New fractional identities, associated novel fractional inequalities with applications to means and error estimations for quadrature formulas

*Correspondence: poom.kum@kmutt.ac.th 4Center of Excellence in Theoretical and Computational Science (TaCS-CoE) & KMUTT Fixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand 5Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Full list of author information is available at the end of the article Abstract


Introduction and preliminaries
Computational and Fractional Analysis are nowadays more and more at the center of mathematics and of other related sciences, either by themselves because of their rapid development, which is based on very old foundations, or because they cover a great variety of applications in the real world. In recent years, fractional calculus (FC) is applied in many phenomena in applied sciences, fluid mechanics, physics and also biology can be described as very effective using the mathematical tools of FC. The fractional derivatives have occurred in many applied sciences equations such as reaction and diffusion processes, system identification, velocity signal analysis, relaxation of damping behavior fabrics and creeping of polymer composites [1,2,26,32]. A set S ∈ R is said to be convex, if (1τ )b 1 + τ b 2 ∈ S, ∀b 1 , b 2 ∈ S, τ ∈ [0, 1].
© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Similarly, a function ϒ : S → R is said to be convex, if Recently, İşcan [3] introduced the class of harmonic convex functions as: A function ϒ : I ⊂ (0, +∞) → R is said to be harmonic convex, if The harmonic property has played a significant role in different fields of pure and applied sciences. In [4] the authors discussed the important role of the harmonic mean in Asian options of stock. Interestingly, harmonic means have applications in electric circuit theory.
To be more precise, the total resistance of a set of parallel resistors is just half of the total resistor's harmonic means. For example, if R 1 and R 2 are the resistances of two parallel resistors, then the total resistance is computed by the formula: which is half of the harmonic mean. Noor [5] showed that the harmonic mean also played a crucial role in developing parallel algorithms for solving nonlinear problems. The author used the harmonic means and harmonic convex functions to suggest some iterative methods for solving linear and nonlinear equations.
The theory of convexity also has a wide range of applications in other areas of pure and applied sciences. It also has a great impact on the development of the theory of inequalities. Several inequalities are consequences of the applications of convex functions. Many generalizations, variants and extensions for the convexity have attracted the attention of many researchers. An interesting result pertaining to convex functions is the trapezium inequality (Hermite-Hadamard inequality) that provides an integral average of a continuous convex function on a compact interval. This result reads as: Let ϒ : I = [b 1 , b 2 ] ⊂ R → R be a convex function, then Over the years, a variety of new generalizations of this classical result have been obtained in the literature. For example, İşcan [3] obtained a new refinement of the trapezium inequality using the class of harmonic convex functions. He derived the following version of the trapezium inequality. Let ϒ : I = [b 1 , b 2 ] ⊂ (0, +∞) → R be an harmonic convex function, then We now recall some useful definitions. For brevity, the set of integrable functions on the The left-and right-sided Riemann-Liouville fractional integrals J α b 1 respectively, and (α) is Gamma function. Also, we define J 0 -ϒ of order α, k > 0 with b 1 ≥ 0 are given as follows: respectively. 3 , z) has the following integral representation
The aim of this paper is to derive some new generalizations of fractional trapezium-like inequalities using the class of harmonic convex functions. In order to establish some of our main results, we derive three new fractional integral identities. These identities will be used as auxiliary results. Moreover, in order to show the efficiency of our main results, some applications to special means for positive different real numbers and error estimations for quadrature formulas will also be obtained. We also discuss special cases that show that our results represent significant generalizations and under suitable conditions one can obtain many other new and known results.
Before moving to the main results, let us recall some previously known concepts and results that will help us to obtain our main results. Let : [0, +∞) → [0, +∞) be a function satisfying the following conditions: 1.

Main results
In this section, before we discuss our main results, let us denote, respectively

Generalized trapezium inequality
We now derive a new generalized fractional trapezium-type integral inequality using the class of harmonic convex functions. For brevity, we denote in the following (τ ) := 1 τ .
where m ∈ N.
Proof Since ϒ is an harmonic convex function, then Multiplying both sides by and integrating with respect to τ on [0, 1], we have This implies Now, we prove the second inequality, for this we have Adding (2.1) and (2.2) and multiplying both sides by and integrating with respect to τ on [0, 1], we have Using generalized fractional integrals, we obtain our second inequality. This completes the proof.

Corollary 2.2 If we choose
For m = 1, we obtain

Corollary 2.3 If we choose
For m = 1, we obtain

Auxiliary results
In this subsection, we derive three new fractional integral identities that will be used in the following.
Proof Consider the right-hand side where Similarly, Substituting the values of I 1 and I 2 in I, we obtain our required result.
Substituting the values of I 3 and I 4 in (2.3), we obtain our required result.
Proof Integrating by parts M i for i = 1, 2, 3, 4, and changing the variables, we have Adding 2 (1) , we obtain our required result.

Further results
Now, utilizing auxiliary results obtained in the previous subsection, we derive some further generalized fractional trapezium-like inequalities using the class of harmonic convex functions.

Theorem 2.5 Let
|ϒ | q be an harmonic convex function with 1 p + 1 q = 1, then Proof Using Lemma 2.2, the modulus property, Hölder's inequality and the harmonic convexity of |ϒ | q , we have After simple calculations, we obtain our required result.

Corollary 2.15 Choosing
Proof Using Lemma 2.2, the modulus property, the power mean inequality and the convexity of |ϒ | q , we have After simple calculations, we obtain our required result.
and |ϒ | q be an harmonic convex function with q ≥ 1 and λ, μ ∈ [0, ∞) with λ + μ = 0, then Proof Using Lemma 2.3, the modulus property, the power mean inequality and the harmonic convexity of of |ϒ | q , we have This completes the proof.
Proof By using Lemma 2.4, the property of modulus, Hölder's inequality and the harmonic convexity of |ϒ | q , we obtain the desired result. We omit here the proof.

Corollary 2.27 Taking
If |ϒ | q is an harmonic convex function with q ≥ 1, then Proof By using Lemma 2.4, the property of modulus, the power mean inequality and the convexity of |ϒ | q we obtain the desired result. We omit here the proof.

Corollary 2.28
Taking (τ ) = τ in Theorem 2.10, then Remark For other suitable choices of function , several new interesting inequalities can be found from our results. We omit here their proofs and the details are left to the interested reader.