{{\Delta _{1}}} \bigl(s,{\sigma _{2}}(t) \bigr){\Delta _{2}}t { \Delta _{1}}s} } } \\ &{}+ {\alpha _{1}}(1 - {\alpha _{2}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {{p}(x,s)f^{{\Delta _{1}}} \bigl(s,{\rho _{2}}(t) \bigr){\nabla _{2}}t { \Delta _{1}}s} } \\ &{} + {\alpha _{2}}(1 - {\alpha _{1}}) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p}(x,s)f^{{\nabla _{1}}} \bigl(s,{\sigma _{2}}(t) \bigr){\Delta _{2}}t { \nabla _{1}}s} } \\ &{}+ ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} )f^{{\nabla _{1}}} \bigl( {s,{\rho _{2}} ( t )} \bigr)} } {\nabla _{2}}t{ \nabla _{1}}s \biggr] \\ &{} + \biggl[ {{\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q} ( {y,t} )f^{{\Delta _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{ \Delta _{1}}s} \\ &{}+ {\alpha _{1}} ( {1 - {\alpha _{2}}} ) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n _{2}}} {{q} ( {y,t} )f^{{\nabla _{2}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \Delta _{1}}s \\ &{} + {\alpha _{2}} ( {1 - {\alpha _{1}}} ) \int _{ {m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q} ( {y,t} )f^{{\Delta _{2}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\Delta _{2}}t{ \nabla _{1}}s \\ &{}+ ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{q} ( {y,t} )f ^{{\nabla _{2}}} \bigl( {{\rho _{1}} ( s ),t} \bigr)} } {\nabla _{2}}t{ \nabla _{1}}s \biggr] \\ &{} + \biggl[ {{\alpha _{1}} {\alpha _{2}} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} ){q} ( {y,t} )f^{{\Delta _{1}}{\Delta _{2}}} ( {s,t} )} } {\Delta _{2}}t} {\Delta _{1}}s \\ &{}+ {\alpha _{1}} ( {1 - {\alpha _{2}}} ) \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {{p} ( {x,s} ){q} ( {y,t} )f^{ {\Delta _{1}}{\nabla _{2}}} ( {s,t} )} } {\nabla _{2}}t{\Delta _{1}}s \\ &{} + {\alpha _{2}} ( {1 - {\alpha _{1}}} ) \int _{ {m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {{p} ( {x,s} ){q} ( {y,t} )f^{{\nabla _{1}}{\Delta _{2}}} ( {s,t} )} } {\Delta _{2}}t{\nabla _{1}}s \\ &{} + ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{ {n_{2}}} {{p} ( {x,s} ){q} ( {y,t} )f^{{\nabla _{1}}{\nabla _{2}}} ( {s,t} )} } {\nabla _{2}}t{\nabla _{1}}s} \biggr], \end{aligned}

and after simplification one gets the required result. □

### Example 2.6

For $$\mathbb{T}_{1}=h_{1}\mathbb{N}$$, $$\mathbb{T}_{2}=h_{2}\mathbb{N}$$, $$h _{1},h_{2}>0$$, (2.8) can be written as follows:

\begin{aligned} k{f} ( {x,y} ) =& {\alpha _{1}} {\alpha _{2}}\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{ \frac{{{n_{2}}}}{{{h_{2}}}} - 1} {{f} \bigl( { ( {i + 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \\ &{} + {\alpha _{1}}(1 - {\alpha _{2}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{{{n_{2}}}}{{{h _{2}}}}} {{f} \bigl( { ( {i + 1} ){h_{1}}, ( {i - 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \\ &{} + {\alpha _{2}}(1 - {\alpha _{1}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} { \sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{\frac{{{n _{2}}}}{{{h_{2}}}} - 1} {{f} \bigl( { ( {i - 1} ){h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \\ &{} + (1 - {\alpha _{1}}) (1 - {\alpha _{2}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{ {{n_{2}}}}{{{h_{2}}}}} {{f} \bigl( { ( {i - 1} ){h_{1}}, ( {i - 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \\ &{} + {\alpha _{2}} \Biggl[ {{\alpha _{1}}\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{ \frac{{{n_{2}}}}{{{h_{2}}}} - 1} {p(x,{i + 1})f ^{\diamondsuit {\alpha _{1}}} \bigl( { ( {i + 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } } \\ &{} + (1 - {\alpha _{1}})\sum_{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{\frac{{{n _{2}}}}{{{h_{2}}}} - 1} {p(x,{i + 1}) f ^{\diamondsuit {\alpha _{1}}} \bigl( { ( {i - 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \Biggr] \\ &{} + (1 - {\alpha _{2}}) \Biggl[ {{\alpha _{1}} \sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{{{n_{2}}}}{{{h _{2}}}}} {p(x,{i - 1})f ^{\diamondsuit {\alpha _{1}}} \bigl( { ( {i + 1} ){h_{1}}, ( {i - 1} ) {h_{2}}} \bigr){h_{1}} {h_{2}}} } } \\ &{} + (1 - {\alpha _{1}})\sum_{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{ {{n_{2}}}}{{{h_{2}}}}} {p(x,{i - 1}) f ^{\diamondsuit {\alpha _{1}}} \bigl( { ( {i - 1} ) {h_{1}}, ( {i - 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \Biggr] \\ &{} + {\alpha _{1}} \Biggl[ {{\alpha _{2}}\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{ \frac{{{n_{2}}}}{{{h_{2}}}} - 1} {q(y,{i + 1})f ^{\diamondsuit {\alpha _{2}}} \bigl( { ( {i + 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } } \\ &{} + (1 - {\alpha _{2}})\sum_{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{{{n_{2}}}}{{{h _{2}}}}} {q(y,{i + 1}) f ^{\diamondsuit {\alpha _{2}}} \bigl( { ( {i + 1} ){h_{1}}, ( {i - 1} ) {h_{2}}} \bigr){h_{1}} {h_{2}}} } \Biggr] \\ &{} + (1 - {\alpha _{1}}) \Biggl[ {{\alpha _{2}} \sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{\frac{{{n _{2}}}}{{{h_{2}}}} - 1} {q(y,{i - 1})f ^{\diamondsuit {\alpha _{2}}} \bigl( { ( {i - 1} ) {h_{1}}, ( {i + 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } } \\ &{} + (1 - {\alpha _{2}})\sum_{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{ {{n_{2}}}}{{{h_{2}}}}} {q(y,{i - 1}) f ^{\diamondsuit {\alpha _{2}}} \bigl( { ( {i - 1} ) {h_{1}}, ( {i - 1} ){h_{2}}} \bigr){h_{1}} {h_{2}}} } \Biggr] \\ &{} + {\alpha _{1}} {\alpha _{2}}\sum _{i = \frac{{{m_{1}}}}{{{h _{1}}}}}^{\frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{ \frac{{{n_{2}}}}{{{h_{2}}}} - 1} {p(x,{i + 1})q(y,{i + 1})f ^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}} \bigl( { ( {i + 1} ){h_{1}}, ( {i + 1} ){h_{2}}} \bigr) {h_{1}} {h_{2}}} } \\ &{} + {\alpha _{1}}(1 - {\alpha _{2}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}}}^{ \frac{{{n_{1}}}}{{{h_{1}}}} - 1} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{{{n_{2}}}}{{{h _{2}}}}} {p(x,{i + 1})q(y,{i - 1})f ^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}} \bigl( { ( {i + 1} ){h_{1}}, ( {i - 1} ){h_{2}}} \bigr) {h_{1}} {h_{2}}} } \\ &{} + {\alpha _{2}}(1 - {\alpha _{1}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} {\sum_{j = \frac{{{m_{2}}}}{{{h_{2}}}}}^{\frac{{{n _{2}}}}{{{h_{2}}}} - 1} {p(x,{i - 1})q(y,{i + 1})f ^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}} \bigl( { ( {i - 1} ){h_{1}}, ( {i + 1} ){h_{2}}} \bigr) {h_{1}} {h_{2}}} } \\ &{} + (1 - {\alpha _{1}}) (1 - {\alpha _{2}})\sum _{i = \frac{{{m_{1}}}}{{{h_{1}}}} + 1}^{\frac{{{n_{1}}}}{{{h _{1}}}}} \sum _{j = \frac{{{m_{2}}}}{{{h_{2}}}} + 1}^{\frac{ {{n_{2}}}}{{{h_{2}}}}} p(x,{i - 1})q(y,{i - 1}) \\ &{}\times f ^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}} \bigl( { ( {i - 1} ){h_{1}}, ( {i - 1} ){h_{2}}} \bigr) {h_{1}} {h_{2}} . \end{aligned}

### Example 2.7

If $${\mathbb{T}_{1}} = q_{1}^{\mathbb{N}}$$, $${\mathbb{T}_{2}} = q_{2} ^{\mathbb{N}}$$, $${q_{1}},{q_{2}} > 1$$, and $$a_{1} = q_{1}^{{m_{1}}}$$, $$b_{1} = q_{1}^{{n_{1}}}$$, $$c_{1} = q_{2}^{{m_{2}}}$$, and $$d_{1} = q _{2}^{{n_{2}}}$$, then for $${m_{1}}, {m_{2}}, {n_{1}}, {n_{2}} \in \mathbb{ N}$$, (2.8) takes the form

\begin{aligned}& \bigl( {q_{1}^{{n_{1}}} - q_{1}^{{m_{1}}}} \bigr) \bigl( {q_{2} ^{{n_{2}}} - q_{2}^{{m_{2}}}} \bigr){f}(x,y) \\& \quad = ({q_{1}} - 1) ( {q_{2}} - 1) \Biggl[ {{\alpha _{1}} {\alpha _{2}}\sum_{{k_{1}} = {m_{1}}}^{{n_{1}} - 1} {\sum_{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {{f} \bigl(q_{1}^{{k_{1}} + 1},q_{2}^{{k_{2}} + 1}} } \bigr)} q_{1}^{{k_{1}}}q_{2}^{{k_{2}}} \\& \qquad {} + {\alpha _{1}} ( {1 - {\alpha _{2}}} )\sum _{{k_{1}} = {m_{1}}}^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {{f} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}}}q_{2}^{{k_{2}} - 1} \\& \qquad {} + {\alpha _{2}} ( {1 - {\alpha _{1}}} )\sum _{{k_{1}} = {m_{1}} + 1}^{{n_{1}}} {\sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {{f} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} + 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}}} \\& \qquad {} + ( {1 - {\alpha _{1}}} ) ( {1 - {\alpha _{2}}} ) \sum_{{k_{1}} = {m_{1}} + 1}^{{n_{1}}} { \sum_{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {{f} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}} - 1} \\& \qquad {} + {\alpha _{2}} \Biggl\{ {{\alpha _{1}}\sum _{{k_{1}} = {m_{1}}} ^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {p \bigl(x,q _{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{1}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{{k_{2}} + 1} \bigr)} q_{1}^{{k_{1}}}q_{2}^{{k_{2}}} \\& \qquad {}+ {(1 - {\alpha _{1}})\sum_{{k_{1}} = {m_{1}} + 1} ^{{n_{1}}} {\sum_{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {p \bigl(x,q_{2} ^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{1}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} + 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}}}} \Biggr\} \\& \qquad {} + (1 - {\alpha _{2}}) \Biggl\{ {{\alpha _{1}} \sum_{{k_{1}} = {m_{1}}}^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {p \bigl(x,q_{2}^{{k_{2}} - 1} \bigr) {{f}^{\diamondsuit {\alpha _{1}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{ {k_{2}} - 1} \bigr)} q_{1}^{{k_{1}}}q_{2}^{{k_{2}} - 1} \\& \qquad {} + {(1 - {\alpha _{1}})\sum_{{k_{1}} = {m_{1}} + 1} ^{{n_{1}}} {\sum_{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {p \bigl(x,q_{2} ^{{k_{2}} - 1} \bigr){{f}^{\diamondsuit {\alpha _{1}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}} - 1}} \Biggr\} \\& \qquad {} + {\alpha _{1}} \Biggl\{ {{\alpha _{2}}\sum _{{k_{1}} = {m_{1}}} ^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {q \bigl(y,q _{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{{k_{2}} + 1} \bigr)} q_{1}^{{k_{1}}}q_{2}^{{k_{2}}} \\& \qquad {} + {(1 - {\alpha _{2}})\sum_{{k_{1}} = {m_{1}}}^{ {n_{1}} - 1} {\sum_{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {q \bigl(y,q _{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}}}q_{2}^{{k_{2}} - 1}} \Biggr\} \\& \qquad {} + (1 - {\alpha _{1}}) \Biggl\{ {{\alpha _{2}} \sum_{{k_{1}} = {m_{1}} + 1}^{{n_{1}}} {\sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {q \bigl(y,q_{2}^{{k_{2}} - 1} \bigr){{f}^{\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} + 1} \bigr)} q _{1}^{{k_{1}} - 1}q_{2}^{{k_{2}}} \\& \qquad {} + {(1 - {\alpha _{2}})\sum_{{k_{1}} = {m_{1}} + 1} ^{{n_{1}}} {\sum_{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {q \bigl(y,q_{2} ^{{k_{2}} - 1} \bigr){{f}^{\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}} - 1}} \Biggr\} \\& \qquad {} + {\alpha _{1}} {\alpha _{2}}\sum _{{k_{1}} = {m_{1}}}^{{n_{1}} - 1} {\sum_{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {p \bigl(x,q_{1}^{{k _{1}} + 1} \bigr)q \bigl(y,q_{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{1}} \diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q_{2} ^{{k_{2}} + 1} \bigr)q_{1}^{{k_{1}}}q_{2}^{{k_{2}}} \\& \qquad {} + {\alpha _{1}}(1 - {\alpha _{2}})\sum _{{k_{1}} = {m_{1}}} ^{{n_{1}} - 1} {\sum _{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {p \bigl(x,q _{1}^{{k_{1}} + 1} \bigr)q \bigl(y,q_{2}^{{k_{2}} - 1} \bigr)){{f}^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} + 1},} } q _{2}^{{k_{2}} - 1} \bigr)q_{1}^{{k_{1}}}q_{2}^{{k_{2}} - 1} \\& \qquad {} + {\alpha _{2}}(1 - {\alpha _{1}})\sum _{{k_{1}} = {m_{1}} + 1} ^{{n_{1}}} { \sum _{{k_{2}} = {m_{2}}}^{{n_{2}} - 1} {p \bigl(x,q_{1} ^{{k_{1}} - 1} \bigr)q \bigl(y,q_{2}^{{k_{2}} + 1} \bigr){{f}^{\diamondsuit {\alpha _{1}} \diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2} ^{{k_{2}} + 1} \bigr)q_{1}^{{k_{1}} - 1}q_{2}^{{k_{2}}} \\& \qquad {} + {(1 - {\alpha _{1}}) (1 - {\alpha _{2}}) \sum_{{k_{1}} = {m_{1}} + 1}^{{n_{1}}} {\sum _{{k_{2}} = {m_{2}} + 1}^{{n_{2}}} {p \bigl(x,q_{1}^{{k_{1}} - 1} \bigr)q \bigl(y,q _{2}^{{k_{2}} - 1} \bigr){{f}^{\diamondsuit {\alpha _{1}}\diamondsuit {\alpha _{2}}}} \bigl(q_{1}^{{k_{1}} - 1},} } q_{2}^{{k_{2}} - 1} \bigr)} \Biggr]. \end{aligned}

### Remark 2.8

For $$\mathbb{T}_{1}=\mathbb{T}_{2}=\mathbb{R}$$, (2.8) coincides with (1.2).

### Corollary 2.9

In addition to the conditions of Theorem 2.5, if the function is monotonically increasing, then we have the following inequality:

\begin{aligned}[b] f ( {x,y} ) &\le \frac{1}{k} \biggl[ { \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr)} } } { \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \\ &\quad {}+ \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )} } {} {f^{{\diamondsuit _{{\alpha _{1}}}}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr){ \diamondsuit _{{\alpha _{2}}}}t {\diamondsuit _{{\alpha _{1}}}}s \\ &\quad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} } {f^{{\diamondsuit _{{\alpha _{2}}}}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr){ \diamondsuit _{{\alpha _{2}}}}t {\diamondsuit _{{\alpha _{1}}}}s \\ &\quad {}+ \int _{{m_{1}}}^{ {n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )} } q ( {y,t} ){f^{{\diamondsuit _{{\alpha _{2}}}}{\diamondsuit _{{\alpha _{1}}}}}} ( {s,t} ){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \biggr]. \end{aligned}
(2.9)

### Proof

Since f is increasing, $${\rho _{i}} ( {{t_{i}}} ) \le {t_{i}} \le {\sigma _{i}} ( {{t_{i}}} )$$, $$\forall {t_{i}} \in {\mathbb{T}_{i}}$$, therefore by replacing $${\rho _{i}}$$ with $${\sigma _{i}}$$, on the right-hand side of (2.8), one gets the required result. □

## Ostrowski type inequality

### Theorem 3.1

Let $$m_{1},n_{1} \in {\mathbb{T}_{1}}$$, $$m_{2},n_{2} \in {\mathbb{T} _{2}}$$, $${f} \in C{C^{1}} ( {{{ [ {{m_{1}},{n_{1}}} ]} _{{\mathbb{T}_{1}}}} \times {{ [ {{m_{2}},{n_{2}}} ]}_{ {\mathbb{T}_{2}}}}, \mathbb{R}} )$$, further assume $$f(\cdot , \cdot )$$ is an increasing function with respect to both parameters, then $$\forall ( {x,y} ) \in { [ {{m_{1}},{n_{1}}} ] _{{\mathbb{T}_{1}}}} \times { [ {{m_{2}},{n_{2}}} ]_{ {\mathbb{T}_{2}}}}$$, one gets

\begin{aligned} & \biggl\vert {f ( {x,y} ) - \frac{1}{k} \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \\ &\quad \le \frac{1}{{ ( {{n_{1}} - {m_{1}}} )}}{M_{1}} \bigl[ { \hat{{h_{2}}} ( {x,{m_{1}}} ) - \hat{{h_{2}}} ( {x,{n_{1}}} )} \bigr] + \frac{1}{{ ( {{n_{2}} - {m_{2}}} )}}{M_{2}} \bigl[ {\hat{{h_{2}}} ( {y,{m_{2}}} ) - \hat{{h_{2}}} ( {y,{n_{2}}} )} \bigr] \\ &\qquad {}+ \frac{1}{k}{M_{3}} \bigl[ { \hat{{h_{2}}} ( {x,{m_{1}}} ) - \hat{{h_{2}}} ( {x, {n_{1}}} )} \bigr] \bigl[ { \hat{{h_{2}}} ( {y,{m_{2}}} ) - \hat{{h_{2}}} ( {y,{n_{2}}} )} \bigr], \end{aligned}

where

\begin{aligned}& \hat{{h_{2}}} ( {x,{m_{1}}} ) = \int _{{m_{1}}}^{x} { ( {s - {m_{1}}} )} {\diamondsuit _{{\alpha _{1}}}}s,\qquad \hat{{h_{2}}} ( {x,{n_{1}}} ) = \int _{{n_{1}}}^{x} { ( {s - {n_{1}}} )} {\diamondsuit _{{\alpha _{1}}}}s, \\& \hat{{h_{2}}} ( {y,{m_{2}}} ) = \int _{{m_{2}}}^{y} { ( {t - {m_{2}}} )} {\diamondsuit _{{\alpha _{2}}}}t,\qquad \hat{{h_{2}}} ( {y,{n_{2}}} ) = \int _{{n_{2}}}^{y} { ( {t - {n_{2}}} )} {\diamondsuit _{{\alpha _{2}}}}t. \end{aligned}

### Proof

Inequality (2.9) can be written as

\begin{aligned} &f ( {x,y} ) - \frac{1}{k} \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ), {\sigma _{2}} ( t )} \bigr)} } { \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \\ & \quad \le \frac{1}{k} \biggl[ { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )} } {f^{{\diamondsuit _{{\alpha _{1}}}}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr){ \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} \\ &\qquad {} + \int _{{m _{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} } {f^{{\diamondsuit _{{\alpha _{2}}}}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr){ \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \\ & \qquad {} + \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}} ^{{n_{2}}} {p ( {x,s} )} } q ( {y,t} ){f^{ {\diamondsuit _{{\alpha _{2}}}}{\diamondsuit _{{\alpha _{1}}}}s}} ( {s,t} ){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s \biggr]. \end{aligned}

By taking absolute value on both sides, one gets

\begin{aligned} & \biggl\vert {f ( {x,y} ) - \frac{1}{k} \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \\ &\quad \le \frac{1}{k} \biggl[ {{M_{1}} \biggl\vert { \int _{{m_{1}}}^{ {n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} ) {\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert } + {M_{2}} \biggl\vert { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} ){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \\ &\qquad {} + {M_{3}} \biggl\vert { \int _{{m_{1}}}^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {p ( {x,s} )q ( {y,t} ) { \diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \biggr], \end{aligned}

where

\begin{aligned}& {M_{1}} = \mathop{\operatorname{Sup}}_{{m_{1}} < s < {n_{1}}} \bigl\vert {{f ^{{\diamondsuit _{{\alpha _{1}}}}}} \bigl( {s,{\sigma _{2}} ( t )} \bigr)} \bigr\vert , \qquad {M_{2}} =\mathop{\operatorname{Sup}} _{{m_{2}} < t < {n_{2}}} \bigl\vert {{f^{{\diamondsuit _{{\alpha _{2}}}}}} \bigl( {{\sigma _{1}} ( s ),t} \bigr)} \bigr\vert \quad \mbox{and} \\& {M_{3}} = \mathop{\operatorname{Sup}} _{{m_{1}} < s < {n_{1}}, {m_{2}} < t < {n_{2}}} \bigl\vert {{f^{{\diamondsuit _{{\alpha _{1}}}}{\diamondsuit _{{\alpha _{2}}}}}} ( {s,t} )} \bigr\vert , \end{aligned}

which gives

\begin{aligned} & \biggl\vert {f ( {x,y} ) - \frac{1}{k} \int _{{m_{1}}} ^{{n_{1}}} { \int _{{m_{2}}}^{{n_{2}}} {f \bigl( {{\sigma _{1}} ( s ),{\sigma _{2}} ( t )} \bigr){\diamondsuit _{{\alpha _{2}}}}t{\diamondsuit _{{\alpha _{1}}}}s} } } \biggr\vert \\ &\quad \le \frac{1}{k} \biggl[ {{{M_{1}}} ( {{n_{2}} - {m_{2}}} )} \int _{{m_{1}}}^{{n_{1}}} {p ( {x,s} )} {\diamondsuit _{{\alpha _{1}}}}s + {M_{2}} ( {{n_{1}} - {m_{1}}} ) \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} {\diamondsuit _{{\alpha _{2}}}}t \\ &\qquad {} + {M_{3}} \int _{{m_{1}}}^{{n_{1}}} {p ( {x,s} )} {\diamondsuit _{{\alpha _{1}}}}s \int _{{m_{2}}}^{{n_{2}}} {q ( {y,t} )} {\diamondsuit _{{\alpha _{2}}}}t \biggr] \\ &\quad = \frac{1}{{ ( {{n_{1}} - {m_{1}}} )}}{M_{1}} \bigl[ {\hat{{h_{2}}} ( {x,{m_{1}}} ) - \hat{{h_{2}}} ( {x,{n_{1}}} )} \bigr] + \frac{1}{{ ( {{n_{2}} - {m_{2}}} )}}{M_{2}} \bigl[ { \hat{{h_{2}}} ( {y,{m_{2}}} ) - \hat{{h_{2}}} ( {y,{n_{2}}} )} \bigr] \\ & \qquad {}+ \frac{1}{k}{M_{3}} \bigl[ { \hat{{h_{2}}} ( {x,{m_{1}}} ) - \hat{{h_{2}}} ( {x,{n_{1}}} )} \bigr] \bigl[ { \hat{{h_{2}}} ( {y,{m_{2}}} ) - \hat{{h_{2}}} ( {y,{n_{2}}} )} \bigr], \end{aligned}

which is the required result. □

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### Acknowledgements

The authors are thankful to the editor and referees for their fruitful time to read and make valuable suggestions to the article.

### Availability of data and materials

The datasets used or analysed during the current study are cited in references and are available from the corresponding author on reasonable request.

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All the authors equally participated to complete this research article. They read and approved the final manuscript.

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Correspondence to Ammara Nosheen.

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