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Bivariate Montgomery identity for alpha diamond integrals
Advances in Difference Equations volume 2019, Article number: 314 (2019)
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.
1 Introduction
Mitrinovic, Pecaric, and Fink proved Montgomery identities for functions defined on the real line [17] 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
where \(p ( {x,t} )\) is called Peano kernel, defined in [17]. 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 [12] in the following form:
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 [12].
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 [7].
Let \(a,b,s,t \in \mathbb{T}\); \(a < b\) and \(f: [ {a,b} ] \to \mathbb{R}\) be differentiable, then
where
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
where
Özkan and Yildrim gave the representation of functions depending on two variables in the form of delta integrals [18]. 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 [19]. The aim of the paper is to extend Montgomery identity by using alpha diamond integrals [2] and to establish respective Ostrowski type inequality for alpha diamond integrals.
1.1 Preliminaries
1.1.1 Alpha diamond derivative [19]
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
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
The product \(f g :{\mathbb{T}} \to {\mathbb{R}}\) is alpha diamond differentiable at \(t \in {\mathbb{T}}\), satisfying
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
1.1.2 Alpha diamond integration [2]
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
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
The following result can be found in [4] and is used in the proof of next results.
1.1.3 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
2 Montgomery identities for function of two variables on time scales
2.1 Montgomery identity I
Lemma 2.1
([18])
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
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:
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.
2.2 Montgomery identity II
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
Proof
Apply (1.3) to \(f(\cdot ,y)\) for fixed \(y \in { [ {{m_{2}}, {n_{2}}} ]_{{\mathbb{T}_{2}}}}\) to get
Apply (1.4) to \(f(s,\cdot )\) for fixed \(s \in { [ {{m_{1}}, {n_{1}}} ]_{{\mathbb{T}_{1}}}}\) to obtain
Again apply (1.4) to \({f^{{\Delta _{1}}}} ( {s,\cdot } )\) for \(s \in { [ {{m_{1}},{n_{1}}} ]_{{\mathbb{T}_{1}}}}\) to get
Substitute (2.4) and (2.5) in (2.3), then use Fubini’s theorem to obtain
which gives the required result. □
2.3 Montgomery identity III
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
Proof
It can be proved accordingly, by shuffling the roles of delta and nabla integrals in Lemma 2.2. □
2.4 Montgomery identity IV
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
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. □
2.5 Montgomery identity for alpha diamond integrals
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
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
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:
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
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:
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. □
3 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
where
Proof
Inequality (2.9) can be written as
By taking absolute value on both sides, one gets
where
which gives
which is the required result. □
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Ahmad, M., Awan, K.M., Hameed, S. et al. Bivariate Montgomery identity for alpha diamond integrals. Adv Differ Equ 2019, 314 (2019). https://doi.org/10.1186/s13662-019-2254-6
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DOI: https://doi.org/10.1186/s13662-019-2254-6
MSC
- 26D15
- 39A13
- 34N05
Keywords
- Time scales
- Montgomery identity
- Delta integrals
- Nabla integrals
- Alpha diamond integrals