Abstract
In this paper, two versions of the Hadamard inequality are obtained by using Caputo fractional derivatives and strongly -convex functions. The established results will provide refinements of well-known Caputo fractional derivative Hadamard inequalities for -convex and convex functions. Also, error estimations of Caputo fractional derivative Hadamard inequalities are proved and show that these are better than error estimations already existing in literature.
1. Introduction
Strongly convex function was introduced by Polyak in [1]. Strong convexity is a strengthening of the notion of convexity; some properties of strongly convex functions are just “stronger versions” of known properties of convex functions. Strongly convex functions have been used for proving the convergence of a gradient-type algorithm for minimizing a function. They play an important role in the optimization theory and mathematical economics.
Definition 1. Let be a convex subset of be a normed space. A function is called strongly convex function with modulus if it satisfies
and .
Many authors have been inventing the properties and applications of strongly convex function, for detailed information, see [2–6].
The concepts of -convex functions and strongly -convex functions were introduced in [7, 8], respectively. Toader [7] gave the idea of -convex functions as follows.
Definition 2. A function is said to be -convex, where , if for every and , we have
Lara et al. introduced strongly -convex functions as follows:
Definition 3 (see [8]). A function is called strongly -convex function with modulus if for and
Nowadays, fractional integral inequalities are in the study of several researchers (see, [9–17] and references therein), where they have used different kinds of well-known functions and fractional integral operators. Here, in this paper, we are interested to produce fractional integral inequalities for Caputo fractional derivatives of strongly -convex functions. The Caputo fractional derivative operators are defined as follows.
Definition 4 (see [18]). Let . The Caputo fractional derivatives of order of are defined as follows: where . If and usual derivative of order exists, then Caputo fractional derivative coincides with , whereas coincides with with exactness to a constant multiplier . In particular, we have where and .
The Hadamard inequality is another interpretation of convex function. It is stated as follows.
Definition 5 (see [19]). Let be a convex function on interval and where . Then, the following inequality holds: If order in (6) is reversed, then it holds for concave function.
Farid et al. [20] have proved the following Hadamard inequality for Caputo fractional derivatives of convex functions:
Theorem 6. Let be the function with and . Also, let be positive and convex function on Then, the following inequality holds for Caputo fractional derivatives:
They also established the following identity.
Lemma 7 (see [20]). Let , be the function such that . Then, the following equality for Caputo fractional derivatives holds:
Farid et al. [20] also proved the following inequality for Caputo fractional derivatives.
Theorem 8. Let be the function with and also let be convex on Then, the following inequality for Caputo fractional derivatives holds:
Kang et al. [21] proved the following version of the Hadamard inequality for Caputo fractional derivatives.
Theorem 9. Let be a positive function with and . If is convex function on , then the following inequality for Caputo fractional derivatives holds: Farid et al. [22] established the following identity.
Lemma 10. Let be a differentiable mapping on with . If , then the following equality for Caputo fractional derivatives holds:
Kang et al. [21] also proved the following inequalities for Caputo fractional derivatives.
Theorem 11. Let be a differentiable mapping on with and . If is convex on for , then the following inequality for Caputo fractional derivatives holds:
Theorem 12 (see [21]). Let be a function with and . If is convex on for , then the following inequality for Caputo fractional derivatives holds: We will study all of the above fractional inequalities for strongly -convex functions and at the same time will obtain their generalizations and refinements. In Section 2, we will give refinements of two versions of the Hadamard inequality for Caputo fractional derivatives. We will connect their particular cases with some well-known results. In Section 3, by applying known identities, we will give refinements of error estimations of the Hadamard inequalities.
2. Main Results
The following result is the generalization of Theorem 6 which in a particular case also provides its refinement.
Theorem 13. Let , be a positive function. If is a strongly -convex function with modulus , then the following inequality for Caputo fractional derivatives holds: with .
Proof. Since is strongly -convex function with modulus , for , we have Let and , . Then, we have By multiplying (16) with on both sides and making integration over , we get By using change of variables and computing the last integral, from (17), we get Further, it takes the following form Since is strongly -convex function with modulus , for , then one has By multiplying (20) with on both sides and making integration over , we get By using change of variables and computing the last integral, from (21), we get Further, it takes the following form Inequalities (19) and (11) constituted the required inequality.
The consequences of Theorem 13 are stated in the following corollary and remark:
Corollary 14. By setting in inequality (14), we will get ([23], Theorem 6)
Remark 15. If and in (14), then we will get the fractional Hadamard inequality stated in Theorem 6.
The upcoming result is the refinement of another version of the Hadamard inequality for Caputo fractional derivatives stated in a theorem in [19].
Theorem 16. Under the assumptions of Theorem 13, the following inequality for Caputo fractional derivatives holds: with .
Proof. Let and , in (15), then we have By multiplying (26) with on both sides and making integration over , we get By using change of variables and computing the last integral, from (27), we get Further, it takes the following form: Since is strongly -convex function and , we have the following inequality: By multiplying (30) with on both sides and making integration over , we get By using change of variables and computing the last integral, from (31), we get Further, it takes the following form From (29) and (33), (25) can be obtained.
Corollary 17. By setting in inequality (25), we will get ([23], Theorem 7)
Remark 18. If and in (25), then we will get the fractional Hadamard inequality stated in Theorem 9.
3. Error Bounds of Fractional Hadamard Inequalities
In this section, we give refinements of the error bounds of fractional Hadamard inequalities for Caputo fractional derivatives.
Theorem 19. Let be a differentiable mapping on . If is a strongly -convex function on , then the following inequality for Caputo fractional derivatives holds: with .
Proof. Since is strongly -convex function on , for , we have By applying Lemma 7 and the strongly -convexity of , we find In the following, we compute integrals appearing on the right side of inequality (37): By putting the values of (38) and (39) in (37), we get (35).
Corollary 20. By setting in inequality (35), we will get ([23], Theorem 8)
Remark 21. If and in (35), we get the fractional Hadamard inequality which is stated in Theorem 8.
By using Lemma (11), we give the following Caputo fractional derivative inequality.
Theorem 22. Let , , be a differentiable mapping on . If is strongly -convex function on , for , then the following inequality for Caputo fractional derivatives holds:
Proof. By taking in Lemma 10 and using the strongly -convexity of , we have Now, applying Lemma 10, using power mean inequality, we have By using strongly -convexity of , we have inequality (41) is obtained.
Corollary 23. By setting in inequality (41), we will get ([23], Theorem 9)
Remark 24. (i) If and in (41), then we will get inequality stated in Theorem 11.
Theorem 25. Let , be a differentiable mapping on . If is a strongly -convex function on , for , then the following inequality for Caputo fractional derivatives holds:
Proof. By taking in Lemma 10 and with the help of modulus property, we get Now applying Holder’s inequality, we get Using strongly -convexity of , we get Here, we have used the fact that , where , . This completes the proof.
Corollary 26. By setting in inequality (46), we will get ([23], Theorem 10)
Remark 27. If and in (46), then we will get inequality stated in Theorem 12.
Data Availability
There is no data required for this paper.
Conflicts of Interest
The authors declare that they have no conflicts of interest.