M. A. Qazi

On the maximum modulus of polynomials

The maximum modulus on lzI = r < 1 of a polynomial of degree n not vanishing in I zI < 1 is estimated in terms of its maximum modulus on izl = 1 and certain of its coefficients. Some other related problems are also considered.

A new property of entire functions of exponential type not vanishing in a half-plane and applications

Let H be the open upper half-plane, and H its closure. If ƒ is a non-constant transcendental entire function of exponential type such that |ƒ(x)| ≥ μ > 0 on the real axis, then it may happen that |ƒ(z)| is not greater than μ anywhere in H even if ƒ(z) in H. We show that if ƒ is an entire function of exponential type not vanishing in H such that inf x∈ℝ|ƒ(x)| = μ > 0, then inf z∈ H |ƒ(z)| = μ if and only if hƒ (π/2) := lim sup y →∞y − log|ƒ(iy) ≥ 0, and that |ƒ(z)| = mu for some z ∈ H only if ƒ 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.

Some Estimates for the Derivatives of Rational Functions

Let Pn be the class of all polynomials of degree at most n. It is known that if f ∈ Pn and ¦f(z)¦ ≤ 1 on the unit circle, then ¦ft′(z)¦ ≤ ¦z¦nt - 1 outside the unit disk. We present an ‘extension’ of this result to rational functions having all their poles in the open unit disk. Some inequalities involving ¦f(z)¦, ¦ft′(z)¦ and ¦ft′(z)¦ are also proved in this paper. The last section contains an L2 inequality for the derivative of a rational function.

Certain extremal problems for polynomials

Extensions of two classical results about polynomials, one due to W. Markov and the other due to Duffin and Schaeffer, are obtained in this paper. An interesting result of S. Bernstein, which went unnoticed until it was rediscovered by P. Erdos, 34 years later, is also generalized. Our results are especially amenable to numerical calculations, and may, therefore, be of some practical importance.

Local Behavior of Entire Functions of Exponential Type

We consider entire functions of exponential type τ > 0, whose modulus is bounded by a constant M at the extrema of sin(τz), and which vanish at the origin. Extending a result of L. Hörmander, we show that if f is any such function, then ¦f(x)¦ ≤ M¦ sin(τx)¦ for all x ∈ (−π/(2τ),π/(2τ)), provided that f(x) = o(x) as x → ± ∞; furthermore, equality holds at any point x ∈ (−π/(2τ), 0) ∪ (0, π/(2τ)) if and only if f(z) ≡ eiγ sin(τz) for some γ ∈ R. This also generalizes a result due to R. P. Boas Jr. about trigonometric polynomials. Besides, we prove some other results for entire functions of order 1 and type τ > 0, one being an analogue of a result of M. Riesz about trigonometric polynomials whose zeros are all real and simple.

Functions of exponential type in a half-plane

Let f be of exponential type τ in the open upper half-plane and continuous in the closed upper half-plane. Assuming that | f(x)| ≤ M on the real axis, we find the sharp upper bound for | f′(z 0)| at any given point z 0 of the open upper half-plane. Our main tool is the classical Schwarz–Pick theorem.

Distribution of the zeros of a polynomial with prescribed lower and upper bounds for its modulus on a compact set

A few years ago, the second named author was asked if he knew the largest open simply connected region containing no zero of any polynomial of degree such that for all . This question is answered in Theorem 3. This required us to first consider a related problem for polynomials on the unit circle, whose solution is given in Theorem 1. The paper contains several other results which are all sharp.

An L 2 inequality for rational functions

We prove an L 2 inequality for rational functions having all their poles in the open unit disc and as a corollary obtain an inequality for functions belonging to the Hardy class H 2 of power series ∑a j z j with ∑|a j |2 < ∞.

Local behaviour of polynomials

jjIn this paper we study the local behaviour of a trigonometric polynomial t(θ):= Σ n v=-n a v e ivθ around any of its zeros in terms of its estimated values at an adequate number of freely chosen points in [0, 2π). The freedom in the choice of sample points makes our results particularly convenient for numerical calculations. Analogous results for polynomials of the form Σ n v=0 a v x v are also proved.