Enhanced Turan-type Inequalities for Polynomials.

If p(z) = Xn ν=0 aνz ν is a polynomial of degree n having all its zeros in |z| ≤ k, k ≥ 1, Jain [Bull. Math. Soc. Sci. Math. Roumania Tome, 59(2016), 339-347] proved max |z|=1 |p′ (z)| ≥ nAk max |z|=1 |p(z)|, where Ak = ( |a0| + |an|k n+1/ |ao|(1 + k n+1) + |an|(k n+1 + k 2n) ) . In this paper, we initially derive a generalized form that not only encompasses but also enhances the aforementioned inequality. Additionally, we extend this formulation to a more comprehensive result, thereby producing an improved outcome for certain established inequalities as a specific instance. [ABSTRACT FROM AUTHOR]