This paper addresses negative definiteness conditions of quadratic functions, common in control problems with time-varying delay systems. Existing geometric conditions, relaxed by partitioning techniques, may lack monotonic convergence, making their optimality questionable. Alternative conditions based on the generalized S-procedure are known to be necessary and sufficient when coefficients are not dependent on uncertain parameters; otherwise, they are sufficient only. We propose new approaches to mitigate these issues, unexplored in previous studies, demonstrated in stability analysis of linear parameter varying sampled-data control systems. First, we rewrite quadratic polynomial inequalities as homogeneous ones, deriving relaxed conditions using Young's inequality or Polya's theorem, offering recursive monotonic convergent relaxations. Then, based on Bézier curve equivalence and the de Casteljau algorithm, we provide further relaxed recursive monotonic convergent conditions. Two numerical examples illustrate the effectiveness and improvements over previous related studies.