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3 Journal of Mathematics and Applications

3 Jain Vinay Kumar

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On the derivative of a polynomial

Jain Vinay Kumar

Journal of Mathematics and Applications

2023

Vol. 46

5--13

For an arbitrary polynomial P(z), let M(P,r) =max|z|=r|P(z)| and m(P,r)=min|z|=r|P(z)|, (r >0). For a polynomial p(z)=Pn [wzór] of degreen, having all its zeros in|z|≤k, (k≥1), with a zero of orders, (s≥0), at 0 and F0, F1, F2, Gn-s, F3, F4, Hn-s, Fn-s, B0, B1, En-1, B2, B3, Dn-1 and Bn-1, as in Theorem, we have obtained a refinement [wzór] of our old result (1997), there by obtaining a new refinement of known results [wzór].

Inequality for polynomials with prescribed zeros

Jain Vinay Kumar

Journal of Mathematics and Applications

2020

Vol. 43

81--85

For a polynomial p(z) of degree n with a zero at β,of order at least k(≥1), it isknown that [wzór]. By considering polynomial p(z) of degree n in the form [wzór], a polynomial of degree n−k, with [wzór] we have obtained [wzór] a generalization of the known result.

On the maximum modulus of a polynomial

Jain Vinay Kumar

Journal of Mathematics and Applications

2019

Vol. 42

109--116

For a polynomial p(z) of degree n, having no zeros in |z| < 1 Ankeny and Rivlin had shown that for R ≥ 1 [wzór]. Using Govil, Rahman and Schmeisser’s refinement of the generalization of Schwarz’s lemma we have obtained a refinement of Ankeny and Rivlin’s result. Our refinement is also a refinement of Dewan and Pukhta’s refinement of Ankeny and Rivlin’s result.