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    Fractional Hermite-Hadamard inequality and error estimates for Simpson's formula through convexity with respect to a pair of functions
    (Univ Miskolc Inst Math, 2023) Ali, Muhammad Aamir; Soontharanon, Jarunee; Budak, Hüseyin; Sitthiwirattham, Thanin; Feckan, Michal
    In this article, we establish two new and different versions of fractional HermiteHadamard type inequality for the convex functions with respect to a pair of functions. Moreover, we establish a new Simpson's type inequalities for differentiable convex functions with respect to a pair of functions. We also prove two more Simpson's type inequalities for differentiable convex functions with respect to a pair of functions using the power mean and Ho & BULL;lder's inequalities. It is also shown that the newly established inequalities are the extension of some existing results. Finally, we add some mathematical examples and their graphs to show the validity of newly established results.
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    Simpson's and Newton's Type Inequalities for (alpha, m)-Convex Functions via Quantum Calculus
    (Mdpi, 2022) Soontharanon, Jarunee; Ali, Muhammad Aamir; Budak, Hüseyin; Nonlaopon, Kamsing; Abdullah, Zoya
    In this paper, we give the generalized version of the quantum Simpson's and quantum Newton's formula type inequalities via quantum differentiable (alpha, m)-convex functions. The main advantage of these new inequalities is that they can be converted into quantum Simpson and quantum Newton for convex functions, Simpson's type inequalities (alpha, m)-convex function, and Simpson's type inequalities without proving each separately. These inequalities can be helpful in finding the error bounds of Simpson's and Newton's formulas in numerical integration. Analytic inequalities of this type as well as particularly related strategies have applications for various fields where symmetry plays an important role.
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    Some New Generalized Fractional Newton's Type Inequalities for Convex Functions
    (Hindawi Ltd, 2022) Soontharanon, Jarunee; Ali, Muhammad Aamir; Budak, Hüseyin; Kösem, Pınar; Nonlaopon, Kamsing; Sitthiwirattham, Thanin
    In this paper, we establish some new Newton's type inequalities for differentiable convex functions using the generalized Riemann-Liouville fractional integrals. The main edge of the newly established inequalities is that these can be turned into several new and existing inequalities for different fractional integrals like Riemann-Liouville fractional integrals, k-fractional integrals, Katugampola fractional operators, conformable fractional operators, Hadamard fractional operators, and fractional operators with the exponential kernel without proving one by one. It is also shown that the newly established inequalities are the refinements of the previously established inequalities inside the literature.