Some inequalities for the polar derivative of a polynomial

Extensions of some polynomial inequalities to the polar derivative

“…The above corollary is an extension and refinement of a result of Aziz [1] and for t = 1, it reduces to a result of Aziz and Shah [5].…”

Inequalities for the polar derivative of a polynomial with restricted zeros

“…For the polar derivative D α p(z), Aziz and Shah [2] proved that if p(z) having all its zeros in |z| ≤ 1, then…”

Differential inequalities for entire functions of finite degree and rational functions with prescribed poles

“…Theorems 2 and 5 ensure the upper estimate for sup x∈R |σf (x) + i(1 − ζ)f (x)|, |ζ| > 1, with f (z) an entire function of finite degree σ > 0 bounded on R and such that h f (π/2) = 0 and f (z) = 0 for Im z > −k, k > 0 (Corollary 2). This estimate contains the following result by Aziz and Shah [12]: if an algebraic polynomial P (z) of degree n does not vanish in the circle |z| < K, K ≥ 1, then max |z|=1 |nP (z) + (α − z)P (z)| ≤ n K + 1…”

“…If τ < σ = h f (−π/2) then Theorem 1 is applicable to f (z) and ω(z) = e iσz . This theorem ensures the required inequality and the equality holds only for the functions of the form (12). In the case of τ < σ = h f (π/2) we apply the assertion proven tof (z).…”