| Abstrakt: |
By using the principle of mathematical induction, it was shown by Singh and Chanam (J. Math. Inequal 15:1663–1675, 2021) that if p(z) is a polynomial of degree n having all its zeros in | z | ≤ 1 , then for all z on | z | = 1 for which p (z) ≠ 0 , ℜ z p ′ (z) p (z) ≥ n + 1 2 - 1 2 | a 0 | | a n | . In this paper, by using simple techniques we generalize the above inequality, thereby give a simple proof of the above inequality. As an application of our result, we obtain improvements of the well-known result due to Malik (J. Lond. Math. Soc 1(2):57–60, 1969). Further, we obtain some sharp refinements of a result due to Aziz and Rather (J. Math. Inequal. Appl 1:231–238, 1998). These results take into account the placement of the coefficients of the underlying polynomial. Moreover, a concrete numerical example is presented in order to graphically illustrate and compare the obtained inequalities with a classical result, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known. [ABSTRACT FROM AUTHOR] |
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