Maximum modulus of derivatives of a polynomial

For an arbitrary entire function f(z), let M(f,R) = \max_{|z|=R} |f(z)| and m(f, r) =\min_{|z|=r} |f(z)|. If P(z) is a polynomial of degree n having no zeros in |z| < k, k \geq ۱, then for ۰ \leq r \leq\rho\leq k, it is proved by Aziz et al. thatM(P',\rho)\leq\frac{n}{\rho+k}\{(\frac{\rho+k}{r+k})^n[۱-\frac{(k-\rho)(n|a_۰|-k|a_۱|)n}{(\rho^۲+k^۲)n|a_۰|+۲k^۲\rho |a_۱|}(\frac{\rho-r}{k+r})(\frac{k+۱}{k+\rho})^{n-۱}]M(P,r)-[\frac{(n|a_۰|\rho+k^۲|a_۱|)(r+k)}{(\rho^۲+k^۲)n|a_۰|+۲k^۲\rho|a_۱|}\times[((\frac{\rho+k}{r+k})^n-۱)-n(\rho-r)]]m(P,k)\}In this paper, we obtain a refinement of the above inequality. Moreover, we obtaina generalization of above inequality for M(P', R), where R\geq k.

Keywords:

Polynomial , inequality , Maximum modulus , Restricted Zeros