Fractional Newton-Type Inequalities Involving h-convex Functions

Fractional Newton-Type Inequalities Involving h-convex Functions Hasan Kara1, Hüseyin Budak2, Fatih Hezenci1 1Department of Mathematics, Faculty of Science and Arts, Duzce University, Türkiye 2Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Türkiye Corresponding author: [email protected] ORCID IDs: First Author: 0000-0002-2075-944X Second Author: 0000-0001-8843-955X Third Author: 0000-0003-1008-5856 DOI : 10.5281/zenodo.18017895 Abstract In this study, Newton-type inequalities involving Riemann-Liouville fractional integrals have been obtained for a class of functions that are twice differentiable and h-convex. The main step has been to derive an identity that connects the second derivative of the function with its fractional integrals. This identity has been presented as the fundamental for proving new inequalities under the assumption that the function satisfies the condition of h-convexity. The method adopted here is based on analyzing the structure of the second derivative together with the properties of h -convex functions. The definition of h -convexity has allowed a variable to choose where classical convexity may be seen as a special case. By assuming that the second derivative is h-convex on a given interval, estimates have been established that generalize some known results in the literature related to Newton-type inequalities. Tools from classical analysis, such as Hölder’s inequality and integral inequalities, have been used in the results obtained. Keywords: Newton-type inequalities, h-convex functions, fractional integral 1 Introduction The basic principle of Simpson’s second rule is the three-point Newton-Cotes quadrature rule. Results for three-step quadratic kernel computations are generally described as Newton-type results. These results are recognized to be Newton-type inequalities from the literature. There have been some mathematicians who have been studied to Newton-type inequalities. For instance, several Newton-type inequalities are proved for the case of functions whose second derivatives are convex in paper [ 1 ]. Newton-type inequalities are established by harmonic convex and p -harmonic convex functions in [ 2 ] and [ 3 ], respectively. In paper [ 4 ], some Newton-type inequalities are acquired by post-quantum integrals. Moreover, in paper [ 5 ], some Newton-type inequalities are presented for the case of quantum differentiable convex functions. Furthermore, in paper [ 6 ], some error estimates of Newton-type quadrature formula are given by bounded variation and M21-1 KOSC-2025 Proceedings Lipschitzian mappings. See the references cited in [ 7 , 8 , 9 ] for some recent results associating with Newton-type inequalities. The popularity of fractional calculus has enhanced in recent years since its broad range of applications in various fields of science. Given the importance of fractional calculus, some operators for fractional integrals can be taken into consideration. For example, in paper [ 12 ], some Newton-type inequalities are obtained for the case of functions whose first derivative in absolute value at certain power are arithmetically-harmonically convex. In addition, some Newton-type inequalities are established by using Riemann-Liouville fractional integrals for differentiable convex functions and some Riemann-Liouville fractional Newton-type inequalities are given for functions of bounded variation in paper [ 13 ]. Moreover, several Newton-type inequalities are established by means of the well-known Riemann-liouville fractional integrals for differentiable convex functions in [15]. 2 Preliminaries Simpson-type inequality is a variation on inequality that is derived from Simpson’s rules. It can be expressed as follows: i. Simpson’s 1/3rule, or Simpson’s quadrature formula: Zb a f(x)dx ≈b−a 6f(a) + 4fa+b 2+f(b). ii. The Simpson’s second formula, often known as the Simpson’s 3 / 8rule, or the Newton-Cotes quadrature formula: Zb a f(x)dx ≈b−a 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b). Three-point Simpson-type inequality is the most widely used Newton-Cotes quadrature, and it corresponds to this: Theorem 2.1. Let f : [a, b]→R be a four times differentiable and continuous function on (a, b), and let   f(4)  ∞= sup x∈(a,b)f(4)(x)<∞.Then, the following inequality holds:  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 Simpson 3/8rule is a classical closed type quadrature rule is as follows: Theorem 2.2. Consider that f : [a, b]→R is a four times differentiable, continuous 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. M21-2 2nd Kocaeli Science Congress, November 19-21, 2025 Definition 2.1 (See [ 17 ]).Assume that I is an interval of real numbers. Then, a function f:I→Ris said to be convex, if f(tx +(1−t)y)≤tf (x)+(1−t)f(y) is valid ∀x, y ∈Iand ∀t∈[0,1]. Definition 2.2. Let h : J⊆R→R , where (0 , 1) ⊆J , be a nonnegative function with h = 0. We say that f : I⊆R→R is an h -convex function if f is nonnegative and for all x, y ∈I , α∈ (0 , 1) we have f(αx +(1−α)y)≤h(α)f(x)+h(1 −α)f(y).(1) If the inequality in (1) is reversed, then fis said to be h-concave. By setting •h(λ)=λ, Definition 2.2 reduces to that of the classical convex function. •h(λ)=1, Definition 2.2 reduces to that of P-functions [11,18]. •h(λ)=λs, Definition 2.2 reduces to that of s-convex functions [10]. •h ( λ ) = 1 nPn k=1 λ1/k , Definition 2.2 reduces to that of polynomial n -fractional convex functions [16]. Definition 2.3 (See [ 19 , 20 ]).Let us consider f∈L1 [ a, b ], a, b ∈R with a<b . The RiemannLiouville fractional integrals Jα a+fand Jα b−fof order α > 0are defined by Jα a+f(x) = 1 Γ(α)Zx a (x−t)α−1f(t)dt, x > a (2) and Jα b−f(x) = 1 Γ(α)Zb x (t−x)α−1f(t)dt, x < b, (3) respectively. Here, Γdenotes the Gamma function defined by Γ(α) = Z∞ 0 e−uuα−1du. Lemma 2.1. [ 14 ] If f : [ a, b ] →R is an absolutely continuous function ( a, b )such that f′∈ L1[a, b], then the identity 1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) =b−a 4[I1+I2] 2nd Kocaeli Science Congress, November 19-21, 2025 M21-3 KOSC-2025 Proceedings is valid. Here,            I1= 2 3 R0tα−1 4hf′t 2b+2−t 2a−f′t 2a+2−t 2bidt, I2= 1 R2 3 (tα−1) hf′t 2b+2−t 2a−f′t 2a+2−t 2bidt. 3 Fractional Newton-type inequalities by h-convex functions Firstly, some Newton-type inequalities are obtained for differentiable h -convex functions by using Riemann-Liouville fractional integrals. Theorem 3.1. Assume that the assumptions of Lemma 2.1 hold and the function |f′| is h -convex on the interval [a, b]. Then, one can prove fractional Newton-type inequality  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) (4) ≤b−a 4ht 2+h2−t 2f′(a)+f′(b). Proof. Let us first consider the absolute value in Lemma 2.1. Then, one can directly have  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) (5) ≤b−a 4       2 3 Z 0 tα−1 4 f′t 2b+2−t 2a−f′t 2a+2−t 2b dt + 1 Z 2 3 |tα−1| f′t 2b+2−t 2a−f′t 2a+2−t 2b dt       . Based on the h-convexity of |f′|, one can obtain  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤b−a 4       2 3 Z 0 tα−1 4ht 2f′(b)+h2−t 2f′(a)+ht 2f′(a)+h2−t 2f′(b)dt + 1 Z 2 3 (1 −tα)ht 2f′(b)+h2−t 2f′(a)+ht 2f′(a)+h2−t 2f′(b)dt       =b−a 4ht 2+h2−t 2f′(a)+f′(b). M21-4 2nd Kocaeli Science Congress, November 19-21, 2025 Remark 3.1. If we choose h(t) = tin Theorem 3.1, and then we have  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) (6) ≤b−a 4(Ω1(α)+Ω2(α)) f′(a)+f′(b). Here, Ω1(α) = 2 3 Z 0 tα−1 4 dt =           2α α+1 1 41+ 1 α+1 α+1 2 3α+1 −1 6,0< α < ln(1 4) ln(2 3), 1 6−1 α+1 2 3α+1 ,ln(1 4) ln(2 3)< α and Ω2(α) = 1 Z 2 3 (1 −tα)dt =1 3−1 α+ 1 +1 α+ 1 2 3α+1 . This result was obtained by Hezenci and Budak [14]. Remark 3.2. If we assign α= 1 in Remark 3.1, then one can get Newton-type inequality  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−1 b−a b Z a f(t)dt ≤25 (b−a) 576 f′(a)+f′(b), which is proved by Sitthiwirattham et al. in paper [13, Remark 3]. Theorem 3.2. Consider that the assumptions in Lemma 2.1 and the function |f′|q , q > 1is h-convex on [a, b]. Then, the Newton-type inequality  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)(7) −2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤b−a 4             2 3 Z 0 tα−1 4 p dt    1 p         1 Z 2 3 ht 2f′(b) q+h2−t 2f′(a) qdt    1 q +    2 3 Z 0ht 2f′(a) q+h2−t 2f′(b) qdt    1 q     +    1 Z 2 3 (1 −tα)pdt    1 p         1 Z 2 3 ht 2f′(b) q+h2−t 2f′(a) qdt    1 q 2nd Kocaeli Science Congress, November 19-21, 2025 M21-5 KOSC-2025 Proceedings +    1 Z 2 3 ht 2f′(a) q+h2−t 2f′(b) qdt    1 q         is valid. Here, 1 p+1 q= 1. Proof. By utilizing Hölder’s inequality to (5), one can obtain  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b) −2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤b−a 4             2 3 Z 0 tα−1 4 p dt    1 p    2 3 Z 0 f′t 2b+2−t 2a q dt    1 q +    2 3 Z 0 tα−1 4 p dt    1 p    2 3 Z 0 f′t 2a+2−t 2b q dt    1 q +    1 Z 2 3 |tα−1|pdt    1 p    1 Z 2 3  f′t 2b+2−t 2a q dt    1 q +    1 Z 2 3 |tα−1|pdt    1 p    1 Z 2 3  f′t 2a+2−t 2b q dt    1 q         . From the facts of the h-convexity |f′|q, one can readily have  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b) −2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤b−a 4             2 3 Z 0 tα−1 4 p dt    1 p    2 3 Z 0ht 2f′(b) q+h2−t 2f′(a) qdt    1 q +    2 3 Z 0 tα−1 4 p dt    1 p    2 3 Z 0ht 2f′(a) q+h2−t 2f′(b) qdt    1 q +    1 Z 2 3 (1 −tα)pdt    1 p    1 Z 2 3 ht 2f′(b) q+h2−t 2f′(a) qdt    1 q M21-6 2nd Kocaeli Science Congress, November 19-21, 2025 +    1 Z 2 3 (1 −tα)pdt    1 p    1 Z 2 3 ht 2f′(a) q+h2−t 2f′(b) qdt    1 q         =b−a 4             2 3 Z 0 tα−1 4 p dt    1 p ×         2 3 Z 0ht 2f′(b) q+h2−t 2f′(a) qdt    1 q +    2 3 Z 0ht 2f′(a) q+h2−t 2f′(b) qdt    1 q     +    1 Z 2 3 (1 −tα)pdt    1 p . ×         1 Z 2 3 ht 2f′(b) q+h2−t 2f′(a) qdt    1 q +    1 Z 2 3 ht 2f′(a) q+h2−t 2f′(b) qdt    1 q              . Remark 3.3. If we choose h(t) = tin Theorem 3.2, and then we have  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) (8) ≤b−a 4             2 3 Z 0 tα−1 4 p dt    1 p  |f′(b)|q+ 5 |f′(a)|q 9!1 q + |f′(a)|q+ 5 |f′(b)|q 9!1 q  +    1 Z 2 3 (1 −tα)pdt    1 p  5|f′(b)|q+ 7 |f′(a)|q 36 !1 q + 5|f′(a)|q+ 7 |f′(b)|q 36 !1 q          is valid. Here, 1 p+1 q= 1. This result was esatblished by Hezenci and Budak [14]. 2nd Kocaeli Science Congress, November 19-21, 2025 M21-7 KOSC-2025 Proceedings Remark 3.4. Consider that α= 1 in Remark 3.3. Then, we obtain  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−1 b−a b Z a f(t)dt ≤b−a 4   1 p+ 1 "1 4p+1 +5 12p+1#!1 p ×  |f′(b)|q+ 5 |f′(a)|q 9!1 q + |f′(a)|q+ 5 |f′(b)|q 9!1 q  + 1 p+ 1 1 3p+1!1 p  5|f′(b)|q+ 7 |f′(a)|q 36 !1 q + 5|f′(a)|q+ 7 |f′(b)|q 36 !1 q    . This result was esatblished by Hezenci and Budak [14]. Theorem 3.3. If the assumptions of Lemma 2.1 hold and the function |f′|q , q≥ 1is h -convex on [a, b],then the following Newton-type inequality satisfies:  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)(9) −2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤b−a 4                  2 3 Z 0 tα−1 4ht 2f′(b) q+h2−t 2f′(a) qdt    1 q +    2 3 Z 0 tα−1 4ht 2f′(a) q+h2−t 2f′(b) qdt    1 q     +    1 Z 2 3 (1 −tα)dt    1−1 q    1 Z 2 3 |tα−1|ht 2f′(b) q+h2−t 2f′(a) qdt    1 q +    1 Z 2 3 |tα−1|ht 2f′(a) q+h2−t 2f′(b) qdt    1 q         . Proof. If we first apply (5) to the power-mean inequality, then it yields  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤b−a 4             2 3 Z 0 tα−1 4 dt    1−1 q    2 3 Z 0 tα−1 4 f′t 2b+2−t 2a q dt    1 q M21-8 2nd Kocaeli Science Congress, November 19-21, 2025 +    2 3 Z 0 tα−1 4 dt    1−1 q    2 3 Z 0 tα−1 4 f′t 2a+2−t 2b q dt    1 q +    1 Z 2 3 |tα−1|dt    1−1 q    1 Z 2 3 |tα−1| f′t 2b+2−t 2a q dt    1 q +    1 Z 2 3 |tα−1|dt    1−1 q    1 Z 2 3 |tα−1| f′t 2a+2−t 2b q dt    1 q         . It is known that |f′|qis h-convex. Then, one can obtain  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) ≤b−a 4             2 3 Z 0 tα−1 4 dt    1−1 q    2 3 Z 0 tα−1 4ht 2f′(b) q+h2−t 2f′(a) qdt    1 q +    2 3 Z 0 tα−1 4 dt    1−1 q    2 3 Z 0 tα−1 4ht 2f′(a) q+h2−t 2f′(b) qdt    1 q +    1 Z 2 3 (1 −tα)dt    1−1 q    1 Z 2 3 |tα−1|ht 2f′(b) q+h2−t 2f′(a) qdt    1 q +    1 Z 2 3 (1 −tα)dt    1−1 q    1 Z 2 3 |tα−1|ht 2f′(a) q+h2−t 2f′(b) qdt    1 q         . Remark 3.5. If we choose h(t) = tin Theorem 4, and then we have  1 8f(a) + 3f2a+b 3+ 3fa+ 2b 3+f(b)−2α−1Γ (α+ 1) (b−a)αJα a+b 2−f(a) + Jα a+b 2+f(b) (10) ≤b−a 4(Ω1(α))1−1 qΩ3(α)f′(b) q+ Ω4(α)f′(a) q1 q +Ω3(α)f′(a) q+ Ω4(α)f′(b) q1 q + (Ω2(α))1−1 qΩ5(α)f′(b) q+ Ω6(α)f′(a) q1 q +Ω5(α)f′(a) q+ Ω6(α)f′(b) q1 q, 2nd Kocaeli Science Congress, November 19-21, 2025 M21-9