Estimate For Second Bullen-Type Inequalities Using Various Classes and Their Applications

This paper introduces new estimates for the recently developed second Bullen-type inequality, which refines the classical Bullen inequality by incorporating additional evaluations of convex functions. The study establishes integral identities that quantify the deviation between the integral mean and discrete approximations, expressed in terms of the derivative and kernel functions. These identities serve as the foundation for deriving bounds under various conditions, including convexity, Lipschitz continuity and boundedness of the derivative. Additionally, we explore applications involving Bernoulli numbers and special functions, highlighting the relevance of Bullen-type inequalities in approximation theory and analytic number theory. The results provide sharper bounds and suggest new directions for research in integral inequalities and their applications.