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On Sendov's conjecture about critical points of a polynomial
Ishfaq Nazir,Mohammad Ibrahim Mir,Irfan Ahmad Wani 강원경기수학회 2021 한국수학논문집 Vol.29 No.4
The derivative of a polynomial $p(z)$ of degree $n$, with respect to point $\alpha$ is defined by $D_{\alpha}p(z)=np(z)+(\alpha-z)p'(z)$. Let $p(z)$ be a polynomial having all its zeros in the unit disk $|z| \leq 1$. The Sendov conjecture asserts that if all the zeros of a polynomial $p(z)$ lie in the closed unit disk, then there must be a zero of $p'(z)$ within unit distance of each zero. In this paper, we obtain certain results concerning the location of the zeros of $D_{\alpha}p(z)$ with respect to a specific zero of $p(z)$ and a stronger result than Sendov conjecture is obtained. Further, a result is obtained for zeros of higher derivatives of polynomials having multiple roots.
On the zeros of generalized derivative of a polynomial
Ishfaq Nazir,Mohammad Ibrahim Mir,Irfan Ahmad Wani 강원경기수학회 2023 한국수학논문집 Vol.31 No.1
In this paper, we obtain some results concerning the location of zeros of generalized derivatives of polynomials which are analogous to those for the ordinary derivative of polynomials.
LP−TYPE INEQUALITIES FOR DERIVATIVE OF A POLYNOMIAL
Irfan Ahmad Wani,Mohammad Ibrahim Mir,Ishfaq Nazir 강원경기수학회 2021 한국수학논문집 Vol.29 No.4
For the polynomial $P(z)$ of degree $n$ and having all its zeros in $|z| \leq k$, $k \geq 1$, Jain \cite{j} proved that \begin{align*} \max_{|z|=1} |P^{\prime}(z)|\geq n \frac{|c_0| + |c_n|k^{n+1}}{|c_0|(1+k^{n+1}) + |c_n| ( k^{n+1} + k^{2n})} \max_{|z|=1}|P(z)| . \end{align*} In this paper, we extend above inequality to its integral analogous and there by obtain more results which extended the already proved results to integral analogous.
SOME BOUNDS FOR THE ZEROS OF POLYNOMIALS
( Mahnaz Shafi Chishti ),( Mohammad Ibrahim Mir ),( Vipin Kumar Tyagi ) 한국수학교육학회 2023 純粹 및 應用數學 Vol.30 No.1
In this paper, we find a bound for all the zeros of a polynomial in terms of its coefficients similar to the bound given by Montel (1932) and Kuneyida (1916) as an improvement of Cauchy's classical theorem. In fact, we use a generalized version of Hölder's inequality for obtaining various interesting bounds for all the zeros of a polynomial as function of their coefficients.
GENERALIZATION OF SOME INEQUALITIES TO THE CLASS OF GENERALIZED DERIVATIVE
( Irfan Ahmad Wani ),( Mohammad Ibrahim Mir ),( Ishfaq Nazir ) 한국수학교육학회 2021 純粹 및 應用數學 Vol.28 No.4
In this paper, we obtain some inequalities concerning the class of generalized derivative and generalized polar derivative which are analogous respectively to the ordinary derivative and polar derivative of polynomials.
Some Bounds for Zeros of a Polynomial with Restricted coefficients
Mahnaz Shafi Chishti,Vipin Kumar Tyagi,Mohammad Ibrahim Mir 한국수학교육학회 2024 純粹 및 應用數學 Vol.31 No.1
For a Polynomial P(z)= SMALLSUM _{j=0}^{n} `a _{j} z ^{j} with a_j ≥ a_j−1, a_0 > 0 (j = 1, 2, ..., n), a classical result of Enestrom-Kakeya says that all the zeros of P(z) lie in |z| ≤ 1. This result was generalized by A. Joyal et al. [3] where they relaxed the non-negative condition on the coefficents. This result was further generalized by Dewan and Bidkham [9] by relaxing the monotonicity of the coefficients. In this paper, we use some techniques to obtain some more generalizations of the results [3], [8], [9].