Summary By employing a novel idea and simple techniques, we substantially generalize the Turán type inequality for rational functions with real zeros and prescribed poles established by Min  to include Lp spaces for 1≤ p ≤ ∞ while loosing the restriction ρ > 2 at the same time.
Turán’s book , in Section 19.4, refers to the following result of Gábor Halász. Let a0, a1, ..., an−1 be complex numbers such that the roots α1, ⋯, αn of the polynomial xn + an−1xn−1 + ⋯ + a1x + a0 satisfy minj Re αj ≧ 0 and let function Y(t) be a solution of the linear differential equation Y(n) + an−1Y(n−1) + ⋯ + a1Y′ + a0Y = 0. Then
Re αj ≧ 0.
In this paper we improve the exponent 5 on the right-hand side to the best possible value (which is 2) and prove an analogous
inequality where the integration domain is symmetric to the origin.