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    Approximation properties of a modified Szász-Mirakyan operator with enhanced convergence and comparative numerical examples
    (Springer, 2026) Ansari, Khursheed J.; Civelek, Muhammet; Kunt, Mehmet; Usta, Fuat
    Sz & aacute;sz-Mirakyan operators constitute an important class of positive linear operators in approximation theory due to their ability to approximate continuous functions on unbounded intervals. Motivated by the search for operators with improved approximation behaviour, we introduce a modified form of the classical Sz & aacute;sz-Mirakyan operators and investigate their fundamental properties. We first establish the connection between the proposed operators and the classical Sz & aacute;sz-Mirakyan operators and derive the corresponding moments. Using tools of classical approximation theory, including the modulus of continuity and direct estimates, we study their approximation behaviour and obtain quantitative estimates for the rate of convergence. The obtained results show that the modified operators provide improved approximation characteristics compared with the classical Sz & aacute;sz-Mirakyan operators under suitable conditions. These findings contribute to the theory of positive linear operators and provide a useful framework for approximation processes on unbounded intervals.
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    (p, q)-Hermite-Hadamard inequalities and (p, q)-estimates for midpoint type inequalities via convex and quasi-convex functions
    (Springer-Verlag Italia Srl, 2018) Kunt, Mehmet; İşcan, İmdat; Alp, Necmettin; Sarıkaya, Mehmet Zeki
    In this paper, we prove the correct (p, q)-Hermite-Hadamard inequality, some new (p, q)-Hermite-Hadamard inequalities, and generalized (p, q)-Hermite-Hadamard inequality. By using the left hand part of the correct (p, q)-Hermite-Hadamard inequality, we have a new equality. Finally using the new equality, we give some (p, q)-midpoint type integral inequalities through (p, q)-differentiable convex and (p, q)-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works.
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    PROOF PERTAINING TO SOME IMPORTANT PROPERTIES OF q-INTEGRAL
    (Rocky Mt Math Consortium, 2025) Alp, Necmettin; Sarikaya, Mehmet Zeki; Kunt, Mehmet; Budak, Huseyin
    In most studies on quantum integral inequalities, the properties of the Riemann integral were also provided for the quantum integral. Due to the misuse of these properties, many studies include some mistakes. In this paper, counterexamples are given first; the properties that cause mistakes are not provided. Moreover, the correct versions of these properties are proved under some conditions. We believe that this study will be an important reference for authors who study quantum integral inequalities.
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    q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions
    (Elsevier Science Bv, 2018) Alp, Necmettin; Sarıkaya, Mehmet Zeki; Kunt, Mehmet; İşcan, İmdat
    In this paper, we prove the correct q-Hermite-Hadamard inequality, some new q-Hermite-Hadamard inequalities, and generalized q-Hermite-Hadamard inequality. By using the left hand part of the correct q-Hermite-Hadamard inequality, we have a new equality. Finally using the new equality, we give some q-midpoint type integral inequalities through q-differentiable convex and q-differentiable quasi-convex functions. Many results given in this paper provide extensions of others given in previous works. (C) 2016 The Authors. Production and hosting by Elsevier B.V.