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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.
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.
Shamsul Hayat,Sangeeta Yadav,Arif Shafi Wani,Mohammad Irfan,Mohammad Naseer Alyemini,Aqil Ahmad 한국원예학회 2012 Horticulture, Environment, and Biotechnology Vol.53 No.5
The present study was carried out to elucidate the protective role of sodium nitroprusside (SNP) against salt (NaCl) stress given as seed soaking in tomato (Solanum lycopersicum Mill. cv. K-21). For this experiment prior to sowing, the surface sterilized tomato seeds were soaked in different concentrations of NaCl solution (50, 100, and 150 mM) for 8 hours. Some NaCl soaked seeds were then transferred to the 10-5 M of SNP solution for 8 hours. The plants were sampled at 60 DAS to assess the nitrate reductase activity, proline content and the activities of antioxidant enzymes. The antioxidant enzyme activity and proline content increased in the plants where SNP was applied as a follow up treatment of the NaCl as seed soaking over the control plants thereby providing stress tolerance to the plants. In case of the nitrate reductase activity the toxic effects generated by the lowest concentration of NaCl (50 mM) was completely neutralized by SNP and partly at higher concentrations (100/150 mM). These results suggest that nitric oxide can be used as a stress alleviator in plants which are grown in the soil contaminated with salinity stress.
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.