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Entire functions of exponential type not vanishing in the half-plane \(\Im z > k\), where \(k > 0\)

100%

Hachani M. A.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2017

tom 71

nr 1

Let \(P (z)\) be a polynomial of degree \(n\) having no zeros in \(|z| < k\), \(k \leq 1\), and let \(Q (z) := z^n \overline{P (1/{\overline {z}})}\). It was shown by Govil that if \(\max_{|z| = 1} |P^\prime (z)|\) and \(\max_{|z| = 1} |Q^\prime (z)|\) are attained at the same point of the unit circle \(|z| = 1\), then \[\max_{|z| = 1} |P'(z)| \leq \frac{n}{1 + k^n} \max_{|z| = 1} |P(z)|.\]The main result of the present article is a generalization of Govil's polynomial inequality to a class of entire functions of exponential type.

On the convergence of certain integrals

100%

Hachani M. A.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2019

tom 73

nr 1

Let \(M(r) := \max_{|z|=r} |f(z)|\), where \(f(z)\) is an entire function. Also let \(\alpha> 0\) and \(\beta>1\). We discuss the behavior of the integrand \(M(r)e^{-\alpha(log r)^\beta}\) as \(r \to \infty\) if \(\int_1^\infty M(r)e^{-\alpha(log r)^\beta}dr\) is convergent.

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2 Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2 Hachani M. A.

1 2019

1 2017

Wyniki wyszukiwania

Entire functions of exponential type not vanishing in the half-plane \(\Im z > k\), where \(k > 0\)

On the convergence of certain integrals