SOME LOWER BOUND ESTIMATES FOR THE GENERALIZED DERIVATIVE OF A POLYNOMIAL
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If P(z) is a polynomial of degree n having all its zeros in |z| ≤ k,k ≤ 1, then Rather et al. ( Some inequalities for polynomials with restricted zeros, Ann. Univ. Ferrara, 67 (2021), 183-189.) proved that for all z on |z| = 1 for which P(z) ̸ = 0, Re(zP′(z)/P(z)) ≥ n/(1+k){1+k/n((k^n|a_n|−|a_0|)/(k_n|a_n| + |a_0|))}. In this paper, we extend this inequality to the generalised derivative by taking sfolded zeros at origin. As an application, we obtain some lower bound estimates for the generalized derivative and generalized polar derivative of a polynomial with restricted zeros, which include various results due to Tur´an, Malik, Dubinin, Aziz, Rather and Govil as special cases.
Abstract
If P(z) is a polynomial of degree n having all its zeros in |z| ≤ k,k ≤ 1, then Rather et al. ( Some inequalities for polynomials with restricted zeros, Ann. Univ. Ferrara, 67 (2021), 183-189.) proved that for all z on |z| = 1 for which P(z) ̸ = 0, Re(zP′(z)/P(z)) ≥ n/(1+k){1+k/n((k^n|a_n|−|a_0|)/(k_n|a_n| + |a_0|))}. In this paper, we extend this inequality to the generalised derivative by taking sfolded zeros at origin. As an application, we obtain some lower bound estimates for the generalized derivative and generalized polar derivative of a polynomial with restricted zeros, which include various results due to Tur´an, Malik, Dubinin, Aziz, Rather and Govil as special cases.