Fractional Simpson-Like Inequalities for Thrice Differentiable Functions

Fractional Simpson-Like Inequalities for Thrice Differentiable Functions Hasan Kara1, Hüseyin Budak2, Ather Qayyum3 1Department of Mathematics, Faculty of Science and Arts, Duzce University, Türkiye 2Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Türkiye 3Institute of Mathematical Sciences, Universiti Malaya, Malaysia Corresponding author: dratherqa[email protected]y ORCID IDs: First Author: 0000-0002-2075-944X Second Author: 0000-0001-8843-955X Third Author: 0000-0003-4149-5305 DOI : 10.5281/zenodo.18017837 Abstract In this investigation, new fractional Simpson-like inequalities for thrice differentiable functions are established by using bounded functions. Moreover, with the aid of Lipschitzian functions, some new fractional Simpson-like inequalities for thrice differentiable functions are also obtained. In addition, by considering specific parameter values, some known results are recovered as special cases of the main theorems, particularly those corresponding to the Riemann–Liouville fractional integral operator. Keywords: Simpson type inequalities, Convex functions, Fractional integrals, Third derivative 1 Introduction and Preliminaries The inequality theory is a considerable topic and remains an interesting research area with numerous number of applications in many mathematical fields. In addition, convex functions have also a significant place in the theory of inequality. Many inequalities have been investigated for convex functions but the most prominent is the Simpson type inequality, because of the its rich geometrical importance and applications. The following inequality is one of well-known outcome in the literature as the classical Simpson type inequality for four times continuously differentiable functions. Theorem 1.1. Let f : [a, b]→R denote a four times continuously differentiable function on (a, b), and let   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞.Then, one has the inequality  1 6f(a)+4fa+b 2+f(b)−1 b−aZb a f(x)dx ≤1 2880   f(4)  ∞(b−a)4. The convex theory is an impressive method to solve a large number of problems from varied branches of mathematics. Hence, many papers are established the Simpson type inequalities for M20-1 KOSC-2025 Proceedings convex function. For instance, Sarikaya et al. proved the new variants of Simpson type inequalities with the aid of differentiable convex function in the papers [ 32 ]. For results with respect to these type of inequalities one can see Refs. [ 25 , 12 ] and the references therein. In addition to these, Simpson type inequalities for various convex classes have been studied extensively by many authors (see, [23,17,27] and the references therein). Twice differentiable convex functions have been established by many authors to get significant inequalities. For example, some Simpson type inequalities were presented for functions whose absolute values of derivatives are convex in [ 31 ]. Moreover, J. Park proved new estimates on generalization of Hadamard, Ostrowski and Simpson type inequalities to the case of functions whose second derivatives in absolute value at certain powers are convex and quasi–convex functions in [ 28 ]. Furthermore, it was proved some fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex in [ 7 ]. It can be referred to [15,3,34,9] for further information about twice differentiable functions. Some inequalities of Simpson type for functions whose three derivatives in absolute value are the class of ( α, m )-geometric-arithmetically-convex functions established in the paper [ 20 ]. In addition to this, some applications to special means of positive real numbers were given in this paper. In [ 1 ], some inequalities of Simpson type for quasi-convex functions in terms of third derivatives are presented and applications to Simpson numerical quadrature rule is also established. Furthermore, Ozdemir et. al. presented some inequalities by s -convex and s -concave functions in [ 26 ]. In the paper [ 11 ], the authors proved new inequalities of Simpson type for functions whose third derivatives are extended s-convex functions, and apply these inequalities to provide some inequalities of special means. In the paper [ 29 ], J. Park established some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex and s− convex in the second sense. For further information related to these subject, we refer to reader to [35,10] and the references therein. Mathematical preliminaries of fractional calculus theory, which will be used throughout this paper, will be presented as follows: The well-known Gamma function and Beta function are defined by Γ(x) := ∞ Z 0 tx−1e−tdt and β(x, y) := 1 Z 0 tx−1(1 −t)y−1dt =Γ(x)Γ(y) Γ(x+y), respectively for 0< x, y < ∞and x, y ∈R. Let us consider f∈L1 [ a, b ] . The Riemann–Liouville integrals Jα a+f and Jα b−f of order α > 0 with a≥0are defined by Jα a+f(x) = 1 Γ(α)Zx a (x−t)α−1f(t)dt, x > a M20-2 2nd Kocaeli Science Congress, November 19-21, 2025 and Jα b−f(x) = 1 Γ(α)Zb x (t−x)α−1f(t)dt, x < b, respectively. Let us note that J0 a+f ( x ) = J0 b−f ( x ) = f ( x ) . Let us also note that α = 1 in above. Then, the fractional integral becomes to the classical integral. The fractional integral inequalities and applications have been established with the aid of the Riemann–Liouville fractional integral. For instance, Sarikaya et al. established some Simpson type inequalities to the case of functions whose second derivatives are convex [ 33 ]. Moreover, Iqbal et. al. generalized the Simpson type inequalities based on differentiable functions to Riemann-Liouville fractional integrals in the paper [ 16 ]. Some Simpson type inequalities using s -( α, m ) − convex function by Riemann–Liouville fractional integrals were given in [ 21 ]. The reader is referred to [5,6,13,14,24] and the references therein for more information and unexplained subjects about several properties of Riemann–Liouville fractional integrals and various fractional integral operators. Whereas Simpson type inequalities for Riemann–Liouville fractional integrals have been considered by the authors, some mathematicians have also established the Simpson type inequalities for other type of fractional integrals such as k–fractional integral Conformable fractional integrals, Katugampola fractional integrals, etc. Concerning some papers related to the these subjects see [19,18,30,22,2], and references therein. In [ 8 ], Hezenci and Budak previously established Simpson-like inequalities for thrice differentiable functions by employing the Riemann–Liouville fractional integrals. In their study, several inequalities and related consequences were derived by utilizing the convexity of the involved functions together with Hölder’s inequality and the power mean inequality. Before obtaining these results, the following lemma was first established. Lemma 1.1. [ 8 ] Let us note that f : [ a, b ] →R is a three times differentiable function ( a, b ) such that f′′′ ∈L1([a, b]). Then, the following equality holds: (1) 1 (α+ 1)(α+ 2) f(a) + α2+ 3αfa+b 2+f(b)−2α−1Γ(α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) =(b−a)3 16(α+ 1)(α+ 2) 1 Z 0tα+2 −t2f′′′ t 2b+2−t 2a−f′′′ t 2a+2−t 2bdt. The aim of this study is to establish several new Simpson-like inequalities for thrice differentiable functions involving Riemann–Liouville fractional integrals by employing the lemma presented above. The obtained lemma is utilized throughout the subsequent sections to derive the main results of the paper. The overall organization of the study is arranged as follows. In Section 2, inequalities are developed for the class of bounded functions. In Section 3, further inequalities are obtained for Lipschitzian functions, where the use of the established lemma enables the derivation of sharper estimates. By considering particular choices of the parameters, additional corollaries and new consequences are also produced. Finally, in Section 4, concluding remarks and possible directions for future research are discussed. 2nd Kocaeli Science Congress, November 19-21, 2025 M20-3 KOSC-2025 Proceedings 2 Bounded functions: Simpson like inequalities with fractional integrals In this section, we deal with some Simpson like inequalities for bounded functions via fractional integrals. Theorem 2.1. Note that the conditions of Lemma 1.1 hold. If there exist m, M ∈R such that m≤f′′′ (t)≤Mfor t∈(a, b),then it follows  1 (α+ 1)(α+ 2) f(a) + α2+ 3αfa+b 2+f(b)(2) −2α−1Γ(α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤(b−a)3α 48(α+ 1)(α+ 2)(α+ 3) (M−m). Proof. By using the Lemma 1.1, we have 1 (α+ 1)(α+ 2) f(a) + α2+ 3αfa+b 2+f(b)(3) −2α−1Γ(α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) =(b−a)3 16(α+ 1)(α+ 2)    1 Z 0tα+2 −t2f′′′ t 2b+2−t 2a−m+M 2dt + 1 Z 0tα+2 −t2m+M 2−f′′′ t 2a+2−t 2bdt. Through the absolute value of (3), we get  1 (α+ 1)(α+ 2) f(a) + α2+ 3αfa+b 2+f(b) −2α−1Γ(α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤(b−a)3 16(α+ 1)(α+ 2)    1 Z 0tα+2 −t2 f′′′ t 2b+2−t 2a−m+M 2 dt + 1 Z 0tα+2 −t2 m+M 2−f′′′ t 2a+2−t 2b dt. It is known that m≤f′′′(t)≤Mfor t∈(a, b).Then, we have  f′′′ t 2b+2−t 2a−m+M 2 ≤M−m 2,(4)  m+M 2−f′′′ t 2a+2−t 2b ≤M−m 2.(5) M20-4 2nd Kocaeli Science Congress, November 19-21, 2025 With the help of the (4) and (5), we obtain  1 (α+ 1)(α+ 2) f(a) + α2+ 3αfa+b 2+f(b) −2α−1Γ(α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤(b−a)3 16(α+ 1)(α+ 2) ×   1 Z 0tα+1 −t2 M−m 2dt + 1 Z 0tα+1 −t2 M−m 2dt   =(b−a)3 16(α+ 1)(α+ 2) (M−m) 1 Z 0tα+1 −t2dt =(b−a)3α 48(α+ 1)(α+ 2)(α+ 3) (M−m). Corollary 2.1. If we select α= 1 in Theorem 2.1, then we get  1 6f(a) + 4fa+b 2+f(b)−1 b−aZb a f(t)dt ≤(b−a)3 1152 (M−m). Corollary 2.2. Under assumption of Theorem 2.1, if there exist M∈R+ such that |f′(t)| ≤ M for all t∈(a, b),then we have  1 (α+ 1)(α+ 2) f(a) + α2+ 3αfa+b 2+f(b)(6) −2α−1Γ(α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤(b−a)3α 24(α+ 1)(α+ 2)(α+ 3)M. Corollary 2.3. Let us consider α= 1 in Corollary 2.2. Then, the following inequality holds:  1 6f(a) + 4fa+b 2+f(b)−1 b−aZb a f(t)dt ≤(b−a)3 576 M. 3 Lipschitzian functions: Fractional Simpson like inequalities In this section, we give some fractional Simpson like inequalities for Lipschitzian functions. Theorem 3.1. Assume that the assumptions of Lemma 1.1 are valid. If f′′′ isaL-Lipschitzian 2nd Kocaeli Science Congress, November 19-21, 2025 M20-5 KOSC-2025 Proceedings function on [a, b],then the following inequality holds:  1 (α+ 1)(α+ 2) f(a) + α2+ 3αfa+b 2+f(b) −2α−1Γ(α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤(b−a)4α(α+ 7) 192(α+ 1)(α+ 2)(α+ 3)(α+ 4)L. Proof. With the aid of Lemma 1.1 and since f′′′ is L-Lipschitzian function, we have  1 (α+ 1)(α+ 2) f(a) + α2+ 3αfa+b 2+f(b) −2α−1Γ(α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤(b−a)3 16(α+ 1)(α+ 2) ×   1 Z 0tα+2 −t2f′′′ t 2b+2−t 2a−f′′′ t 2a+2−t 2b dt   ≤(b−a)3 16(α+ 1)(α+ 2)    1 Z 0 t2|tα−1| f′′′ t 2b+2−t 2a−f′′′ t 2a+2−t 2b dt   ≤(b−a)3 16(α+ 1)(α+ 2)    1 Z 0 t2|tα−1|L(b−a)|t−1|dt   =(b−a)4 16(α+ 1)(α+ 2)L  1 Z 0 t2|tα−1|dt − 1 Z 0 t3|tα−1|dt  =(b−a)4α(α+ 7) 192(α+ 1)(α+ 2)(α+ 3)(α+ 4)L Corollary 3.1. Consider α = 1 in Theorem 3.1. Then, the following Simpson like inequality holds:  1 6f(a) + 4fa+b 2+f(b)−1 b−aZb a f(t)dt ≤(b−a)4 2880 L. 4 Conclusion In this investigation, new fractional Simpson-like inequalities for thrice differentiable functions have been established by employing bounded functions. 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