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Öğe Advances in Ostrowski-Mercer Like Inequalities within Fractal Space(Mdpi, 2023) Vivas-Cortez, Miguel; Awan, Muhammad Uzair; Asif, Usama; Javed, Muhammad Zakria; Budak, HueseyinThe main idea of the current investigation is to explore some new aspects of Ostrowski's type integral inequalities implementing the generalized Jensen-Mercer inequality established for generalized s-convexity in fractal space. To proceed further with this task, we construct a new generalized integral equality for first-order local differentiable functions, which will serve as an auxiliary result to restore some new bounds for Ostrowski inequality. We establish our desired results by employing the equality, some renowned generalized integral inequalities like Holder's, power mean, Yang-Holder's, bounded characteristics of the functions and considering generalized s-convexity characteristics of functions. Also, in support of our main findings, we deliver specific applications to means, and numerical integration and graphical visualization are also presented here.Öğe Bounds for the Error in Approximating a Fractional Integral by Simpson's Rule(Mdpi, 2023) Budak, Hueseyin; Hezenci, Fatih; Kara, Hasan; Sarıkaya, Mehmet ZekiSimpson's rule is a numerical method used for approximating the definite integral of a function. In this paper, by utilizing mappings whose second derivatives are bounded, we acquire the upper and lower bounds for the Simpson-type inequalities by means of Riemann-Liouville fractional integral operators. We also study special cases of our main results. Furthermore, we give some examples with graphs to illustrate the main results. This study on fractional Simpson's inequalities is the first paper in the literature as a method.Öğe Bullen-type inequalities for twice-differentiable functions by using conformable fractional integrals(Springer, 2024) Hezenci, Fatih; Budak, HueseyinIn this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Holder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems.Öğe Conformable fractional Newton-type inequalities with respect to differentiable convex functions(Springer, 2023) uenal, Cihan; Hezenci, Fatih; Budak, HueseyinThe authors propose a new method of investigation of an integral identity according to conformable fractional operators. Moreover, some Newton-type inequalities are considered for differentiable convex functions by taking the modulus of the newly established equality. In addition, we prove several Newton-type inequalities with the aid of Holder and power-mean inequalities. Furthermore, several new results are given by using special choices of the obtained inequalities. Finally, we give several inequalities of conformable fractional Newton-type for functions of bounded variation.Öğe Exploring Quantum Simpson-Type Inequalities for Convex Functions: A Novel Investigation(Mdpi, 2023) Iftikhar, Sabah; Awan, Muhammad Uzair; Budak, HueseyinThis study seeks to derive novel quantum variations of Simpson's inequality by primarily utilizing the convexity characteristics of functions. Additionally, the study examines the credibility of the obtained results through the presentation of relevant numerical examples and graphs.Öğe Fractional Simpson-Type Inequalities for Twice Differentiable Functions(Univ Maragheh, 2023) Budak, Hueseyin; Kara, Hasan; Hezenci, FatihIn the literature, several papers are devoted to inequal-ities of Simpson-type in the case of differentiable convex functions and fractional versions. Moreover, some papers are focused on in-equalities of Simpson-type for twice differentiable convex functions. In this research article, we obtain an identity for twice differentiable convex functions. Then, we prove several fractional inequalities of Simpson-type for convex functions.Öğe Generalization of quantum calculus and corresponding Hermite-Hadamard inequalities(Springer Basel Ag, 2024) Akbar, Saira Bano; Abbas, Mujahid; Budak, HueseyinThe aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called (phi-h) integrals and (phi-h) derivatives, respectively. Then we investigate some implicit integral inequalities for (phi-h) integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite-Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and & hstrok;-convex functions defined on the non-negative part of the real line.Öğe Generalized Hermite-Hadamard inclusions for a generalized fractional integral(Rocky Mt Math Consortium, 2023) Budak, Hueseyin; Kara, Hasan; Hezenci, FatihWe introduce new generalized fractional integrals for interval-valued functions. Then we prove generalized Hermite-Hadamard type inclusions for interval-valued convex functions using these newly defined generalized fractional integrals. We also show that these results generalize several results obtained in earlier works.Öğe Hermite-Hadamard-Mercer type inequalities for fractional integrals: A study with h-convexity and ψ-Hilfer operators(Springer, 2025) Azzouz, Noureddine; Benaissa, Bouharket; Budak, Hueseyin; Demir, IzzettinIn this paper, we first prove a generalized fractional version of Hermite-Hadamard-Mercer type inequalities using h-convex functions by means of psi-Hilfer fractional integral operators. Then, we give new identities of this type with special functions depending on psi. Moreover, we establish some new fractional integral inequalities connected with the right- and left-hand sides of Hermite-Hadamard-Mercer inequalities involving differentiable mappings whose absolute values of the derivatives are h-convex. For the development of these novel integral inequalities, we utilize h-Mercer inequality and H & ouml;lder's integral inequality. These results offer the significant advantage of being convertible into classical integral inequalities and Riemann-Liouville fractional integral inequalities for convex functions, s-convex functions, and P-convex functions.Öğe Hermite-Hadamard-Type Inequalities Arising from Tempered Fractional Integrals Including Twice-Differentiable Functions(Springer, 2025) Hezenci, Fatih; Budak, Hueseyin; Latif, Muhammad AmerWe propose a new method for the investigation of integral identities according to tempered fractional operators. In addition, we prove the midpoint-type and trapezoid-type inequalities by using twice-differentiable convex functions associated with tempered fractional integral operators. We use the well-known H & ouml;lder inequality and the power-mean inequality in order to obtain inequalities of these types. The resulting Hermite-Hadamard-type inequalities are generalizations of some investigations in this field, involving Riemann-Liouville fractional integrals.Öğe The Multi-Parameter Fractal-Fractional Inequalities For Fractal (P,M)-Convex Functions(World Scientific Publ Co Pte Ltd, 2024) Yuan, Xiaoman; Budak, Hueseyin; Du, TingsongLocal fractional calculus theory and parameterized method have greatly assisted in the advancement of the field of inequalities. To continue its enrichment, this study investigates the multi-parameter fractal-fractional integral inequalities containing the fractal (P,m)-convex functions. Initially, we formulate the new conception of the fractal (P,m)-convex functions and work on a variety of properties. Through the assistance of the fractal-fractional integrals, the 2l-fractal identity with multiple parameters is established, and from that, integral inequalities are inferred regarding twice fractal differentiable functions which are fractal (P,m)-convex. Furthermore, a few typical and novel outcomes are discussed and visualized for specific parameter values, separately. It concludes with some applications in respect of the special means, the quadrature formulas and random variable moments, respectively.Öğe New Quantum Hermite-Hadamard-Type Inequalities for p-Convex Functions Involving Recently Defined Quantum Integrals(Springer, 2024) Gulshan, Ghazala; Budak, Hueseyin; Hussain, Rashida; Ali, Muhammad AamirWe develop new Hermite-Hadamard-type integral inequalities for p-convex functions in the context of q-calculus by using the concept of recently defined T-q-integrals. Then the obtained Hermite-Hadamard inequality for p-convex functions is used to get a new Hermite-Hadamard inequality for coordinated p-convex functions. Furthermore, we present some examples to demonstrate the validity of our main results. We hope that the proposed ideas and techniques may stimulate further research in this field.Öğe New Results on Bullen-Type Inequalities for Coordinated Convex Functions Obtained by using Conformable Fractional Integrals(Springer, 2025) Hezenci, Fatih; Kara, Hasan; Budak, HueseyinOur aim is to investigate novel Bullen-type inequalities for coordinated convex mappings by employing conformable fractional integrals. Initially, an identity incorporating the conformable fractional integrals was established to serve for this purpose. By using this identity, new inequalities are derived expanding the scope of previously established results obtained with the help of Riemann-Liouville integrals by making specific choices of the variable and applying the H & ouml;lder inequality and the power-mean inequality.Öğe New Versions of Midpoint Inequalities Based on Extended Riemann-Liouville Fractional Integrals(Mdpi, 2023) Hyder, Abd-Allah; Budak, Hueseyin; Barakat, Mohamed A.This study aims to prove some midpoint-type inequalities for fractional extended Riemann-Liouville integrals. Crucial equality is proven to build new results. Using this equality, several midpoint-type inequalities are established via differentiable convex functions and the proposed extended fractional operators. To be more specific, the well-known Holder, Jensen, and power mean integral inequalities are employed in the demonstrated inequalities. Additionally, many remarks based on specific selections of the main results are presented. Moreover, to illustrate the key conclusions, a few instances are provided.Öğe New versions of the Hermite-Hadamard inequality for (φ-h)-integrals(Springer, 2024) Akbar, Saira Bano; Abbas, Mujahid; Nazeer, Waqas; Budak, HueseyinIn this paper, we prove some new versions of the Hermite-Hadamard inequality for (phi-h)-integrals. For this aim, we use the tangent and secant lines at the same special points. Moreover, we investigate the relations between the newly obtained results and earlier published papers. We also present some new results by special choices of phi and h.Öğe Novel Fractional Boole's-Type Integral Inequalities via Caputo Fractional Operator and Their Implications in Numerical Analysis(Mdpi, 2025) Haider, Wali; Mateen, Abdul; Budak, Hueseyin; Shehzadi, Asia; Ciurdariu, LoredanaThe advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) orders. This generalization allows for more flexible and accurate modeling of complex phenomena that cannot be adequately described using integer-order derivatives. Motivated by its applications in various scientific disciplines, this paper establishes novel n-times fractional Boole's-type inequalities using the Caputo fractional derivative. For this, a fractional integral identity is first established. Using the newly derived identity, several novel Boole's-type inequalities are subsequently obtained. The proposed inequalities generalize the classical Boole's formula to the fractional domain. Further extensions are presented for bounded functions, Lipschitzian functions, and functions of bounded variation, providing sharper bounds compared to their classical counterparts. To demonstrate the precision and applicability of the obtained results, graphical illustrations and numerical examples are provided. These contributions offer valuable insights for applications in numerical analysis, optimization, and the theory of fractional integral equations.Öğe On Fractional Ostrowski-Mercer-Type Inequalities and Applications(Mdpi, 2023) Ramzan, Sofia; Awan, Muhammad Uzair; Vivas-Cortez, Miguel; Budak, HueseyinThe objective of this research is to study in detail the fractional variants of Ostrowski-Mercer-type inequalities, specifically for the first and second order differentiable s-convex mappings of the second sense. To obtain the main outcomes of the paper, we leverage the use of conformable fractional integral operators. We also check the numerical validations of the main results. Our findings are also validated through visual representations. Furthermore, we provide a detailed discussion on applications of the obtained results related to special means, q-digamma mappings, and modified Bessel mappings.Öğe On Hermite-Hadamard-Fejer-Type Inequalities for ?-Convex Functions via Quantum Calculus(Mdpi, 2023) Arunrat, Nuttapong; Nonlaopon, Kamsing; Budak, HueseyinIn this paper, we use qa- and qb-integrals to establish some quantum Hermite-Hadamard-Fejer-type inequalities for ?-convex functions. By taking q & RARR;1, our results reduce to classical results on Hermite-Hadamard-Fejer-type inequalities for ?-convex functions. Moreover, we give some examples for quantum Hermite-Hadamard-Fejer-type inequalities for ?-convex functions. Some results presented here for ?-convex functions provide extensions of others given in earlier works for convex and ?-convex functions.Öğe On the Fractional Inequalities of the Milne Type(Wiley, 2025) Budak, Hueseyin; Kara, Hasan; Ogunmez, HasanOur investigations in this paper revolve around exploring fractional variants of inequalities of Milne type by applying twice differentiable convex mappings. Based on some principles of convexity, H & ouml;lder inequality, and power-mean inequality, novel inequalities are derived. The acquired inequalities are supported by illustrative examples, which are calculated via their proofs. Additionally, graphical representations are to verify the examples visually. Furthermore, this investigation unveils fresh findings within the realm of inequalities.Öğe Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables(Springer, 2021) Ali, Muhammad Aamir; Chu, Yu-Ming; Budak, Hueseyin; Akkurt, Abdullah; Yildirim, Hueseyin; Zahid, Manzoor AhmedIn this investigation, we demonstrate the quantum version of Montgomery identity for the functions of two variables. Then we use the result to derive some new Ostrowski-type inequalities for the functions of two variables via quantum integrals. We also consider the particular cases of the key results and offer some new integral inequalities.
