Extensions of Newton-Type Inequalities via Multiplicative Conformable Fractional Integrals

Extensions of Newton-Type Inequalities via Multiplicative Conformable Fractional Integrals Hüseyin Budak1, Büşra Betül Ergün1 1Department of Mathematics, Faculty of Science and Arts, Kocaeli University, Kocaeli 41001, Türkiye Corresponding author: ergn[email protected] ORCID IDs: First Author: 0000-0001-8843-955X Second Author: 0009-0009-4038-4487 DOI : 10.5281/zenodo.18012198 Abstract In this study, a novel form of Newton-type inequality for multiplicative convex functions is derived using multiplicative conformable fractional integrals, which has not been previously addressed in the literature. To prove the main results of the study, a well-structured integral identity is first established. This identity is then combined with multiplicative conformable fractional integrals to construct a new inequality structure. In addition, a special Newton-type inequality is presented for functions whose multiplicative derivatives are bounded. To support the validity of the obtained inequality and the derived identity, various specific cases are provided, thereby demonstrating the generality and flexibility of the results. In this respect, the study offers a new perspective to the field of multiplicative analysis and lays a foundation for future applications to different types of fractional integral operators or function classes. Keywords: Multiplicative calculus, Newton inequality, fractional integrals, bounded function. 1 Introduction Fractional calculus has been widely employed to describe nonlocal and memory-driven phenomena, leading to the development of various fractional integral operators with strong modeling capabilities [12,1]. M11-1 KOSC-2025 Proceedings As an alternative to classical calculus, multiplicative calculus has emerged as an alternative analytical framework based on multiplication rather than addition, making it particularly effective for processes governed by exponential growth, decay, or proportional changes[ 15 ].The foundations of multiplicative derivatives and integrals were formalized in[ 6 ],enabling the adaptation of classical inequalities to the multiplicative setting.Within this framework, several fundamental inequalities (such as Hermite–Hadamard and Ostrowski inequalities) have been extended to their multiplicative counterparts [ 2 , 8 , 4 ]. Chasreechai et al. obtained Newton-type inequalities for multiplicative integrals[ 11 ]. Subsequently, Ali presented Newton-type inequalities for the multiplicative Riemann–Liouville fractional integrals[ 2 ]. More recently, the multiplicative conformable fractional integral (MCFI) was introduced as a hybrid operator combining the advantages of both conformable and multiplicative calculus structures[ 9 ]. For more information, please refer to[11,7,10,5]. This article consists of five sections, including the introduction. In Section 2, multiplicative derivatives and multiplicative integrals are reviewed, and the properties related to these concepts are presented. Then, the definitions of Riemann–Liouville fractional integrals, conformable fractional integrals, multiplicative Riemann–Liouville fractional integrals, and multiplicative conformable fractional integrals are introduced. In Section 3, an identity necessary for presenting the main findings is obtained. In Section 4, it is aimed to develop Newton-type inequalities for multiplicative differentiability and multiplicative convex functions with the help of MCFI. Finally, in Section 5, the results of our study are evaluated, and potential directions for future research are highlighted. 2 Preliminaries 2.1 Multiplicative Calculus Proposition 2.1. [ 6 ] If f is positive and Riemann integrable on [ a, b ], then f is multiplicative integrable on [a, b]and b Z a (f(x))dx = exp    b Z a ln(f(x))dx   . Proposition 2.2. [ 6 ] Under the condition that the functions f and g , being positive,are multiplicative integrable on the interval [ a, b ], it is evident that the properties given below are applicable. M11-2 2nd Kocaeli Science Congress, November 19-21, 2025 2.1 Multiplicative Calculus 1. b R a ((f(x))p)dx = b R a (f(x))dx!p , p ∈R, 2. b R a (f(x)g(x))dx =b R a (f(x))dx. b R a (g(x))dx, 3. b R af(x) g(x)dx = b Ra (f(x))dx b Ra (g(x))dx , 4. b R a (f(x))dx =c R a (f(x))dx. b Rc (f(x))dx, a ≤c≤b, 5. a R a (f(x))dx = 1 and b R a (f(x))dx = a Rb (f(x))dx!−1 . Definition 2.1. [ 6 ] Consider the function f : I→R+ and assume that its *derivative exists. The notation f∗ symbolizes multiplicative derivative of f (shortly *derivative of the function f), which is expressed as f∗(x) = d∗f(x) dx = exp n(ln f(x))′o. Proposition 2.3. [ 6 ] Presuming that the functions f, g : I→R+ are *differentiable, and the function h : I→R is differentiable on I◦ . If the constant c > 0 , then the functions cf, f + g, fg, f g, fh and f◦h are all *differentiable on I◦ as well, and the listed below properties holds: 1. (cf)∗(x)=f∗(x), 2. (f+g)∗(x)=[f∗(x)] f(x) f(x)+g(x)[g∗(x)] g(x) f(x)+g(x), 3. (fg)∗(x)=f∗(x)g∗(x), 4. f g∗(x) = f∗(x) g∗(x), 5. fh∗(x)=f∗(x)h(x)f(x)h′(x), 6. (f◦h)∗(x)=f∗(h(x))h′(x). 2nd Kocaeli Science Congress, November 19-21, 2025 M11-3 KOSC-2025 Proceedings Definition 2.2 (Multiplicative absolute value).[ 14 ] Let x∈R∗ . The multiplicative absolute value is defined as follows |x|∗=       x, if x ≥1, 1 x, if 0<x<1. Remark 2.1. The classical absolute value and the multiplicative one are linked by the following relation |exp{x}|∗= exp |x| and |ln(x)|= ln |x|∗. The following theorems give the integral-by-part formulas for multiplicative integrals,which will be used in the main results. Theorem 2.1. [ 6 ] Let f : [ a, b ] →R be multiplicative differentiable and g : [ a, b ] →R be differentiable so the fgis multiplicative integrable. Then b Z a(f∗(x))g(x)dx =(f(b))g(b) (f(a))g(a).1 b R a(f(x))g′(x)dx . Theorem 2.2. [ 4 ] Let f : [ a, b ] →R be multiplicative differentiable, let g : [ a, b ] →R and h:I⊂R→[a, b]be two differentiable functions. Then we have b Z a(f∗(h(x)))g(x)h′(x)dx =f(h(b))g(b) f(h(a))g(a).1 b R a(f(h(x)))g′(x)dx . 2.2 Some Definitions and Several Newton Inequalities Definition 2.3. [ 22 ] A non-empty set K is said to be convex, if for every a, b ∈K we have a+t(b−a)∈K, ∀t∈[0,1]. Definition 2.4. [22] A function fis said to be convex function set K, if f(tx +(1−t)y)≤tf(x)+(1−t)f(y) for all a, b ∈Kand all t∈[0,1]. M11-4 2nd Kocaeli Science Congress, November 19-21, 2025 2.2 Some Definitions and Several Newton Inequalities Definition 2.5. [ 19 ] A function f is said to be log or multiplicatively convex function on set K, if f(ta +(1−t)b)≤[f(x)]t[f(y)](1−t) for all a, b ∈Kand all t∈[0,1]. Remark 2.2. If a positive function f is a multiplicatively convex, then the function ln f is a convex function. Definition 2.6. The Euler Gamma function, Beta function and Incomplete Beta function are defined by Γ(x) := Z∞ 0 ξx−1e−tdξ B(x, y) := Z1 0 ξx−1(1 −ξ)y−1dξ and B(x, y, r) := Zr 0 ξx−1(1 −ξ)y−1dξ respectively for 0< x, y < ∞and r∈[0,1]. Definition 2.7. [ 18 ] Let the function f∈L1 ([ a, b ]). For the order β > 0, the RiemannLiouville Fractional Integrals (RLFI) Jβ a+f(x)and Jβ b−f(x)are defined by Jβ a+f(x) = 1 Γ(β)Zx a (x−t)β−1f(t)dt, x > a and Jβ b−f(x) = 1 Γ(β)Zb x (t−x)β−1f(t)dt, b > x respectively. Here Γrepresents the Euler Gamma function. Definition 2.8. [ 8 ] The multiplicative left Riemann-Liouville Fractional Integral aIβ ∗f ( x ) of order β > 0starting from βis defined by aIβ ∗f(x) = exp n(Jβ a+(ln ◦f))(x)o and the multiplicative right one ∗Iβ bf(x)is defined by ∗Iβ bf(x) = exp n(Jβ b−(ln ◦f))(x)o. 2nd Kocaeli Science Congress, November 19-21, 2025 M11-5 KOSC-2025 Proceedings Definition 2.9. [ 16 ] Let the function f∈L1 ([ a, b ]). For the order β > 0and α∈ (0 , 1] , the Conformable Fractional Integrals (CFI) β +Iα af ( x )and β −Iα bf ( x ), correspondingly, are defined by β +Iα af(x) = 1 Γ(β) x Z a(x−a)α−(t−a)α αβ−1 (t−a)α−1f(t)dt, x > a and β −Iα bf(x) = 1 Γ(β) b Z x(b−x)α−(b−t)α αβ−1 (b−t)α−1f(t)dt, b > x respectively. Here , Γrepresents the Euler Gamma function. Definition 2.10. [ 10 ] The multiplicative left Conformable Fractional Integral ( β aIα ∗f )( x ) of order β > 0and α∈(0,1] by (β aIα ∗f)(x) = exp nβ +Iα a((ln ◦f)(x))o = exp    1 Γ(β) x Z a(x−a)α−(t−a)α αβ−1(ln ◦f)(t) (t−a)1−αdt, x > a   , and the multiplicative right Conformable Fractional Integral (β ∗Iα bf)(x)is defined by (β ∗Iα bf)(x) = exp nβ −Iα b((ln ◦f)(x))o = exp    1 Γ(β) b Z x(b−x)α−(b−t)α αβ−1(ln ◦f)(t) (b−t)1−αdt, b > x   . Here, Γis the Euler Gamma function. M11-6 2nd Kocaeli Science Congress, November 19-21, 2025 3 An Essential Identity MCFI Lemma 3.1. Let f : [ a, b ] →R+ be multiplicative differentiable function over ( a, b ) . If f∗is multiplicative integrable on [a, b], then the following equality holds: f(a).f 2a+b 33.f a+2b 33.f(b)1 8 hβ ∗Iα bfa+b 2.β aIα ∗fa+b 2iαβ2αβ−1Γ(β+1) (b−a)αβ = [T1×T2×T3×T4]αβ(b−a) 4.(1) Here, T1: = 1 3 Z 0   f∗1+t 2b+1−t 2a1−(1−t)α αβ   dt , T2: = 1 3 Z 0   f∗1+t 2a+1−t 2b−1−(1−t)α αβ   dt , T3: = 1 Z 1 3   f∗1+t 2b+1−t 2a1−(1−t)α αβ −3 4αβ   dt , T4: = 1 Z 1 3   f∗1+t 2a+1−t 2b3 4αβ−1−(1−t)α αβ   dt , and Γis Euler Gamma function. Proof. By using Theorem 2.2, we have T1= 1 3 Z 0   f∗1+t 2b+1−t 2a1−(1−t)α αβ   dt (2) =    1 3 Z 0   f∗1+t 2b+1−t 2a1−(1−t)α αβ(b−a 2)   dt    2 b−a 2nd Kocaeli Science Congress, November 19-21, 2025 M11-7 KOSC-2025 Proceedings =hf2b+a 3i(2 b−a)3α−2α α3αβ fa+b 20.1   1 3 R0 hf1+t 2b+1−t 2aiβ(1−t)α−11−(1−t)α αβ−1!dt  2 b−a =hf2b+a 3i(2 b−a)3α−2α α3αβ exp    2 b−a 1 3 R0 β(1 −t)α−11−(1−t)α αβ−1ln f1+t 2b+1−t 2adt   =hf2b+a 3i(2 b−a)3α−2α α3αβ exp    2 b−a a+2b 3 R a+b 2 β(b−u)α−12 b−aα−11−(2 b−a)α(b−u)α αβ−1 ln f(u)2 b−adu   =hf2b+a 3i(2 b−a)3α−2α α3αβ exp    2αβ+1Γ(β+1) (b−a)αβ+1Γ(β) a+2b 3 R a+b 2 (b−u)α−1(b−a 2)α −(b−u)α αβ−1 ln f(u)du   , T2= 1 3 Z 0   f∗1+t 2a+1−t 2b−1−(1−t)α αβ   dt (3) =hf2a+b 3i(2 b−a)3α−2α α3αβ exp    2αβ+1Γ(β+1) (b−a)αβ+1Γ(β) a+b 2 R 2a+b 3 (k−a)α−1(b−a 2)α −(k−a)α αβ−1 ln f(k)dk   , T3= 1 Z 1 3   f∗1+t 2b+1−t 2a1−(1−t)α αβ −3 4αβ   dt (4) =    1 Z 1 3   f∗1+t 2b+1−t 2a1−(1−t)α αβ −3 4αβ(b−a 2)   dt    2 b−a M11-8 2nd Kocaeli Science Congress, November 19-21, 2025 =[f(b)]1 αβ−3 4αβ(2 b−a) hfa+2b 3i 2 b−ah(3α−2α α3α)β −3 4αβi.1   1 R1 3 hf1+t 2b+1−t 2aiβ(1−t)α−11−(1−t)α αβ−1!dt  2 b−a =[f(b)] 1 2(b−a)αβ hfa+2b 3i 2 b−ah(3α−2α α3α)β −3 4αβi.exp    2 b−a 1 R1 3 β(1 −t)α−11−(1−t)α αβ−1ln f1+t 2b+1−t 2adt   =[f(b)] 1 2(b−a)αβ.hfa+2b 3i 2 b−ah3 4αβ−3α−2α α3αβi exp    2 b−a b R a+2b 3 β(b−u)α−12 b−aα−11−(2 b−a)α(b−u)α αβ−1 ln f(u)2 b−adu   =[f(b)] 1 2(b−a)αβ.hfa+2b 3i 2 b−ah3 4αβ−3α−2α α3αβi exp    2αβ+1Γ(β+1) (b−a)αβ+1Γ(β) b R a+2b 3 (b−u)α−1(b−a 2)α −(b−u)α αβ−1 ln f(u)du   and similarly T4= 1 Z 1 3   f∗1+t 2a+1−t 2b3 4αβ−1−(1−t)α αβ   dt (5) =[f(a)] 1 2(b−a)αβ.hf2a+b 3i 2 b−ah3 4αβ−3α−1 α3αβi exp    2αβ+1Γ(β+1) (b−a)αβ+1Γ(β) 2a+b 3 R a (k−a)α−1(b−a 2)α −(k−a)α αβ−1 ln f(k)dk   . By multiplying the results of (2),(3),(4) and (5) and raising both sides of the obtained identity to the power of (b−a)αβ 4, we obtain the equality (1). Corollary 3.1. If we pick α = 1 in (1), then we obtain the next identity holds for MRLFI: f(a).f 2a+b 33.f a+2b 33.f(b)1 8 h∗Iβ bfa+b 2.aIβ ∗fa+b 2i2β−1Γ(β+1) (b−a)β = [K1×K2×K3×K4]b−a 4. 2nd Kocaeli Science Congress, November 19-21, 2025 M11-9 KOSC-2025 Proceedings [17] Kashuri, A., Sahoo, S. K., Aljuaid, M., Tariq, M., & De La Sen, M. (2023). Some new Hermite–Hadamard type inequalities pertaining to generalized multiplicative fractional integrals. Symmetry, 15(4), 868. [18] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier,Amsterdam (2006) [19] Niculescu, C., & Persson, L. E. (2006). Convex functions and their applications (Vol. 23). New York: Springer. [20] Noor, M. A., Noor, K. I., & Iftikhar, S. (2018). Newton inequalities for p-harmonic convex functions. Honam Mathematical Journal, 40(2), 239-250. [21] Özcan, S. (2025). Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions. Demonstratio Mathematica, 58(1), 20240060. [22] Peajcariaac, J. E., & Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Academic Press. [23] Saleh, W., Lakhdari, A., Abdeljawad, T., & Meftah, B. (2023). On fractional biparameterized Newton-type inequalities. Journal of Inequalities and Applications, 2023(1), 122. [24] Zhan, X., Mateen, A., Toseef, M., & Aamir Ali, M. (2024). Some Simpson-and Ostrowski-type integral inequalities for generalized convex functions in multiplicative calculus with their computational analysis. Mathematics, 12(11), 1721. M11-16 2nd Kocaeli Science Congress, November 19-21, 2025