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Öğe Aralık değerli fonksiyonlar için genelleştirilmiş hermite-hadamard eşitsizlikleri(Düzce Üniversitesi, 2024) Kara, Hasan; Budak, HüseyinBu tezde, aralık değerli fonksiyonlar için genelleştirilmiş Hermite-Hadamard eşitsizlikleri elde edilmiştir. Bu tez üç ana bölümden oluşmaktadır. İlk bölüm, tek değişkenli aralık değerli fonksiyonlar için elde edilen genelleştirilmiş kesirli eşitsizlikleri içermektedir. İkinci bölümde ise iki değişkenli aralık değerli fonksiyonlar için eşitsizlikler elde edilmiştir. Bu kısımda aralık değerli fonksiyonlar için Riemann-Liouville kesirli integralleri, Sarıkaya kesirli integralleri ve başka bir fonksiyona göre kesirli integraller yardımıyla Hermite-Hadamard eşitsizlikleri elde edilmiştir. Üçüncü bölüm, aralık değerli fonksiyonlar için Hermite-Hadamard tipli eşitsizliklerin ağırlıklı versiyonlarından oluşmaktadır. Bu bölümde tek değişkenli ve iki değişkenli fonksiyonların çarpımı için elde edilen eşitsizlikler verilmiştir. Ayrıca, ağırlıklı Jensen eşitsizliği yardımıyla aralıklı tek değişkenli fonksiyonlar ve aralık değerli iki değişkenli fonksiyonlar için eşitsizlikler elde edilmiştir.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 Conformable fractional versions of Hermite-Hadamard-type inequalities for twice-differentiable functions(Springer, 2023) Hezenci, Fatih; Kara, Hasan; Budak, HüseyinIn this paper, new inequalities for the left and right sides of the Hermite-Hadamard inequality are acquired for twice-differentiable mappings. Conformable fractional integrals are used to derive these inequalities. Furthermore, we provide our results by using special cases of obtained theorems.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 Generalizations of some important fractional integral inequalities by using a parameter(Elsevier, 2024) Hezenci, Fatih; Budak, H¨Useyin; Kara, Hasan; Agarwal, PraveenIn the present paper, we first prove an equality for the functions whose second derivatives in absolute value are convex. By using this equality we obtain some parameterized inequalities for twice differentiable functions by using generalized fractional integrals. Finally, we show that our main results reduce to trapezoid, midpoint, and Simpson- and Bullen-type inequalities proved in earlier papers. © 2024 Elsevier B.V., All rights reserved.Öğe Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions(De Gruyter Poland Sp Z O O, 2022) Vivas-Cortez, Miguel J. J.; Kara, Hasan; Budak, Hüseyin; Ali, Muhammad Aamir; Chasreechai, SaowaluckIn this article, we introduce the notions of generalized fractional integrals for the interval-valued functions (IVFs) of two variables. We establish Hermite-Hadamard (H-H) type inequalities and some related inequalities for co-ordinated convex IVFs by using the newly defined integrals. The fundamental benefit of these inequalities is that these can be turned into classical H-H inequalities and Riemann-Liouville fractional H-H inequalities, and new k k -Riemann-Liouville fractional H-H inequalities can be obtained for co-ordinated convex IVFs without having to prove each one separately.Öğe Generalized fractional midpoint type inequalities for co-ordinated convex functions(University of Nis, 2023) Hezenci, Fatih; Budak, Hüseyin; Kara, Hasan; Sarıkaya, Mehmet ZekiIn this research paper, we investigate generalized fractional integrals to obtain midpoint type inequalities for the co-ordinated convex functions. First of all, we establish an identity for twice partially differentiable mappings. By utilizing this equality, some midpoint type inequalities via generalized fractional integrals are proved. We also show that the main results reduce some midpoint inequalities given in earlier works for Riemann integrals and Riemann-Liouville fractional integrals. Finally, some new inequalities for k-Riemann-Liouville fractional integrals are presented as special cases of our results. © 2023, University of Nis. All rights reserved.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, Trapezoid and Midpoint Type Inequalities Involving Generalized Fractional Integrals for Convex Functions(Univ Maragheh, 2023) Kara, Hasan; Erden, Samet; Budak, HüseyinWe first construct new Hermite-Hadamard type in-equalities which include generalized fractional integrals for convex functions by using an operator which generates some significant fractional integrals such as Riemann-Liouville fractional and the Hadamard fractional integrals. Afterwards, Trapezoid and Mid-point type results involving generalized fractional integrals for func-tions whose the derivatives in modulus and their certain powers are convex are established. We also recapture the previous results in the particular situations of the inequalities which are given in the earlier works.Öğe Hermite-Hadamard-Mercer type inclusions for interval-valued functions via Riemann-Liouville fractional integrals(Scientific Technical Research Council Turkey-Tubitak, 2022) Kara, Hasan; Ali, Muhammad Aamir; Budak, HüseyinIn this research, we first establish some inclusions of fractional Hermite???Hadamard???Mercer type for interval -valued functions. Moreover, by special cases of our main results, we show that our main results reduce several inclusions obtained in the earlier works.Öğe Hermite-Hadamard-type inequalities for interval-valued coordinated convex functions involving generalized fractional integrals(Wiley, 2021) Kara, Hasan; Ali, Muhammad Aamir; Budak, HüseyinIn this paper, we define interval-valued left-sided and right-sided generalized fractional double integrals. We establish inequalities of Hermite-Hadamard like for coordinated interval-valued convex functions by applying our newly defined integrals.Öğe HERMITE–HADAMARD TYPE INEQUALITIES FOR INTERVAL-VALUED COORDINATED LR-CONVEXITY VIA GENERALIZED FRACTIONAL INTEGRALS(Rocky Mountain Mathematics Consortium, 2025) Kara, Hasan; Ali, Muhammad Aamir; Budak, H¨UseyinWe first obtain some new Hermite–Hadamard-type inequalities for interval-valued LR-convex functions. Afterwards, we investigate Hermite–Hadamard-type inequalities for interval-valued coordinated LRconvex functions. New results are obtained by making special choices in newly established inequalities in the case of interval-valued LR-convex functions and interval-valued coordinated LR-convex functions. It is also shown that the newly established inequalities are extensions of comparable results in the literature. © 2025 Elsevier B.V., All rights reserved.Öğe New approaches to corrected Euler-Maclaurin-type inequalities involving Riemann-Liouville fractional integrals for different function classes(Springer, 2025) Kara, Hasan; Hezenci, Fatih; Munir, Arslan; Budak, HuseyinThis paper investigates several Corrected Euler-Maclaurin-type inequalities for different function classes using Riemann-Liouville fractional integrals. The results, which are derived from special cases of theorems and illustrative examples, are subsequently presented. Furthermore, the authors have developed fractional Corrected Euler-Maclaurin-type inequalities for bounded functions. In addition, the research has acquired fractional Corrected Euler-Maclaurin-type inequalities for Lipschitzian functions. Finally, the study concludes with the derivation of fractional Corrected Euler-Maclaurin-type inequalities for functions of bounded variation.Öğe A New extension of Hermite Hadamard inequalities associating ψ-Hilfer fractional integral(Palestine Polytechnic University, 2025) Qayyum, Ather; Budak, H¨Useyin; Bat, Umut; Kara, Hasan; Munir, Arslan; Rathour, LaxmiFractional inequalities have been an essential topic in mathematics and have found applications in various domains. In this article, we established some new bounds (below and above) for mid-point type inequality and trapezoidal-type inequality for ψ-Hilfer- fractional integral by utilizing functions whose second derivatives are bounded. We also investigate some new generalization and extension of Hermite-Hadamard type inequalities for ψ-Hilfer fractional integrals for the functions having the condition: (Formula presented). © 2025 Elsevier B.V., All rights reserved.Öğe NEW EXTENSIONS OF THE HERMITE-HADAMARD INEQUALITIES BASED ON ψ-HILFER FRACTIONAL INTEGRALS(Korean Soc Mathematical Education, 2024) Budak, Huseyin; Bas, Umut; Kara, Hasan; Samei, Mohammad EsmaelThis article presents the above and below bounds for Midpoint and Trapezoid types inequalities for 95-Hilfer fractional integrals with the assistance of the functions whose second derivatives are bounded. We also possess some extensions and generalizations of Hermite-Hadamard inequalities via 95-Hilfer fractional integrals with the aid of the functions that have the conditions that will said.Öğe New Extensions of the Parameterized Inequalities Based on Riemann-Liouville Fractional Integrals(Mdpi, 2022) Kara, Hasan; Budak, Hüseyin; Hezenci, FatihIn this article, we derive the above and below bounds for parameterized-type inequalities using the Riemann-Liouville fractional integral operators and limited second derivative mappings. These established inequalities generalized the midpoint-type, trapezoid-type, Simpson-type, and Bullen-type inequalities according to the specific choices of the parameter. Thus, a generalization of many inequalities and new results were obtained. Moreover, some examples of obtained inequalities are given for better understanding by the reader. Furthermore, the theoretical results are supported by graphs in order to illustrate the accuracy of each of the inequalities obtained according to the specific choices of the parameter.Öğe NEW EXTENSIONS VERSION OF HERMITE-HADAMARD TYPE INEQUALITIES BY MEANS OF CONFORMABLE FRACTIONAL INTEGRALS(Univ Miskolc Inst Math, 2024) Bas, Umut; Budak, Huseyin; Kara, HasanIn the current investigation, we acquire the upper and lower bounds for inequalities of midpoint-type and trapezoid-type involving conformable fractional integral operators with the help of the mappings whose second derivatives are bounded. We support the established inequalities with examples. Moreover, we use graphs to demonstrate the correctness of the given examples. What's more, we prove the Hermite-Hadamard inequality, which includes conformable fractional integrals, with the aid of condition f ' (a + b t) f ' (t) >= 0, t is an element of a, a+b than the convexity of function.Öğe New midpoint type inequalities for generalized fractional integral(Univ Tabriz, 2022) Budak, Hüseyin; Kara, Hasan; Kapucu, RabiaIn this paper, we first establish two new identities for differentiable function involving generalized fractional integrals. Then, by utilizing these equalities, we obtain some midpoint type inequalities involving generalized fractional integrals for mappings whose derivatives in absolute values are convex. We also give several results as special cases of our main results.Öğe New parameterized inequalities for twice differentiable functions(Univ Nis, Fac Sci Math, 2023) Budak, Hüseyin; Kara, Hasan; Hezenci, Fatih; Sarıkaya, Mehmet ZekiThe present paper first establishes that an identity involving generalized fractional integrals is proved for twice differentiable functions by using a parameter. By using this equality, we obtain some parameterized inequalities for the functions whose second derivatives in absolute value are convex. Finally, we show that our main results reduce to trapezoid, midpoint Simpson and Bullen-type inequalities which are proved in earlier published papers.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.
