REFINEMENTS OF SOME INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL

If P(z) = a(n)z(n) + Sigma(n)(j=mu) a(n-j)z(n-j), 1 <= mu <= n, is a polynomial of degree n having all its zeros in |z| <= k, k <= 1, then it was recently claimed by K. K. Dewan, Naresh Singh, Abdullah Mir [Extensions of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009), 807-815] that for every real or complex number alpha, with |alpha| >= k k(mu), [GRAPHIC] where m = min(|z|=k) |P(z)|, D alpha P (z) is a polar derivative of P (z) with respect to the point alpha epsilon C and A(mu) is given by (1.11). The proof of this result is not correct. In this paper, we present certain more refined results which not only provides a correct proof of above inequality as a special case but also yields a refinement of above and other related result.