In 2009, Maksa and Páles established an extension of the decomposition theorem of Ng in the context of higher-order convexity notions. They proved that a real function is Wright convex of order n if and only if it can be decomposed as the sum of a convex function of order n and a polynomial function of order at most n. Their proof was based on transfinite tools in the background. The main purpose of Chapter one is to adopt the methods of a paper of Páles published in 2020 and establish a new and elementary proof for the theorem of Maksa and Páles. The main purpose of Chapter two is to introduce various convexity concepts in terms of a positive Chebyshev system ω and give a systematic investigation of the relations among them. We generalize a celebrated theorem of Bernstein-Doetsch to the setting of ω-Jensen convexity. We also give sufficient conditions for the existence of discontinuous ω-Jensen affine functions. The concept of Wright convexity is extended to the setting of Chebyshev systems, as well, and it turns out to be an intermediate convexity property between ω-convexity and ω-Jensen convexity. For certain Chebyshev systems, we generalize the decomposition theorems of Wright convex and higher-order Wright convex functions obtained by C. T. Ng in 1987 and by Maksa and Páles in 2009, respectively. The main goal of Chapter three is to show that if a real valued function defined on a groupoid satisfies a certain Levi-Civita-type functional equation, then it also fulfills a Cauchy-Schwarz-type functional inequality. In particular, if the groupoid is the multiplicative structure of commutative ring, then we can establish the existence of nontrivial additive functions satisfying inequalities connected to the multiplicative structure.