For every $1leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.

Introduction. As we have shown several years ago [Y2], zeros of L(s,∆) and L(s,∆) can be calculated quite efficiently by a certain experimental method. Here ∆ denotes the cusp form of weight 12 with respect to SL(2,Z) and L(s,∆) (resp. L(s,∆)) denotes the standard (resp. symmetric square) L-function attached to ∆. The purpose of this paper is to show that this method can be applied to a wide cl...

‎let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎in this paper we obtain some new results about the dependence of $|p(rz)|$ on $|p(rz)| $ for $r^2leq rrleq k^2$‎, ‎$k^2 leq rrleq r^2$ and for $rleq r leq k$‎. ‎our results refine and generalize certain well-known polynomial inequalities‎.

The goals of the proposed research are: 1) To obtain additional concrete computational evidence §2.1 of the unknown structure on the imaginary parts of the non-trivial zeros of the Riemann zeta function, called herein the Riemann spectrum, following the method from [1]; specifically, the PI will compute correlations between histograms of random variables Xp (§2); 2) To prove the that the densit...

We describe a new, rigorous algorithm for efficiently and simultaneously computing many values of the Riemann zeta function on the critical line by exploiting the fast Fourier transform (FFT). We apply this to locating non-trivial zeros of zeta to high precision which are in turn used as input to our own implementation of the Lagarias and Odlyzko analytic algorithm to compute π(x), the prime co...

The paper presents an $L^{r}-$ analogue of an inequality regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle of arbitrary radius but greater or equal to one. Our result provides improvements and generalizations of some well-known polynomial inequalities.

For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...