Approximation inequalities for modified $(p,q)$-Balázs-type BBH operators via almost A-statistical convergence

In this paper, we introduce a sequence of modified \((p,q)\) -Balázs-type Bleimann-Butzer-Hahn (BBH) operators depending on two auxiliary positive sequences \((bₘ)\) and \((cₘ)\) . The proposed construction includes the classical BBH, the q -BBH, and the \((p,q)\) -BBH operators as particular cases. We investigate the approximation properties of these operators within the framework of almost A -statistical convergence, which also applies to parameter sequences that do not converge in the ordinary sense. First, we establish a Korovkin-type approximation theorem for sequences of positive linear operators acting from the weighted function space \(Hω(I)\) into \(CB(I)\) , where \(I=[0,∞ )\) is the nonnegative half-line. We subsequently apply this theorem to the proposed operators under appropriate conditions on the parameter sequences \((pₘ)\) , \((qₘ)\) , \((bₘ)\) , and \((cₘ)\) . Moreover, we derive quantitative estimates for the rate of approximation using the modulus of continuity and obtain sharper bounds within a Lipschitz-type maximal function space. We also establish a Jensen-type inequality for the rational test function \(ẽ₁\) , where the factor \(bₘ\) appears explicitly in the estimate. Finally, numerical examples and graphical illustrations are presented to illustrate the convergence behaviour and provide a quantitative comparison with the classical BBH operators.