Abstarct: Given the importance and wide applications of polynomial inequalities in various fields of mathematics, this thesis focuses on studying and improving some results related to the absolute value growth and derivatives of polynomials. In particular, we also aim to generalize several classical inequalities. In particular, we discuss results concerning the maximum modulus principle, showing that a non-constant polynomial attains its maximum value on the boundary of its domain. The non-existence of an exact maximum modulus theorem provides a method for estimating such values approximately. Hence, various studies and research efforts have been devoted to this topic. In this thesis, we take a further step toward generalizing and improving these results. The first chapter presents the necessary preliminaries and basic concepts. In the second chapter, we study the growth of the absolute value of the derivative of polynomials in Lq spaces. In Chapter 3, a Zygmund-type derivative inequality in the space Lq is established for polynomials. Some known results are improved and extended in this context. In Chapter 4, a Laguerre-type inequality for the s-th derivative of lacunary polynomials in the space is investigated and analyzed in detail.