A Comparative Study of Some Tur\(a\)n-type Inequalities in the Realm of Complex Polynomials - Robinson Soraisam; Barchand Chanam (Mathematics and Computer Science: Contemporary Developments Vol. 1, 2024)
Let \(p(z)\) be a polynomial of degree \(n\) having all its zeros in \(|z| ≤ 1\), then famous inequality due to Turán [Compos. Math. 7 (1939), 89-95] is \[_ |z|=1 |p^′(z)| ≥ n2 _|z|=1|p(z)| . \] This Turán’s inequality was generalized for the first time by Malik [J. London Math. Soc., 1(1969),57-60] that if \(p(z)\) is a polynomial of degree \(n\) having all its zeros in \(|z| ≤ k, k ≤ 1\), then \[_ |z|=1 |p^′(z)| ≥ n1+k _|z|=1|p(z)| . \] While for the case \(k ≥ 1\), Govil [Proc. Amer. Math. Soc. 41(1973), 543-546] prove that \[_ |z|=1 |p^′(z)| ≥ n1+k^n _|z|=1|p(z)| . \] The above inequalities play a vital role in approximation theory. Frequently, further progress in this theory has depended on first obtaining a corresponding generalization or analogue of these inequalities. In this article, we discuss in brief some of the recent improvements of the above inequalities particularly the later type and make a comparative study of them using an example with detail graphical illustrations.