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Finding the sharp estimate of max | z | = 1 | p ′ (z) | in terms of max | z | = 1 | p (z) | for the class of polynomials p(z) satisfying p (z) ≡ z n p (1 / z) has been a well-known open problem for a long time and many papers in this direction have appeared. The earliest result is due to Govil, Jain and Labelle [9] who proved that for polynomials p(z) satisfying p (z) ≡ z n p (1 / z) and having all the zeros either in left half or right half-plane, the inequality max | z | = 1 | p ′ (z) | ≤ n 2 max | z | = 1 | p (z) | holds. A question was posed whether this inequality is sharp. In this paper, we answer this question in the negative by obtaining a bound sharper than n 2 . We also conjecture that for such polynomials max | z | = 1 | p ′ (z) | ≤ ( n 2 - 2 - 1 4 (n - 2)) max | z | = 1 | p (z) | and provide evidence in support of this conjecture. [ABSTRACT FROM AUTHOR]

Let P(z) = ∑nv=0 avzv be a polynomial of degree n: Then as a generalization of a well-known result of Turán [18], it was proved by Govil [5] that if P(z) is a polynomial of degree n having all its zeros in |z| ≤ K, K ≥ 1, then (0.1) max |z|=1 |P'(z)|≥ n/1+Kn max |z|=1 |P(z)|. In this paper, we prove a polar derivative generalization of this inequality, which as a corollary gives a sharpening of this inequality (0.1). [ABSTRACT FROM AUTHOR]

The problem of obtaining the smallest possible region containing all the zeros of a polynomial has been attracting more and more attention recently, and in this paper, we obtain several results providing the annular regions that contain all the zeros of a complex polynomial. Using MATLAB, we construct specific examples of polynomials and show that for these polynomials our results give sharper regions than those obtainable from some of the known results. [ABSTRACT FROM AUTHOR]

Let f be an entire function of exponential type satisfying the condition $$ f(z) \equiv e^{i\gamma } e^{i\tau z} \overline {f(\bar z)} $$ for some real γ. Lower and upper estimates for ∫| f′( x)| p d x in terms of ∫| f( x)| p d x, for such a function f belonging to L p ( R), have been known in the case where p ∊ [1, ∞) and γ = 0. In this paper, these estimates are shown to hold for any p ∊ (0, ∞) and any real γ. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n and for a complex number α, let Dαp(z) np(z) + (α - z)p'(z) denote the polar derivative of the polynomial p(z) with respect to α. In this paper, we obtain inequalities for the polar derivative of a polynomial having all its zeros in /z/ ≤ K. Our results generalize and sharpen a famous inequality of Turán and some other known results in this direction. [ABSTRACT FROM AUTHOR]

Let H be the open upper half-plane, and its closure. If f is a non-constant transcendental entire function of exponential type such that on the real axis, then it may happen that is not greater than μ anywhere in H even if in H . We show that if f is an entire function of exponential type not vanishing in such that , then if and only if , and that for some z ∈ H only if f is a constant. This result, which can be seen as a 'minimum modulus principle' for entire functions of exponential type not vanishing in a half-plane, helps us to obtain generalizations of two inequalities of Boas, and one of Turán. [ABSTRACT FROM AUTHOR]