New Milne-type Inequalities for Lipschitzian Functions by Conformable Fractional Operators

New Milne-type Inequalities for Lipschitzian Functions by Conformable Fractional Operators Fatih Hezenci Department of Mathematics, Faculty of Science and Arts, Duzce University, Duzce 81620, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0003-1008-5856 DOI : 10.5281/zenodo.18017673 Abstract In this paper, new Milne-type inequalities are proved by conformable fractional operators for Lipschitzian functions. Moreover, some results are presented by using specific functions and suitable parameters of the obtained theorem. This shows how the general results reduce to simpler and more familiar inequalities. Furthermore, these results may be further generalized by examining other classes of convex functions or by employing different fractional integral operators. Keywords: Milne-type inequalities, Lipschitzian functions, Conformable fractional operators. 1 Introduction Many mathematicians have investigated numerical integration formulas and their corresponding error bounds using various analytical methods. In this context, numerous mathematical inequalities have been established for different classes of functions, such as convex, bounded, and Lipschitz functions. For instance, Dragomir and Agarwal [ 1 ] derived error bounds for the trapezoidal inequalities by employing the properties of convex functions. In another study, Dragomir [ 2 ] is introduced Simpson-type inequalities and examined their applications to quadrature formulas within numerical analysis. Moreover, Budak et al. [ 3 ] studied several forms of Simpson-type inequalities for differentiable convex functions in the framework of generalized fractional integrals. The three-point Newton-Cotes quadrature rule, commonly known as Simpson’s second rule, has a significant role in numerical integration. Since it employs three-point quadratic kernels, these formulas are often referred to as Newton-type results. Milne-type inequalities can be viewed as extensions of Newton-type formulas, using higher-order or generalized kernels to achieve better approximation accuracy. Recently, several studies have focused on deriving and extending Milne-type inequalities by using convexity properties and fractional integral operators, which provide wider applications in numerical analysis. For instance, in [ 4 ], some error bounds for Newton-type inequalities in numerical integration have been established by employing convex M17-1 KOSC-2025 Proceedings functions. Likewise, certain Newton-type inequalities based on convexity are presented in [ 5 ], along with applications to specific cases of real functions. Several new estimates of Milne’s quadrature rule have been derived by Djenaoui and Meftah [ 6 ], particularly for functions whose first derivative is s -convex. The paper [ 7 ] presents error bounds for Milne-type inequalities applied to functions of bounded variation. Moreover, in [ 8 ], fractional Milne-type inequalities are derived using differentiable convex functions. For more details, the reader is referred to the paper [9]. 2 Preliminaries Let us put forth some preliminaries which will be utilized in the sequel. i. Simpson’s quadrature formula (Simpson’s 1/3rule) is formulated as follows: Zb a f(x)dx ≈b−a 6f(a) + 4fa+b 2+f(b).(1) ii. Simpson’s second formula or Newton-Cotes quadrature formula (Simpson’s 3 / 8rule (cf. [10])) is formulated as follows: Zb a f(x)dx ≈b−a 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b).(2) Formulae (1) and (2) are provided for any function f with continuous 4 th derivative on [ a, b ] . The most popular Newton-Cotes quadrature involving three-point is Simpson-type inequality is as follows: Theorem 2.1. If f : [a, b]→R is a four times continuously differentiable function on (a, b), and   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞,then one has the following 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. One of the classical closed type quadrature rules is the Simpson 3 / 8rule based on the Simpson 3/8inequality as follows: Theorem 2.2 (See [ 10 ]).If f : [a, b]→R is a four times continuously differentiable function on (a, b),and   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞,then one has the inequality  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−1 b−aZb a f(x)dx ≤1 6480   f(4)  ∞(b−a)4. By using the Newton-Cotes formulas, the Milne’s formula which is of open type is parallel to the Simpson’s formula which is of closed type, since they are held under the same conditions. M17-2 2nd Kocaeli Science Congress, November 19-21, 2025 Theorem 2.3 (See [ 11 ]).Let f : [a, b]→R denote a four times continuously differentiable mapping on (a, b),and let   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞.Then, one has the inequality  1 32f(a)−fa+b 2+ 2f(b)−1 b−aZb a f(x)dx ≤7 (b−a)4 23040   f(4)  ∞.(3) For building our main results, the main definitions of Riemann-Liouville integrals and conformable integrals, which are well known in the literature, are presented as follows: Definition 2.1. The gamma function, beta function, and incomplete beta function are defined Γ (x) := ∞ Z 0 tx−1e−tdt, B(x, y) := 1 Z 0 tx−1(1 −t)y−1dt, and B(x, y, r) := r Z 0 tx−1(1 −t)y−1dt, respectively for x, y ∈R+. Definition 2.2 (See [ 12 ]).The Riemann-Liouville integrals Jβ a+f ( x )and Jβ b−f ( x )of order β > 0 are given by Jβ a+f(x) = 1 Γ(β)Zx a (x−t)β−1f(t)dt, x > a (4) and Jβ b−f(x) = 1 Γ(β)Zb x (t−x)β−1f(t)dt, x < b, (5) respectively for f∈L1 [ a, b ]. The Riemann-Liouville integrals equals to the classical integrals for the case of β= 1. Jarad et al. investigated the fractional conformable integral operators in paper [ 13 ]. They also provided certain characteristics and relationships between these operators and several other fractional operators. The fractional conformable integral operators are defined as follows: Definition 2.3 (See [ 13 ]).The fractional conformable integral operator βJα a+f ( x )and βJα b−f ( x ) of order β∈C,Re(β)>0and α∈(0,1] are described by βJα a+f(x) = 1 Γ(β)Zx a(x−a)α−(t−a)α αβ−1f(t) (t−a)1−αdt, t > a (6) and βJα b−f(x) = 1 Γ(β)Zb x(b−x)α−(b−t)α αβ−1f(t) (b−t)1−αdt, t < b, (7) respectively for f∈L1[a, b]. 2nd Kocaeli Science Congress, November 19-21, 2025 M17-3 KOSC-2025 Proceedings If we choose α = 1, then the fractional integral in (6) and (7) coincides with the RiemannLiouville fractional integral in (4) and (5) , respectively. Some recent results according to fractional integral inequalities, see [15,16] and the references cited therein. 3 An essential identity Lemma 3.1 (See [ 14 ]).Let us note that f : [ a, b ] →R is an absolutely continuous function ( a, b ) so that f′∈L1[a, b]. Then, the following equality holds: 1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4−αβΓ (β+ 1) 2 (b−a)αβ hβJα b−f(a) + βJα a+f(b)i(8) =αβ(b−a) 2 4 X i=1 Ii, where                                                      I1= 1 4 R01−(1−t)α αβ[f′(tb + (1 −t)a)−f′(ta + (1 −t)b)] dt, I2= 1 2 R1 41−(1−t)α αβ−2 3αβ[f′(tb + (1 −t)a)−f′(ta + (1 −t)b)] dt, I3= 3 4 R1 21−(1−t)α αβ−1 3αβ[f′(tb + (1 −t)a)−f′(ta + (1 −t)b)] dt, I4= 1 R3 41−(1−t)α αβ−1 αβ[f′(tb + (1 −t)a)−f′(ta + (1 −t)b)] dt. 4 Milne-type inequalities for Lipschitzian functions In this section, we consider some Milne-type inequalities for Lipschitzian functions by using conformable fractional integrals. Theorem 4.1. Suppose that the assumptions of Lemma 3.1 are valid. If f′ is a L -Lipschitzian function on [a, b],then the following inequality holds:  1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4−αβΓ (α+ 1) 2 (b−a)αβ hβJα b−f(a) + βJα a+f(b)i ≤αβ(b−a)2 2L{Ω1(α, β)−2Ω5(α, β) + Ω2(α, β)−2Ω6(α, β) +2Ω7(α, β)−Ω3(α, β) + 2Ω8(α, β)−Ω4(α, β)}. M17-4 2nd Kocaeli Science Congress, November 19-21, 2025 Here,                                                      Ω1(α, β) = 1 4 R01−(1−t)α αβ dt =1 αβ+1 hB1 α, β + 1−B1 α, β + 1,3 4αi, Ω2(α, β) = 1 2 R1 41−(1−t)α αβ−2 3αβ dt, Ω3(α, β) = 3 4 R1 21−(1−t)α αβ−1 3αβ dt, Ω4(α, β) = 1 R3 41−(1−t)α αβ−1 αβ dt =1 4αβ−1 αβ+1 B1 α, β + 1,1 4α, and                                                                            Ω5(α, β) = 1 4 R01−(1−t)α αβ tdt =1 αβ+1 hB1 α, β + 1−B1 α, β + 1,3 4α−B2 α, β + 1+B2 α, β + 1,3 4αi Ω6(α, β) = 1 2 R1 41−(1−t)α αβ−2 3αβ tdt, Ω7(α, β) = 3 4 R1 21−(1−t)α αβ−1 3αβ tdt, Ω8(α, β) = 1 R3 41−(1−t)α αβ−1 αβ tdt =7 32αβ−1 αβ+1 hB1 α, β + 1,1 4α−B2 α, β + 1,1 4αi. Proof. With the aid of Lemma 3.1 and since f′is L-Lipschitzian function, we have  1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4 −αβΓ (α+ 1) 2 (b−a)αβ hβJα b−f(a) + βJα a+f(b)i = αβ(b−a) 2 ×       1 4 Z 01−(1 −t)α αβ f′(tb + (1 −t)a)−f′(ta + (1 −t)b)dt 2nd Kocaeli Science Congress, November 19-21, 2025 M17-5 KOSC-2025 Proceedings + 1 2 Z 1 4 "1−(1 −t)α αβ −2 3αβ#f′(tb + (1 −t)a)−f′(ta + (1 −t)b)dt + 3 4 Z 1 2 "1−(1 −t)α αβ −1 3αβ#f′(tb + (1 −t)a)−f′(ta + (1 −t)b)dt + 1 Z 3 4 "1−(1 −t)α αβ −1 αβ#f′(tb + (1 −t)a)−f′(ta + (1 −t)b)dt        ≤αβ(b−a) 2       1 4 Z 01−(1 −t)α αβf′(tb + (1 −t)a)−f′(ta + (1 −t)b)dt + 1 2 Z 1 4 1−(1 −t)α αβ −2 3αβf′(tb + (1 −t)a)−f′(ta + (1 −t)b)dt + 3 4 Z 1 2 1−(1 −t)α αβ −1 3αβf′(tb + (1 −t)a)−f′(ta + (1 −t)b)dt + 1 Z 3 4 1−(1 −t)α αβ −1 αβf′(tb + (1 −t)a)−f′(ta + (1 −t)b)dt       ≤αβ(b−a) 2       1 4 Z 01−(1 −t)α αβ L(1 −2t) (b−a)dt + 1 2 Z 1 4 1−(1 −t)α αβ −2 3αβ L(1 −2t) (b−a)dt M17-6 2nd Kocaeli Science Congress, November 19-21, 2025 REFERENCES + 3 4 Z 1 2 1−(1 −t)α αβ −1 3αβ L(2t−1) (b−a)dt + 1 Z 3 4 1−(1 −t)α αβ −1 αβ L(2t−1) (b−a)dt       =αβ(b−a)2 2L × {Ω1(α, β)−2Ω5(α, β) + Ω2(α, β)−2Ω6(α, β) +2Ω7(α, β)−Ω3(α, β) + 2Ω8(α, β)−Ω4(α, β)} Remark 4.1. By taking α= 1 in Theorem 4.1, the following Milne-type inequality holds:  1 32fa+ 3b 4−fa+b 2+ 2f3a+b 4−Γ (β+ 1) 2 (b−a)βhJβ a+f(b) + Jβ b−f(a)i ≤(b−a)2 2L{Ω1(1, β)−2Ω5(1, β) + Ω2(1, β)−2Ω6(1, β) +2Ω7(1, β)−Ω3(1, β) + 2Ω8(1, β)−Ω4(1, β)}. This is established by Hezenci and Budak in paper [17, Theorem 5]. Remark 4.2. Let us consider β = 1 and α = 1 in Theorem 4.1. Then, the following Milne-type inequality holds:  1 32f3a+b 4−fa+b 2+ 2fa+ 3b 4−1 b−aZb a f(t)dt ≤(b−a)2 24 L. This is established by Hezenci and Budak in paper [ 17 , Corollary 4]. This inequality helps us to establish the error bound for Milne’s rule. References [1] Dragomir, S.S.; Agarwal, R. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 1998, 11, 91–95. [2] Dragomir S.S.; Agarwal R.P.; Cerone P. On Simpson’s inequality and applications, J. Inequal. Appl., 5, 533–579, 2000. [3] Budak H.; Hezenci F.; Kara H. On parametrized inequalities of Ostrowski and Simpson type for convex functions via generalized fractional integral, Math. Methods Appl. Sci., 44(30), 12522–12536, 2021. 2nd Kocaeli Science Congress, November 19-21, 2025 M17-7 KOSC-2025 Proceedings [4] Iftikhar, S.; Kumam, P.; Erden, S. Newton’s-type integral inequalities via local fractional integrals. Fractals 2020, 28, 2050037. [5] Gao, S.; and Shi, W. On new inequalities of Newton’s type for functions whose second derivatives absolute values are convex, International Journal of Pure and Applied Mathematics, 74(1), 33–41, 2012. [6] Djenaoui, M. and Meftah, B. (2022). Milne type inequalities for differentiable s-convex functions, Honam Mathematical J. 44, No. 3, pp. 325-338. [7] Alomari, M.: New error estimations for the Milne’s quadrature formula in terms of at most first derivatives. Konuralp J. Math. 1(1), 17–23 (2013) [8] Ali, M.A., Zhang, Z., Feckan, M. On Some Error Bounds for Milne’s Formula in Fractional Calculus. Mathematics 2023, 11, 146. https://doi.org/10.3390/math11010146 [9] Fatih Hezenci and Hüseyin Budak, Fractional Milne-type inequalities for functions of bounded variation, The 9th International Conference on Control and Optimization with Industrial Applications, (COIA-2024), Turkiye, 2024, 27–29 August, 343–346. [10] Davis P.J.; Rabinowitz P.; Methods of numerical integration, Academic Press, New York-San Francisco-London, 1975. [11] Booth, A. D., (1966). Numerical methods, 3rd Ed., Butterworths, California. [12] Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J. (2006). Theory and applications of fractional differential equations (Vol. 204). Elsevier, Amsterdam. [13] F. Jarad, E. Uğurlu, T. Abdeljawad, and D. Baleanu, On a new class of fractional operators, Advances in Difference Equations, (2017), 2017, 247. [14] Fatih Hezenci and Huseyin Budak, On Error Bounds for Milne’s Formula in Conformable Fractional Operators, Ukrainian Mathematical Journal, 76, pages 1214–1232. [15] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279, 2015, 57-66. [16] F. Jarad, T. Abdeljawad, D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl. 10(5), 2017, 2607-2619. [17] Fatih Hezenci and Huseyin Budak, Milne-Type Inequalities for Various Function Classes, 6th International Conference on Mathematical and Related Sciences: Current Trends and Developments (ICMRS 2023) Turkiye, 2023, 20–22 November, 20–31. M17-8 2nd Kocaeli Science Congress, November 19-21, 2025