Citation: Humaira Kalsoom, Muhammad Idrees, Artion Kashuri, Muhammad Uzair Awan, Yu-Ming Chu. Some New (p1p2,q1q2)-Estimates of Ostrowski-type integral inequalities via n-polynomials s-type convexity[J]. AIMS Mathematics, 2020, 5(6): 7122-7144. doi: 10.3934/math.2020456
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Calculus is a branch of mathematics which helps us to study the derivatives and integrals. The classical derivative was convoluted with the strength regulation kind kernel and eventually, this gave upward thrust to new calculus referred to as the quantum calculus. Quantum calculus (named q-calculus) is the study of calculus without limits. In recent decades, the quantum calculus has become a powerful tool in numerous branches of mathematics and physics like q-calculus, particularly q-fractional calculus, q-integral calculus, q-transform analysis. Jackson [1] is the first researcher to define the q-analogue of derivative and integral operator as well as provided its applications. It is imperative to mention that quantum integral inequalities are more practical and informative than their classical counterparts. It has been mainly due to the fact that quantum integral inequalities can describe the hereditary properties of the processes and phenomena under investigation. Historically the subject of quantum calculus can be traced back to Euler and Jacobi, but in recent decades it has experienced a rapid development, see [2,3,4,5,6,7]. As a result, new generalizations of the classical concepts of quantum calculus have been initiated and reviewed in many literature. Tariboon and Ntouyas [8,9] proposed the quantum calculus concepts on finite intervals and obtained several q-analogues of classical mathematical objects. This inspired other researchers to establish numerous novel results concerning quantum analogs of classical mathematical results. Noor et al. [10] provided the q-analogues of many known inequalities via the first order q-differentiable convex functions. Humaira et al. [11] obtained a new generalized q1q2-integral identity and established several new q1q2-analogues of first order q1q2-differentiable convex functions over finite rectangles. Wu et al. [12] gave a new corrected q-analogue of the classical Simpson inequality for preinvex function. Deng et al. [13] obtained a new generalized q-integral identity and found several new q-analogues for twice q-differentiable generalized (s,m)-preinvex functions.
The theory of post quantum calculus denoted by (p,q)-calculus is a natural generalization of the quantum calculus denoted by q-calculus, which has been studied extensively. Recently, Tunç and Göv [14] studied the concept of (p,q)-calculus on the intervals [χ1,χ2]⊆R, defined the (p,q)-derivative and (p,q)-integral and established their basic properties and integral inequalities. Integral inequalities [15,16,17,18,19,20,21,22,23,24,25,26,27] play an important role in understanding the universe, and they can be used to find the uniqueness and existence of the linear and nonlinear differential equations. While convexity is an indispensable tool in the study of inequality theory [28,29,30,31,32,33,34,35,36,37,38].
It is well-known that the Hermite-Hadamard inequality [39,40,41,42,43] is one of the most important inequalities in the convex functions theory, which can be stated as follows.
Let K:I⊂R→R be a convex function. Then the double inequality
K(χ1+χ22)≤1χ2−χ1∫χ2χ1K(t)dt≤K(χ1)+K(χ2)2 | (1.1) |
holds for all χ1,χ2∈I with χ1≠χ2.
As the generalization and refinement of the Hermite-Hadamard inequality (1.1), the Ostrowski inequality [44] can be stated in Theorem 1.1.
Theorem 1.1. Let K:[χ1,χ2]⊆R→R be continuous on [χ1,χ2] and differentiable on (χ1,χ2) such that |K′(τ)|≤M for all τ∈(χ1,χ2). Then the inequality
|K(ϱ)−1χ2−χ1∫χ2χ1K(τ)dτ|≤[14+(ϱ−χ1+χ22)2(χ2−χ1)2](χ2−χ1)M | (1.2) |
holds for all ϱ∈[χ1,χ2] with the best possible constant 14.
Inequality (1.2) can be rewritten in its equivalent form
|K(ϱ)−1χ2−χ1∫χ2χ1K(τ)dτ|≤Mχ2−χ1[(ϱ−χ1)2+(χ2−ϱ)22]. |
Recently, the generalizations, variants and applications of the Ostrowski inequality have attracted the attention of many researchers.
Now, we discuss some connections between the class of convex functions and s-type convex functions.
Definition 1.2. Let s∈[0,1]. Then the function K:I→R is said to be a s-type convex function on I if the inequality
K(χϱ+(1−χ)ρ)≤[1−s(1−χ)]K(ϱ)+[1−sχ]K(ρ) | (1.3) |
holds for all ϱ,ρ∈I and χ∈[0,1].
Remark 1. Definition 1.2 leads to the conclusion that
(1) If we choose s=1, then we get classical convex function.
(2) If we take s=0, then we have the definition of P-function [45].
(3) If K is s-type convex function on I, then the range of K is [0,∞).
Indeed, let ϱ∈I. Then it follows from the s-type convexity of K that
K(χη1+(1−χ)ϱ)≤[1−s(1−χ)]K(η1)+[1−sχ]K(ϱ) |
for all η1∈I and χ∈[0,1]. If we choose χ=1, then we get
K(η1)≤K(η1)+(1−s)K(ϱ) |
⇒(1−s)K(ρ)≥0⇒K(ϱ)≥0 |
for all ϱ∈I.
Proposition 1. Every non-negative convex function is also s-type convex function.
Proof. The proof is clearly due to
s(1−χ)≤(1−χ),χ≥sχ |
for all χ∈[0,1] and s∈[0,1].
Next, We recall the definition of n-polynomial s-type convex function.
Definition 1.3. Let s∈[0,1] and n∈N. Then the function K:I→R is said to be a n-polynomial s-type convex function on I if the inequality
K(χϱ+(1−χ)ρ)≤1nn∑i=1[1−(s(1−χ))i]K(ϱ)+1nn∑i=1[1−(sχ)i]K(ρ) | (1.4) |
holds for ϱ,ρ∈I and χ∈[0,1].
Remark 2. From Definition 1.3, one has
(1) If we choose s=0, then we get the P-function [45].
(2) If we take s=1, then we obtain the Definition given in [46].
(3) If we choose n=1, then we obtain Definition 1.2.
(4) If K is a n-polynomial s-type convex function, then the range of K is [0,∞).
Remark 3. Every non-negative n-polynomial convex function is also n-polynomial s-type convex function. Indeed
1nn∑i=1[1−(s(1−χ))i]≥1nn∑i=1[1−(1−χ)i] |
and
1nn∑i=1[1−χi]≤1nn∑i=1[1−(sχ)i] |
for all χ∈[0,1],n∈N and s∈[0,1].
In what follows, we suppose that χ1,χ2,χ3,χ4∈R with χ1<χ2 and χ3<χ4, 0<qk<pk≤1(k=1,2) are constants, N=[χ1,χ2]×[χ3,χ4]⊆R2 is a rectangle and N∘=(χ1,χ2)×(χ3,χ4) is the interior of N.
Definition 1.4. Let K:N→R be a continuous function. Then the partial (p1,q1)-, (p2,q2)- and (p1p2,q1q2)-derivatives at (z,w)∈N are respectively defined by
χ1∂p1,q1K(z,w)χ1∂p1,q1z=K(p1z+(1−p1)χ1,w)−K(q1z+(1−q1)χ1,w)(p1−q1)(z−χ1)(z≠χ1),χ3∂p2,q2K(z,w)χ3∂p2,q2w=K(z,p2w+(1−p2)χ3)−K(z,q2w+(1−q2)χ3)(p2−q2)(w−χ3)(w≠χ3),χ1,χ3∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ3∂p2,q2w=1(p1−q1)(p2−q2)(z−χ1)(w−χ3)[K(q1z+(1−q1)χ1,q2w+(1−q2)χ3)−K(q1z+(1−q1)χ1,p2w+(1−p2)χ3)−K(p1z+(1−p1)χ1,q2w+(1−q2)χ3)+K(p1z+(1−p1)χ1,p2w+(1−p2)χ3)]. |
Definition 1.5. Let K:N→R be a continuous function. Then the definite (p1p2,q1q2)-integral on N is defined by
∫tχ3∫sχ1K(z,w)χ1dp1,q1zχ3dp2,q2w=(p1−q1)(p2−q2)(s−χ1)(t−χ3)×∞∑m=0∞∑n=0qn1qm2pn+11pm+12K(qn1pn+11s+(1−qn1pn+11)χ1,qm2pm+12t+(1−qm2pm+12)χ3) | (1.5) |
for (s,t)∈N.
Definition 1.6. For any real number n, the (p,q)-analogue is defined by
[n]p,q=pn−qnp−q, |
where 0<q<p≤1 are constants.
In the present paper, we introduce new Definitions 1.7 and 1.8 for (p1p2,q1q2)-differentiable function, and (p,q)χ4χ1,(p,q)χ2χ3 and (p,q)χ2,χ4 integrals for two variables mappings over finite rectangles by using convex set. These new definitions will open new doors for convexity and (p,q)-calculus for two variables functions over the finite rectangles in the plane R2. We drive a key lemma for (p1p2,q1q2)-integral. By making use of new identity, we will obtain estimates of integral inequality whose twice partial (p1p2,q1q2)-derivatives in absolute value at certain powers are n-polynomial s-type convex function. We also discuss some new special cases of the main results. A briefly conclusion is given at the end.
Definition 1.7. Let K:N→R be a continuous function. Then the partial (p1,q1)-, (p2,q2)- and (p1p2,q1q2)-derivatives at (z,w)∈N are respectively defined by
χ2∂p1,q1K(z,w)χ2∂p1,q1z=K(p1z+(1−p1)χ2,w)−K(q1z+(1−q1)χ2,w)(p1−q1)(χ2−z)(χ2≠z),χ4∂p2,q2K(z,w)χ4∂p2,q2w=K(z,p2w+(1−p2)χ4)−K(z,q2w+(1−q2)χ4)(p2−q2)(χ4−w)(χ4≠w),χ4χ1∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ4∂p2,q2w=1(p1−q1)(p2−q2)(z−χ1)(χ4−w)[K(q1z+(1−q1)χ1,q2w+(1−q2)χ4)−K(q1z+(1−q1)χ1,p2w+(1−p2)χ4)−K(p1z+(1−p1)χ1,q2w+(1−q2)χ4)+K(p1z+(1−p1)χ1,p2w+(1−p2)χ4)](z≠χ1,χ4≠w),χ2χ3∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ3∂p2,q2w=1(p1−q1)(p2−q2)(χ2−z)(w−χ3)[K(q1z+(1−q1)χ2,q2w+(1−q2)χ3)−K(q1z+(1−q1)χ2,p2w+(1−p2)χ3)−K(p1z+(1−p1)χ2,q2w+(1−q2)χ3)+K(p1z+(1−p1)χ2,p2w+(1−p2)χ3)](χ2≠z,w≠χ3),χ2,χ4∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ4∂p2,q2w=(p1−q1)(p2−q2)(χ2−z)(χ4−w)[K(q1z+(1−q1)χ2,q2w+(1−q2)χ4)−K(q1z+(1−q1)χ2,p2w+(1−p2)χ4)−K(p1z+(1−p1)χ2,q2w+(1−q2)χ4)+K(p1z+(1−p1)χ2,p2w+(1−p2)χ4)](χ2≠z,χ4≠w). |
Definition 1.8. Let K:N→R be a continuous function. Then the definite (p,q)χ4χ1,(p,q)χ2χ3 and (p,q)χ2,χ4 integrals on [χ1,χ2]×[χ3,χ4] are respectively defined by
χ4∫ts∫χ1K(z,w)χ1dp1,q1zχ3dp2,q2w=(p1−q1)(p2−q2)(s−χ1)(χ4−t)×∞∑m=0∞∑n=0qn1qm2pn+11pm+12K(qn1pn+11s+(1−qn1pn+11)χ1,qm2pm+12t+(1−qm2pm+12)χ4), |
t∫χ3χ2∫sK(z,w)χ2dp1,q1zχ3dp2,q2w(p1−q1)(p2−q2)(χ2−s)(t−χ3)×∞∑m=0∞∑n=0qn1qm2pn+11pm+12K(qn1pn+11s+(1−qn1pn+11)χ2,qm2pm+12t+(1−qm2pm+12)χ3) |
and
χ4∫tχ2∫sK(z,w)χ2dp1,q1zχ4dp2,q2w(p1−q1)(p2−q2)(χ2−s)(χ4−t)×∞∑m=0∞∑n=0qn1qm2pn+11pm+12K(qn1pn+11s+(1−qn1pn+11)χ2,qm2pm+12t+(1−qm2pm+12)χ4) |
for (s,t)∈[χ1,χ2]×[χ3,χ4].
Lemma 2.1. Let K:N→R be a twice partial (p1p2,q1q2)-differentiable function on No such that the partial (p1p2,q1q2)-derivatives χ1,χ3∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ3∂p2,q2w, χ4χ1∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ4∂p2,q2w, χ2χ3∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ3∂p2,q2w and χ2,χ4∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ4∂p2,q2w are continuous and integrable on N. Then one has the equality
K(ϱ,ρ)−1p1(χ2−χ1)[∫p1ϱ+(1−p1)χ1χ1K(u,ρ)χ1dp1,q1u+∫χ2p1ϱ+(1−p1)χ2K(u,ρ)χ2dp1,q1u]−1p2(χ4−χ3)[∫p2ρ+(1−p2)χ3χ3K(ϱ,v)χ3dp2,q2v+∫χ4p2ρ+(1−p2)χ4K(ϱ,v)χ4dp2,q2v]+A=Δ{(ϱ−χ1)2(ρ−χ3)2∫10∫10zwχ1,χ3∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w0dp1,q1z0dp2,q2w+(ϱ−χ1)2(χ4−ρ)2∫10∫10zwχ4χ1∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ4)χ1∂p1,q1zχ4∂p2,q2w0dp1,q1z0dp2,q2w+(χ2−ϱ)2(ρ−χ3)2∫10∫10zwχ2χ3∂2p1p2,q1q2K(zϱ+(1−z)χ2,wρ+(1−w)χ3)χ3∂p1,q1zχ2∂p2,q2w0dp1,q1z0dp2,q2w+(χ2−ϱ)2(χ4−ρ)2∫10∫10zwχ2,χ4∂2p1p2,q1q2K(zϱ+(1−z)χ2,wρ+(1−w)χ4)χ2∂p1,q1zχ4∂p2,q2w0dp1,q1z0dp2,q2w}, | (2.1) |
where
A=1p1p2(χ2−χ1)(χ4−χ3)∫p1ϱ+(1−p1)χ1χ1∫p2ρ+(1−p2)χ3χ3K(u,v)χ1dp1,q1uχ3dp2,q2v+1p1p2(χ2−χ1)(χ4−χ3)∫p1ϱ+(1−p1)χ1χ1∫χ4p2ρ+(1−p2)χ4K(u,v)χ1dp1,q1uχ4dp2,q2v+1p1p2(χ2−χ1)(χ4−χ3)∫χ2p1ϱ+(1−p1)χ2∫p2ρ+(1−p2)χ3χ3K(u,v)χ2dp1,q1uχ3dp2,q2v+1p1p2(χ2−χ1)(χ4−χ3)∫χ2p1ϱ+(1−p1)χ2∫χ4p2ρ+(1−p2)χ4K(u,v)χ2dp1,q1uχ4dp2,q2v |
for all ϱ,ρ∈N and Δ=q1q2(χ2−χ1)(χ4−χ3).
Proof. We consider the integral
∫10∫10zwχ1,χ3∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w0dp1,q1z0dp2,q2w. |
By the definition of partial (p1p2,q1q2)-derivative and definite (p1p2,q1q2)χ1,χ3-integral, we have
∫10∫10zwχ1,χ3∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w0dp1,q1z0dp2,q2w=1(1−q1)(1−q2)(ϱ−χ1)(ρ−χ3)×[∫10∫10K(zq1ϱ+(1−zq1)χ1,wq2ρ+(1−wq2)χ3)0dp1,q1z0dp2,q2w−∫10∫10K(zq1ϱ+(1−zq1)χ1,wp2ρ+(1−wp2)χ3)0dp1,q1z0dp2,q2w−∫10∫10K(zp1ϱ+(1−zp1)χ1,wq2ρ+(1−wq2)χ3)0dp1,q1z0dp2,q2w+∫10∫10K(zp1ϱ+(1−zp1)χ1,wp2ρ+(1−wp2)χ3)0dp1,q1z0dp2,q2w]=1(ϱ−χ1)(ρ−χ3)×[∞∑n=0∞∑m=0qn1qm2pn+11pm+12K(qn+11pn+11ϱ+(1−qn+11pn+11)χ1,qm+12pm+12ρ+(1−qm+12pm+12)χ3)−∞∑n=0∞∑m=0qn1qm2pn+11pm+12K(qn+11pn+11ϱ+(1−qn+11pn+11)χ1,qm2pm2ρ+(1−qm2pm2)χ3)−∞∑n=0∞∑m=0qn1qm2pn+11pm+12K(qn1pn+11ϱ+(1−qn1pn+11)χ1,qm+12pm+12ρ+(1−qm+12pm+12)χ3)+∞∑n=0∞∑m=0qn1qm2pn+11pm+12K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3)]=1q1q2(ϱ−χ1)(ρ−χ3)∞∑n=1∞∑m=1qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3)−1p2q1(ϱ−χ1)(ρ−χ3)∞∑n=1∞∑m=0qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3)−1p1q2(ϱ−χ1)(ρ−χ3)∞∑n=0∞∑m=1qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3)+1p1p2(ϱ−χ1)(ρ−χ3)∞∑n=0∞∑m=0qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3). | (2.2) |
Note that
1q1q2(ϱ−χ1)(ρ−χ3)∞∑n=1∞∑m=1qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3)=−K(ϱ,ρ)q1q2(ϱ−χ1)(ρ−χ3)−1q1q2(ϱ−χ1)(ρ−χ3)∞∑n=0qn1pn1K(qn1pn1ϱ+(1−qn1pn1)χ1,ρ)−1q1q2(ϱ−χ1)(ρ−χ3)∞∑m=0qm2pm2K(ϱ,qm2pm2ρ+(1−qm2pm2)χ3)+1q1q2(ϱ−χ1)(ρ−χ3)×∞∑n=0∞∑m=0qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3), | (2.3) |
−1p2q1(ϱ−χ1)(ρ−χ3)∞∑n=1∞∑m=0qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3)=1p2q1(ϱ−χ1)(ρ−χ3)∞∑m=0qm2pm2K(ϱ,qm2pm2ρ+(1−qm2pm2)χ3)−1p2q1(ϱ−χ1)(ρ−χ3)×∞∑n=0∞∑m=0qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3) | (2.4) |
and
−1p1q2(ϱ−χ1)(ρ−χ3)∞∑n=0∞∑m=1qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3)=1p1q2(ϱ−χ1)(ρ−χ3)∞∑n=0qn1pn1K(qn1pn1ϱ+(1−qn1pn1)χ1,ρ)−1p1q2(ϱ−χ1)(ρ−χ3)×∞∑n=0∞∑m=0qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3). | (2.5) |
Utilizing (2.3)–(2.5) in (2.2), we get
∫10∫10zwχ1,χ3∂2p1p2,q1q2χ1∂p1,q1zχ3∂p2,q2wK(zϱ+(1−z)χ1,wρ+(1−w)χ3)0dp1,q1z0dp2,q2w=−K(ϱ,ρ)q1q2(ϱ−χ1)(ρ−χ3)−(p2−q2)(ρ−χ3)p2q1q2(ϱ−χ1)(ρ−χ3)2∞∑m=0qm2pm2K(ϱ,qm2pm2ρ+(1−qm2pm2)χ3)−(p1−q1)(ϱ−χ1)p1q1q2(ϱ−χ1)2(ρ−χ3)∞∑n=0qn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,ρ)+(p1−q1)(p2−q2)(ϱ−χ1)(ρ−χ3)p1p2q1q2(ϱ−χ1)2(ρ−χ3)2∞∑n=0∞∑m=0qn1qm2pn1pm2K(qn1pn1ϱ+(1−qn1pn1)χ1,qm2pm2ρ+(1−qm2pm2)χ3) |
and
∫10∫10zwχ1,χ3∂2p1p2,q1q2χ1∂p1,q1zχ3∂p2,q2wK(zϱ+(1−z)χ1,wρ+(1−w)χ3)0dp1,q1z0dp2,q2w=−K(ϱ,ρ)q1q2(ϱ−χ1)(ρ−χ3)−1p2q1q2(ϱ−χ1)(ρ−χ3)2∫p2ρ+(1−p2)χ3χ3K(ϱ,v)0dp2,q2v−1p1q1q2(ϱ−χ1)2(ρ−χ3)∫p1ϱ+(1−p1)χ1χ1K(u,ρ)0dp1,q1u+1p1p2q1q2(ϱ−χ1)2(ρ−χ3)2∫p1ϱ+(1−p1)χ1χ1∫p2ρ+(1−p2)χ3χ3K(u,v)0dp1,q1u0dp2,q2v. | (2.6) |
Multiplying both sides of equality (2.6) by Δ(ϱ−χ1)2(ρ−χ3)2 leads to
Δ(ϱ−χ1)2(ρ−χ3)2∫10∫10zwχ1,χ3∂2p1p2,q1q2χ1∂p1,q1zχ3∂p2,q2wK(zϱ+(1−z)χ1,wρ+(1−w)χ3)0dp1,q1z0dp2,q2w=−(ϱ−χ1)(ρ−χ3)(χ2−χ1)(χ4−χ3)K(ϱ,ρ)−ϱ−χ1p2(χ2−χ1)(χ4−χ3)∫p2ρ+(1−p2)χ3χ3K(ϱ,v)0dp2,q2v−ρ−χ3p1(χ2−χ1)(χ4−χ3))∫p1ϱ+(1−p1)χ1χ1K(u,ρ)0dp1,q1u+1p1p2(χ2−χ1)(χ4−χ3)∫p1ϱ+(1−p1)χ1χ1∫p2ρ+(1−p2)χ3χ3K(u,v)0dp1,q1u0dp2,q2v. | (2.7) |
Similarly, calculating the remaining integrals and by using definition 1.8 we get
Δ(ϱ−χ1)2(χ4−ρ)2∫10∫10zwχ4χ1∂2p1p2,q1q2χ1∂p1,q1zχ4∂p2,q2wK(zϱ+(1−z)χ1,wρ+(1−w)χ4)0dp1,q1z0dp2,q2w=−(ϱ−χ1)(χ4−ρ)(χ2−χ1)(χ4−χ3)K(ϱ,ρ)−ϱ−χ1p2(χ2−χ1)(χ4−χ3)∫χ4p2ρ+(1−p2)χ4K(ϱ,v)χ4dp2,q2v−χ4−ρp1(χ2−χ1)(χ4−χ3))∫p1ϱ+(1−p1)χ1χ1K(u,ρ)χ1dp1,q1u+1p1p2(χ2−χ1)(χ4−χ3)∫p1ϱ+(1−p1)χ1χ1∫χ4p2ρ+(1−p2)χ4K(u,v)χ1dp1,q1uχ4dp2,q2v, | (2.8) |
Δ(χ2−ϱ)2(ρ−χ3)2∫10∫10zwχ2χ3∂2p1p2,q1q2χ2∂p1,q1zχ3∂p2,q2wK(zϱ+(1−z)χ1,wρ+(1−w)χ3)0dp1,q1z0dp2,q2w=−(χ2−ϱ)(ρ−χ3)(χ2−χ1)(χ4−χ3)K(ϱ,ρ)−χ2−ϱp2(χ2−χ1)(χ4−χ3)∫p2ρ+(1−p2)χ3χ3K(ϱ,v)χ3dp2,q2v−ρ−χ3p1(χ2−χ1)(χ4−χ3))∫χ2p1ϱ+(1−p1)χ2K(u,ρ)χ2dp1,q1u+1p1p2(χ2−χ1)(χ4−χ3)∫p1ϱ+(1−p1)χ1χ2∫χ2p2ρ+(1−p2)χ2K(u,v)χ2dp1,q1uχ3dp2,q2v, | (2.9) |
Δ(χ2−ϱ)2(χ4−ρ)2∫10∫10zwχ2,χ4∂2p1p2,q1q2χ2∂p1,q1zχ4∂p2,q2wK(zϱ+(1−z)χ2,wρ+(1−w)χ4)0dp1,q1z0dp2,q2w=−(χ2−ϱ)(χ4−ρ)(χ2−χ1)(χ4−χ3)K(ϱ,ρ)−χ2−ϱp2(χ2−χ1)(χ4−χ3)∫χ4p2ρ+(1−p2)χ4K(ϱ,v)χ4dp2,q2v−χ4−ρp1(χ2−χ1)(χ4−χ3))∫χ2p1ϱ+(1−p1)χ2K(u,ρ)χ2dp1,q1u+1p1p2(χ2−χ1)(χ4−χ3)∫χ4p1ϱ+(1−p1)χ2∫χ4p2ρ+(1−p2)χ4K(u,v)χ2dp1,q1uχ4dp2,q2v. | (2.10) |
From (2.7)–(2.10) and (2.1), we derive the desired result of Lemma 2.1.
Remark 4. Taking pk=1 for k=1,2 we get
K(ϱ,ρ)−1(χ2−χ1)[∫ϱχ1K(u,ρ) χ1dq1u+∫χ2ϱK(u,ρ) χ2dq1u]−1(χ4−χ3)[∫ρχ3K(ϱ,v) χ3dq2v+∫χ4ρK(ϱ,v) χ4dq2v]+W=Δ{(ϱ−χ1)2(ρ−χ3)2∫10∫10zw χ1,χ3∂2q1,q2K(zϱ+(1−z)χ1,wρ+(1−w)χ3) χ1∂q1z χ3∂q2w 0dq1z 0dq2w+(ϱ−χ1)2(χ4−ρ)2∫10∫10zw χ4χ1∂2q1,q2K(zϱ+(1−z)χ1,wρ+(1−w)χ4) χ1∂q1z χ4∂q2w 0dq1z 0dq2w+(χ2−ϱ)2(ρ−χ3)2∫10∫10zw χ2χ3∂2q1,q2K(zϱ+(1−z)χ2,wρ+(1−w)χ3) χ3∂q1z χ2∂q2w 0dq1z 0dq2w+(χ2−ϱ)2(χ4−ρ)2∫10∫10zw χ2,χ4∂2q1,q2K(zϱ+(1−z)χ2,wρ+(1−w)χ4) χ2∂q1z χ4∂q2w 0dq1z 0dq2w}, | (2.11) |
where
W=1(χ2−χ1)(χ4−χ3)∫ϱχ1∫ρχ3K(u,v)χ1dq1uχ3dq2v+1(χ2−χ1)(χ4−χ3)∫ϱχ1∫χ4ρK(u,v)χ1dq1uχ4dq2v+1(χ2−χ1)(χ4−χ3)∫χ2ϱ∫ρχ3K(u,v)χ2dq1uχ3dq2v+1(χ2−χ1)(χ4−χ3)∫χ2ϱ∫χ4ρK(u,v)χ2dq1uχ4dq2v. |
In this section, we introduce (p1p2,q1q2)-Ostrowski inequalities by using n-polynomial s-type convex function on the co-ordinates.
Theorem 3.1. Suppose that n∈N, s∈[0,1] and all the assumptions of Lemma 2.1 are true. If |χ1,χ3∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ3∂p2,q2w|τ2, |χ4χ1∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ4∂p2,q2w|τ2, |χ2χ3∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ3∂p2,q2w|τ2 and |χ2,χ4∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ4∂p2,q2w|τ2 are n-polynomial s-type convex functions on the co-ordinates on N for τ1,τ2>1 with 1τ1+1τ2=1 and |χ1,χ3∂2p1p2,q1q2K(ϱ,ρ)χ1∂p1,q1zχ3∂p2,q2w|≤M, |χ4χ1∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ4∂p2,q2w|≤M, |χ2χ3∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ3∂p2,q2w|≤M, |χ2,χ4∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ4∂p2,q2w|≤M, ϱ,ρ∈N, then the following inequality holds
|K(ϱ,ρ)−1p1(χ2−χ1)[∫p1ϱ+(1−p1)χ1χ1K(u,ρ)χ1dp1,q1u+∫χ2p1ϱ+(1−p1)χ2K(u,ρ)χ2dp1,q1u]−1p2(χ4−χ3)[∫p2ρ+(1−p2)χ3χ3K(ϱ,v)χ3dp2,q2v+∫χ4p2ρ+(1−p2)χ4K(ϱ,v)χ4dp2,q2v]+A|≤ΔMτ2√(Cp1,q1+Dp1,q1)(Cp2,q2+Dp2,q2)[(ϱ−χ1)2+(χ2−ϱ)2][(ρ−χ3)2+(χ4−ρ)2]τ1√[1+τ1]p1,q1[1+τ1]p2,q2, | (3.1) |
where
Cpk,qk=1−(pk−qk)nn∑i=1∞∑e=0siqekpe+1k(1−qekpe+1k)i,Dpk,qk=1−1nn∑i=1si(pk−qkpi+1k−qi+1k) |
for k=1,2 and Δ,A are defined in Lemma 2.1.
Proof. Taking absolute value on both sides of (2.1), by applying Hölder inequality for double integrals and utilizing the fact that |∂2p1p2,q1q2K∂p1,q1z∂p2,q2w|τ2 is n-polynomial s-type convex on co-ordinates, we get the following inequality
|K(ϱ,ρ)−1p1(χ2−χ1)[∫p1ϱ+(1−p1)χ1χ1K(u,ρ)χ1dp1,q1u+∫χ2p1ϱ+(1−p1)χ2K(u,ρ)χ2dp1,q1u]−1p2(χ4−χ3)[∫p2ρ+(1−p2)χ3χ3K(ϱ,v)χ3dp2,q2v+∫χ4p2ρ+(1−p2)χ4K(ϱ,v)χ4dp2,q2v]+A|≤Δ(∫10∫10zτ1wτ10dp1,q1z0dp2,q2w)1τ1×{(ϱ−χ1)2(ρ−χ3)2(∫10∫10|χ1,χ3∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ20dp1,q1z0dp2,q2w)1τ2 |
+(ϱ−χ1)2(χ4−ρ)2(∫10∫10|χ4χ1∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ4)χ1∂p1,q1zχ4∂p2,q2w|τ20dp1,q1z0dp2,q2w)1τ2+(χ2−ϱ)2(ρ−χ3)2(∫10∫10|χ2χ3∂2p1p2,q1q2K(zϱ+(1−z)χ2,wρ+(1−w)χ3)χ2∂p1,q1zχ3∂p2,q2w|τ20dp1,q1z0dp2,q2w)1τ2+(χ2−ϱ)2(χ4−ρ)2(∫10∫10|χ2,χ4∂2p1p2,q1q2K(zϱ+(1−z)χ2,wρ+(1−w)χ4)χ2∂p1,q1zχ4∂p2,q2w|τ20dp1,q1z0dp2,q2w)1τ2}. |
Considering first integral
∫10∫10|χ1,χ3∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ20dp1,q1z0dp2,q2w≤∫10{∫10[1n∑ni=1[1−(s(1−z))i]|χ1,χ3∂2p1p2,q1q2K(ϱ,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ2+1n∑ni=1[1−(sz)i]|χ1,χ3∂2p1p2,q1q2K(χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ1∂p2,q2w|τ2]0dp1,q1z}0dp2,q2w. | (3.2) |
Computing the (p1,q1)-integral on the right-hand side of (3.2), we have
≤∫10[1n∑ni=1[1−(s(1−z))i]|χ1,χ3∂2p1p2,q1q2K(ϱ,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ2+1n∑ni=1[1−(sz)i]|χ1,χ3∂2p1p2,q1q2K(χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ1∂p2,q2w|τ2]0dp1,q1z. |
In view of the Definitions 1.4 for k=1,2, we get
Cpk,qk=1nn∑i=1∫10[1−(s(1−z))i]0dpk,qkz=1−(pk−qk)nn∑i=1∞∑e=0siqekpe+1k(1−qekpe+1k)i,Dpk,qk=1nn∑i=1∫10[1−(sz)i]0dpk,qkz=1−1nn∑i=1si(pk−qkpi+1k−qi+1k). |
Putting the above calculations into (3.2), we obtain
≤∫10[Cp1,q1|χ1,χ3∂2p1p2,q1q2K(ϱ,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ2+Dp1,q1|χ1,χ3∂2p1p2,q1q2K(χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ2]0dp2,q2w. | (3.3) |
Similarly, by computing the (p2,q2)-integral, utilizing the fact |χ1,χ3∂2p1p2,q1q2K(ϱ,ρ)χ1∂p1,q1zχ3∂p2,q2w|≤M,ϱ,ρ∈N on the right-hand side of (3.3), we have
∫10∫10|χ1,χ3∂2p1p2,q1q2K(zϱ+(1−w)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ20dp1,q1z0dp2,q2w≤Mτ2(Cp1,q1+Dp1,q1)(Cp2,q2+Dp2,q2). | (3.4) |
Analogously, we get
∫10∫10|χ4χ1∂2p1p2,q1q2K(zϱ+(1−w)χ1,χ4+wρ+(1−w)χ4)χ1∂p1,q1zχ4∂p2,q2w|τ20dp1,q1z0dp2,q2w≤Mτ2(Cp1,q1+Dp1,q1)(Cp2,q2+Dp2,q2) | (3.5) |
∫10∫10|χ2χ3∂2p1p2,q1q2K(zϱ+(1−z)χ2,wρ+(1−w)χ3)χ2∂p1,q1zχ3∂p2,q2w|τ20dp1,q1z0dp2,q2w≤Mτ2(Cp1,q1+Dp1,q1)(Cp2,q2+Dp2,q2) | (3.6) |
and
∫10∫10|χ2,χ4∂2p1p2,q1q2K(zϱ+(1−z)χ2,χ4+wρ+(1−w)χ4)χ2∂p1,q1zχ4∂p2,q2w|τ20dp1,q1z0dp2,q2w≤Mτ2(Cp1,q1+Dp1,q1)(Cp2,q2+Dp2,q2). | (3.7) |
Now by making use of the inequalities (3.4)–(3.7) and using the fact that
∫10∫10zτ1wτ10dp1,q1z0dp2,q2w=1[1+τ1]p1,q1[1+τ1]p2,q2, |
we get the desired inequality (3.1). This completes the proof.
Corollary 1. I. Taking pk=1 for k=1,2 in Theorem 3.1, we get
|K(ϱ,ρ)−1(χ2−χ1)[∫ϱχ1K(u,ρ)χ1dq1u+∫χ2ϱK(u,ρ)χ2dq1u]−1(χ4−χ3)[∫ρχ3K(ϱ,v)χ3dq2v+∫χ4ρK(ϱ,v)χ4dq2v]+W|≤ΔMτ2√(Cq1+Dq1)(Cq2+Dq2)[(ϱ−χ1)2+(χ2−ϱ)2][(ρ−χ3)2+(χ4−ρ)2]τ1√[1+τ1]q1[1+τ1]q2, |
where
Cqk=1−(1−qk)nn∑i=1∞∑e=0siqek(1−qek)i,Dqk=1−1nn∑i=1si(1−qk1−qi+1k) |
and Δ,W are defined in Remark 4.
II. Taking qk→1− for k=1,2 in part I, we get
|K(ϱ,ρ)+1(χ2−χ1)(χ4−χ3)∫χ2χ1∫χ4χ3K(u,v)dvdu−Q|≤Mτ2√(2n∑ni=1(i+1−sii+1))2τ1√(1+τ1)2[(ϱ−χ1)2+(χ2−ϱ)2χ2−χ1][(ρ−χ3)2+(χ4−ρ)2χ4−χ3], |
where
Q=1χ2−χ1∫χ2χ1K(u,ρ)du+1χ4−χ3∫χ4χ3K(ϱ,v)dv. |
Remark 5. Taking n=s=1 in part II of Corollary 1, we obtain Theorem 4 of [47].
Theorem 3.2. Suppose that n∈N, s∈[0,1] and all the assumptions of Lemma 2.1 holds. If |χ1,χ3∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ3∂p2,q2w|τ, |χ4χ1∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ4∂p2,q2w|τ, |χ2χ3∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ3∂p2,q2w|τ and |χ2,χ4∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ4∂p2,q2w|τ are n-polynomial s-type convex functions on the co-ordinates on N for τ≥1, and |χ1,χ3∂2p1p2,q1q2K(ϱ,ρ)χ1∂p1,q1zχ3∂p2,q2w|≤M, |χ4χ1∂2p1p2,q1q2K(z,w)χ1∂p1,q1zχ4∂p2,q2w|≤M, |χ2χ3∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ3∂p2,q2w|≤M, |χ2,χ4∂2p1p2,q1q2K(z,w)χ2∂p1,q1zχ4∂p2,q2w|≤M, ϱ,ρ∈N, then the following inequality holds
|K(ϱ,ρ)−1p1(χ2−χ1)[∫p1ϱ+(1−p1)χ1χ1K(u,ρ)χ1dp1,q1u+∫χ2p1ϱ+(1−p1)χ2K(u,ρ)χ2dp1,q1u]−1p2(χ4−χ3)[∫p2ρ+(1−p2)χ3χ3K(ϱ,v)χ3dp2,q2v+∫χ4p2ρ+(1−p2)χ4K(ϱ,v)χ4dp2,q2v]+A|≤ΔMτ√(Ap1,q1+Bp1,q1)(Ap2,q2+Bp2,q2)[(ϱ−χ1)2+(χ2−ϱ)2][(ρ−χ3)2+(χ4−ρ)2][(p1+q1)(p2+q2)]1−1τ, | (3.8) |
where
Apk,qk=1pk+qk−(pk−qk)nn∑i=1∞∑e=0siq2ekp2e+2k(1−qekpe+1k)i,Bpk,qk=1pk+qk−1nn∑i=1si(pk−qkpi+2k−qi+2k) |
for k=1,2 and Δ,A are defined in Lemma 2.1.
Proof. Taking absolute value on both sides of (2.1), by applying power mean inequality for double integrals, we get the following inequality
|K(ϱ,ρ)−1p1(χ2−χ1)[∫p1ϱ+(1−p1)χ1χ1K(u,ρ)0dp1,q1u+∫p1ϱ+(1−p1)χ2χ2K(u,ρ)0dp1,q1u]−1p2(χ4−χ3)[∫p2ρ+(1−p2)χ3χ3K(ϱ,v)0dp2,q2v+∫p2ρ+(1−p2)χ4χ4K(ϱ,v)0dp2,q2v]+A|≤Δ(∫10∫10zw0dp1,q1z0dp2,q2w)1−1τ×{(ϱ−χ1)2(ρ−χ3)2(∫10∫10zw|χ1,χ3∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ0dp1,q1z0dp2,q2w)1τ |
+(ϱ−χ1)2(χ4−ρ)2(∫10∫10zw|χ4χ1∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ4)χ1∂p1,q1zχ4∂p2,q2w|τ0dp1,q1z0dp2,q2w)1τ+(χ2−ϱ)2(ρ−χ3)2(∫10∫10zw|χ2χ3∂2p1p2,q1q2K(zϱ+(1−z)χ2,wρ+(1−w)χ3)χ2∂p1,q1zχ3∂p2,q2w|τ0dp1,q1z0dp2,q2w)1τ+(χ2−ϱ)2(χ4−ρ)2(∫10∫10zw|χ2,χ4∂2p1p2,q1q2K(zϱ+(1−z)χ2,wρ+(1−w)χ4)χ2∂p1,q1zχ4∂p2,q2w|τ0dp1,q1z0dp2,q2w)1τ}. |
Considering first integral
∫10∫10zw|χ1,χ3∂2p1p2,q1q2K(zϱ+(1−z)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ0dp1,q1z0dp2,q2w≤∫10w{∫10z[1n∑ni=1[1−(s(1−z))i]|χ1,χ3∂2p1p2,q1q2K(ϱ,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ+1n∑ni=1[1−(sz)i]|χ1,χ3∂2p1p2,q1q2K(χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ]0dp1,q1z}0dp2,q2w. | (3.9) |
Computing the (p1,q1)-integral on the right-hand side of (3.9), we have
≤∫10z[1n∑ni=1[1−(s(1−z))i]|χ1,χ3∂2p1p2,q1q2K(ϱ,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ+1n∑ni=1[1−(sz)i]|χ1,χ3∂2p1p2,q1q2K(χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ]0dp1,q1z. |
In view of the Definitions 1.5 fpr k=1,2, we get
Apk,qk=1nn∑i=1∫10z[1−(s(1−z))i]0dpk,qkz=1pk+qk−(pk−qk)nn∑i=1∞∑e=0siq2ekp2e+2k(1−qekpe+1k)i,Bpk,qk=1nn∑i=1∫10z[1−(sz)i]0dpk,qkz=1pk+qk−1nn∑i=1si(pk−qkpi+2k−qi+2k). |
Putting the above calculations into (3.9), we obtain
≤∫10w[Ap1,q1|χ1,χ3∂2p1p2,q1q2K(ϱ,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ+Bp1,q1|χ1,χ3∂2p1p2,q1q2K(χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ]0dp2,q2w. | (3.10) |
Similarly, by computing the (p2,q2)-integral, utilizing the fact |χ1,χ3∂2p1p2,q1q2K(ϱ,ρ)χ1∂p1,q1zχ3∂p2,q2w|≤M,ϱ,ρ∈N on the right-hand side of (3.10), we have
∫10∫10zw|χ1,χ3∂2p1p2,q1q2K(zϱ+(1−w)χ1,wρ+(1−w)χ3)χ1∂p1,q1zχ3∂p2,q2w|τ0dp1,q1z0dp2,q2w≤Mτ(Ap1,q1+Bp1,q1)(Ap2,q2+Bp2,q2). | (3.11) |
Analogously, we get
∫10∫10zw|χ4χ1∂2p1p2,q1q2K(zϱ+(1−w)χ1,χ4+wρ+(1−w)χ4)χ1∂p1,q1zχ4∂p2,q2w|τ0dp1,q1z0dp2,q2w≤Mτ(Ap1,q1+Bp1,q1)(Ap2,q2+Bp2,q2), | (3.12) |
∫10∫10zw|χ2χ3∂2p1p2,q1q2K(zϱ+(1−z)χ2,wρ+(1−w)χ3)χ2∂p1,q1zχ3∂p2,q2w|τ0dp1,q1z0dp2,q2w≤Mτ(Ap1,q1+Bp1,q1)(Ap2,q2+Bp2,q2) | (3.13) |
and
∫10∫10zw|χ2,χ4∂2p1p2,q1q2K(zϱ+(1−z)χ2,χ4+wρ+(1−w)χ4)χ2∂p1,q1zχ4∂p2,q2w|τ0dp1,q1z0dp2,q2w≤Mτ(Ap1,q1+Bp1,q1)(Ap2,q2+Bp2,q2). | (3.14) |
Now by making use of the inequalities (3.11)–(3.14) and the fact that
∫10∫10zw0dp1,q1z0dp2,q2w=1(p1+q1)(p2+q2), |
we get the desired inequality (3.8). This completes the proof.
Corollary 2. I. Taking τ=1 in Theorem 3.2, we get
|K(ϱ,ρ)−1p1(χ2−χ1)[∫p1ϱ+(1−p1)χ1χ1K(u,ρ)χ1dp1,q1u+∫χ2p1ϱ+(1−p1)χ2K(u,ρ)χ2dp1,q1u]−1p2(χ4−χ3)[∫p2ρ+(1−p2)χ3χ3K(ϱ,v)χ3dp2,q2v+∫χ4p2ρ+(1−p2)χ4K(ϱ,v)χ4dp2,q2v]+A|≤ΔM(Ap1,q1+Bp1,q1)(Ap2,q2+Bp2,q2)[(ϱ−χ1)2+(χ2−ϱ)2][(ρ−χ3)2+(χ4−ρ)2]. |
II. Taking pk=1 for k=1,2 in Theorem 3.2, we obtain
|K(ϱ,ρ)−1(χ2−χ1)[∫ϱχ1K(u,ρ)χ1dq1u+∫χ2ϱK(u,ρ)χ2dq1u]−1(χ4−χ3)[∫ρχ3K(ϱ,v)χ3dq2v+∫χ4ρK(ϱ,v)χ4dq2v]+W|≤ΔMτ√(Aq1+Bq1)(Aq2+Bq2)[(1+q1)(1+q2)]1−1τ[(ϱ−χ1)2+(χ2−ϱ)2][(ρ−χ3)2+(χ4−ρ)2], |
where
Aqk=11+qk−1−qknn∑i=1∞∑e=0siq2ek(1−qek)i,Bqk=11+qk−1nn∑i=1si(1−qk1−qi+2k), |
and Δ,W are defined in Remark 4.
III. Taking pk=1 for k=1,2, and qk→1− in part II, we get
|K(ϱ,ρ)+1(χ2−χ1)(χ4−χ3)∫χ2χ1∫χ4χ3K(u,v)dvdu−Q|≤M41τ−1(1nn∑i=1i2+5i+6−6si2(i+1)(i+2))2τ[(ϱ−χ1)2+(χ2−ϱ)2χ2−χ1][(ρ−χ3)2+(χ4−ρ)2χ4−χ3], |
where Q is defined in part II of Corollary 1.
IV. Taking n=1=s in part III, we have the following inequality
|K(ϱ,ρ)+1(χ2−χ1)(χ4−χ3)∫χ2χ1∫χ4χ3K(u,v)dvdu−Q|≤M4[(ϱ−χ1)2+(χ2−ϱ)2χ2−χ1][(ρ−χ3)2+(χ4−ρ)2χ4−χ3], |
where Q is defined in part II of Corollary 1.
Let k∈R∖{−1,0}, and φ1 and φ2 be two distinct positive real numbers. Then the generalized logarithmic mean Lk(φ1,φ2) is defined by
Lk(φ1,φ2)=(φk+12−φk+11(k+1)(φ2−φ1))1k. |
Proposition 2. If m,k>1 and χ1,χ2,χ3,χ4 are positive real numbers such that χ1<χ2 and χ3<χ4, then one has
|ϱm×ρk−ρk(χ2−χ1)[Lmm(ϱ,χ1)+Lmm(χ2,ϱ)]−ϱk(χ4−χ3)[Lkk(ρ,χ3)+Lkk(χ4,ρ)]+1(χ2−χ1)(χ4−χ3)(Lmm(ϱ,χ1)+Lmm(χ2,ϱ))(Lkk( ρ,χ3)+Lkk(χ4,ρ))|≤M (1+τ1)2τ1[(ϱ−χ1)2+(χ2−ϱ)2χ2−χ1][(ρ−χ3)2+(χ4−ρ)2χ4−χ3]. |
Proof. Let K(ϱ,ρ)=ϱm×ρk for m,k>1. Then, we have
∫ϱχ1um×ρkχ1dq1u=ρk[1−q11−qm+11](ϱm+1−χm+11ϱ−χ1), |
∫χ2ϱum×ρkχ2dq1u=ρk[1−q11−qm+11](χm+12−ϱm+1χ2−ϱ), |
∫ρχ3ϱm×vkχ3dq2v=ϱm[1−q21−qk+12](ρk+1−χk+13 ρ−χ3), |
∫χ4ρϱm×vkχ4dq2v=ϱm[1−q21−qk+12](χk+14−ρk+1χ4−ρ), |
∫ϱχ1∫ρχ3um×vkχ1dq1uχ3dq2v=[1−q11−qm+11][1−q21−qk+12](ϱm+1−χm+11ϱ−χ1)(ρk+1−χk+13ρ−χ3), |
∫ϱχ1∫χ4ρum×vkχ1dq1uχ4dq2v=[1−q11−qm+11][1−q21−qk+12](ϱm+1−χm+11ϱ−χ1)(χk+14−ρk+1χ4−ρ), |
∫χ2ϱ∫ρχ3um×vkχ2dq1uχ3dq2v=[1−q11−qm+11][1−q21−qk+12](χm+12−ϱm+1χ2−ϱ)(ρk+1−χk+13ρ−χ3) |
and
∫χ2ϱ∫χ4ρum×vkχ2dq1uχ4dq2v=[1−q11−qm+11][1−q21−qk+12](χm+12−ϱm+1χ2−ϱ)(χk+14−ρk+1χ4−ρ). |
It follows from part I of Corollary 1 that
\begin{eqnarray} &&\Big\vert \mathcal{K}\left( \varrho, \rho\right) -\frac{\rho^k}{(\chi_{2} -\chi_{1}) }\left[\frac{1-q_1}{1-q_1^{m+1}}\right]\left[ \left(\frac{\varrho^{m+1}-\chi_1^{m+1}}{\varrho-\chi_1}\right)+\left(\frac{\chi_2^{m+1}-\varrho^{m+1}}{\chi_2-\varrho}\right)\right] \\&&-\frac{\varrho^k}{(\chi_{4} -\chi_{3}) }\left[\frac{1-q_2}{1-q_2^{k+1}}\right]\left[\left(\frac{\rho^{k+1}-\chi_3^{k+1}}{\ \rho-\chi_3}\right)+\left(\frac{\chi_4^{k+1}-\rho^{k+1}}{\chi_4-\rho}\right)\right]+\mathcal{Z} \Big\vert \\ \\ &&\leq\frac{ \Delta\mathcal{M}\sqrt[\tau_2]{\left(\mathcal{C}_{q_{1}}+\mathcal{D}_{q_{1}}\right) (\mathcal{C}_{q_{2}}+\mathcal{D}_{q_{2}})}\left[ \left( \varrho-\chi_{1} \right) ^{2}+\left( \chi_{2} -\varrho\right) ^{2}\right] \left[ \left( \rho-\chi_{3} \right) ^{2}+\left( \chi_{4} -\rho\right) ^{2}\right]}{\sqrt[\tau_1]{[1+\tau_1]_{q_1}[1+\tau_1]_{q_2}}} , \end{eqnarray} |
where
\begin{align*} &&\mathcal{Z} = \frac{1}{(\chi_2-\chi_1)(\chi_4-\chi_3)}\left[\frac{1-q_1}{1-q_1^{m+1}}\right] \left[\frac{1-q_2}{1-q_2^{k+1}}\right]\Bigg[\left(\frac{\varrho^{m+1}-\chi_1^{m+1}}{\varrho-\chi_1}\right) \left(\frac{\rho^{k+1}-\chi_3^{k+1}}{\rho-\chi_3}\right)\\&&+\left(\frac{\varrho^{m+1}-\chi_1^{m+1}} {\varrho-\chi_1}\right)\left(\frac{\chi_4^{k+1}-\rho^{k+1} }{\chi_4-\rho}\right)+\left(\frac{\chi_2^{m+1}-\varrho^{m+1}}{\chi_2-\varrho}\right) \left(\frac{\rho^{k+1}-\chi_3^{k+1}}{\rho-\chi_3}\right)\\ &&+\left(\frac{\chi_2^{m+1}-\varrho^{m+1}}{\chi_2-\varrho}\right)\left(\frac{\chi_4^{k+1}-\rho^{k+1}}{\chi_4-\rho}\right)\Bigg], \end{align*} |
\begin{align*} \mathcal{C}_{q_{k}} & = 1-\frac{(1-q_k)}{n}\sum\limits_{i = 1}^{n}\sum\limits_{e = 0}^{\infty}s^iq_{k}^{e}\left(1-q_{k}^{e}\right)^i, \\ \mathcal{D}_{q_{k}} & = 1-\frac{1}{n}\sum\limits_{i = 1}^{n}s^i\left( \frac{1-q_k}{1-q^{i+1}_k}\right). \end{align*} |
Remark 6. Applying the same idea as in Proposition 2 and using Theorems 3.1, 3.2 and their corresponding corollaries, and choosing suitable functions, for example \mathcal{K}(\varrho, \rho) = \varrho^m\times\rho^{k}, \, m, k > 1 and \varrho, \rho > 0; \; \mathcal{K}(\varrho, \rho) = \frac{1}{\varrho\rho}, \, \varrho, \rho > 0; \; \mathcal{K}(\varrho, \rho) = e^{\varrho+\rho}, \, \varrho, \rho\in \mathbb{R}, and so on, we can obtain other new interesting inequalities for special means. We omit their proofs and the details are left to the interested readers.
In this paper, we have defined several new partial post quantum derivatives and integrals for the functions with two variables, provided some new generalizations in the frame of a new class of convex functions named n -polynomial s -type convex functions, found a new version (p_1p_2, q_1q_2) -Ostrowski type inequality via the class of n -polynomial s -type convex functions on co-ordinates, established a twice partial integral identitity involving (p_1p_2, q_1q_2) -differentiable functions, and generalized the Ostrowski type inequality. Our results are the generalizations of many previous known results, and our ideas and approach may lead to a lot of follow-up research.
The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 61673169, 11701176, 11871202).
The authors declare that they have no competing interests.
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