Abstract
The new outcomes of the present paper are -analogues ( stands for quantum calculus) of Hermite-Hadamard type inequality, Montgomery identity, and Ostrowski type inequalities for -convex mappings. Some new bounds of Ostrowski type functionals are obtained by using Hölder, Minkowski, and power mean inequalities via quantum calculus. Special cases of new results include existing results from the literature.
1. Introduction
Integral inequalities provide a notable role in both pure and applied mathematics in the light of their wide applications in numerous regular and human sociologies, while convexity hypothesis has stayed a significant apparatus in the foundation of the theory of integral inequalities. The classical inequalities are helpful in numerous down-to-earth issues. In recent years, many authors (see [1–12]) proved numerous inequalities associated with the functions of bounded variation, Lipschitzian, monotone, absolutely continuous, convex functions, -convex, -convex, and -times differentiable mappings with error estimates. Integral inequalities have been studied extensively by several researchers either in classical analysis or in the quantum one. In many practical problems, it is important to bound one quantity by another quantity. The classical inequalities including Hermite-Hadamard and Ostrowski type inequalities are very useful for this purpose (see [13–24]). Ostrowski type inequalities are well known to study the upper bounds for approximation of the integral average by the value of the function. In [25], Dragomir and Fitzpatrick have constructed Hermite-Hadamard’s inequality which is specified to -convex functions in the second sense as follows:
Theorem 1. Suppose is an -convex function in second sense, , and suppose . If , then the integral inequality is valid: where .
The following Montgomery equality is established by Alomari (see [26]):
Lemma 2. Assume that is differentiable function on in which for . If , then we have the equality:
for each .
By using Lemma 2, Alomari et al. in [26] had proved the Ostrowski type inequality, which holds for -convex mappings in second sense as follows:
Theorem 3. Assume is a differentiable on and such that for If is -convex mapping in the second sense on unique and , then the following result holds: for each .
Theorem 4. Suppose that is the differentiable on and , where with If absolute value of is -convex function in the second sense in for unique , , and , then following integral inequality holds: for each
Theorem 5. Suppose that is differentiable on and in which for If the absolute value of is -convex function in for static and , then the following integral inequality holds: for each
Theorem 6. Suppose be the differentiable on and in which for If absolute value of is a -convex mapping in second sense on for static and , we have for each
The renowned mathematician Euler started the investigation of -calculus in the eighteenth century by presenting Newton’s work of limitless series. This subject has gotten extraordinary consideration by numerous specialists, and consequently, it is considered an in-corporative subject among math and material science. In the mid-20th century, Jackson (1910) has begun a symmetric investigation of calculus and presented -distinct integrals. The subject of quantum analytic has various applications in different spaces of arithmetic and physical science like number hypothesis, combinatorics, symmetrical polynomials, essential hypermathematical functions, quantum theory, and mechanics and in the hypothesis of relativity. Quantum calculus can be seen as a scaffold among arithmetic and material science. It has been shown that quantum calculus is a subfield of the more general mathematical field of time scales calculus. Time scales provide a unified framework for studying dynamic equations on both discrete and continuous domains. In [27, 28], -Bernoulli and dynamic inequalities associated with Leibniz integral rule on time scales were studied. In studying quantum calculus, we are concerned with a specific time scale, called the -time scale. The study of -integral inequalities is also of great importance. Integral inequalities have been studied extensively by several researchers either in classical analysis or in the quantum one.
The following -Hermite-Hadamard and -Ostrowski type integral inequalities were proved by Tariboon and Ntouyas (see Theorems 3.2 and 3.5 [29]):
Theorem 7. Let be a -differentiable function with continuous on and . Then, we have
Theorem 8. Suppose where is an interval, be a -differentiable in open interval belonging to interior for . If for all and , then the integral inequality is valid:
for all The least value of constant on RHS of inequality (8) is .
The following -Ostrowski type integral inequalities for convex functions were proved by Noor et al. (see [30]):
Theorem 9. Let be -differentiable mapping for and in which for If is convex mapping for some static and , then we have the following -integral inequality: for each
Theorem 10. Assume that is -differentiable mapping on and in which for If is a convex function in second sense on unique , and , then we have the -integral inequality: for each
The aim of this work is to find q-analogues of Hermite-Hadmard and Ostrowski type integral inequalities for functions whose -derivatives are -convex in the second sense. An interesting feature of our results is that they provide new estimates and good approximation on such types of inequalities involving -integrals.
2. Basic Essentials
2.1. Convex Function
Let be the function; it is said to be convex function on interval if holds for all and .
In [31], -convex functions in the second sense have been introduced by Hudzik and Maligranda as follows:
2.2. -Convex Function
A mapping is said to be -convex if for each , and for unique
2.3. -Derivative [32]
For a continuous mapping -derivative at is
Also, for , one may find the following evaluations:
Here, and also, we have
2.4. -Antiderivative [32]
Suppose that be the continuous mapping. Then, -definite integral on is stated as for .
2.5. The Formula of -Integration by Parts [29]
Let be the continuous functions and , Then, the formula of -integration by parts is stated as
Theorem 11. -Hölder Inequality ([4], Theorem 2). Let and be -integrable on and and with ; then, one may obtain the following: Using (19), the following is valid.
2.6. -Minkowski’s Inequality
Let and be a real number then for continuous functions ,
Proof. which gives the required result for positive real numbers such that .
The classical power mean inequality for integrals has the following form for -integral.
2.7. -Power Mean Inequality
Let for real numbers . Let and be continuous functions; then,
Proposition 12. [33]. For each , we have
3. Main Results
3.1. -Hermite-Hadamard Inequality
Theorem 13. Suppose is a -convex mapping in the second sense, in which , and let If then the integral inequality is valid:
Proof. By definition of -convex functions, Hence, Let and in to get From (26) and (28), the desired result is
3.2. -Ostrowski Type Inequalities
To prove some -Ostrowski type inequalities, it needs to establish the following Montgomery identity for -integrals:
Lemma 14. Let be a -differentiable on in which for . If , we have the following -integral equality which is valid:
for each
By using Lemma 14, we have constructed the following Ostrowski type inequalities, which hold for -convex functions in the second sense:
Theorem 15. Let is a -differentiable mapping on and in which for If the absolute value of is -convex in second sense on for unique and is bounded by , , we have been seeing that the following -integral inequality is valid: for each
Proof. Since is -convex function in the second sense on , therefore, Lemma 14 gives the following:
Theorem 16. Suppose that is a -differentiable on and in which for If is a -convex function in second sense on for some static and is bounded by , , then the -integral inequality is valid: for each
Proof. From Lemma 14 and keeping in view the well-known -analogue of Hölder inequality, we have It completes the proof.
Theorem 17. Let is a -differentiable mapping on such as in which for If the absolute value of is a -convex mapping in the second sense on for unique , , and , , we have seen that the -integral inequality is valid: for each
Proof. Lemma 14 and keeping in view the well-known -analogue of power-mean inequality, we have It completes the proof.
Theorem 18. Suppose that is a -differentiable mapping on such that in which for If is -convex function in second sense on for some and and , then the -integral inequality is valid: for each
Proof. Lemma 3.1 and keeping in view the familiar -analogue of Hölder inequality, we have
Remark 19. In Theorem 13, if we choose , then (24) diminishes the inequality (1) of Theorem 1.
Remark 20. In Theorem 13, if we choose , then (24) diminishes the inequality (7) of Theorem 7.
Remark 21. In Theorem 15, if we fixed , then (31) reduces the inequality (3) of Theorem 3.
Remark 22. In Theorem 15, if we take , then (31) diminishes the inequality (9) of Theorem 9.
Remark 23. In Theorem 16, if we take , then (34) reduces the inequality (4) of Theorem 4.
Remark 24. In Theorem 16, if we choose , then (34) diminishes the inequality (10) of Theorem 10.
Remark 25. In Theorem 17, if we take , then (36) diminishes the inequality (5) of Theorem 5.
Remark 26. In Theorem 18, if we take , then (38) diminishes the inequality (6) of Theorem 6.
4. Conclusion
By the virtue of -calculus, some integral inequalities are proved, which provides a method to study more properties of -integrals via other classes of integral inequalities. -Hermite-Hadmard and -Ostrowski type integral inequalities have provided new estimates and good approximations in comparison with existing Hermite-Hadamard and Ostrowski inequalities. In similar fashion, the same methods can be applied to other inequalities, including Simpson’s and trapezoidal inequalities for different classes of -convex functions.
Data Availability
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Conflicts of Interest
The authors declare that there is no conflict of interest.