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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

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2017

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tom 71

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nr 1

EN

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.

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On new multiple extension of Hilbert’s integral inequality

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Sulaiman W. T.

Commentationes Mathematicae

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2006

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tom 46

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nr 2

EN

This paper gives a new multiple extension of Hilbert’s integral inequality generalizing many known results.

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On new multiple extension of Hilbert’s integral inequality (2)

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Sulaiman W. T.

Commentationes Mathematicae

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2007

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tom 47

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nr 1

EN

Via Laplace transformation some new generalization of Hardy-Hilbert’s integral inequality is given.

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Approximation by the Bézier variant of the MKZ-Kantorovich operators in the case α < 1

88%

Zeng X.-M., Gupta V.

Open Mathematics

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2009

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tom 7

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nr 3

550-557

EN

The pointwise approximation properties of the Bézier variant of the MKZ-Kantorovich operators $$ \hat M_{n,\alpha } (f,x) $$ for α ≥ 1 have been studied in [Comput. Math. Appl., 39 (2000), 1-13]. The aim of this paper is to deal with the pointwise approximation of the operators $$ \hat M_{n,\alpha } (f,x) $$ for the other case 0 < α < 1. By means of some new techniques and new inequalities we establish an estimate formula on the rate of convergence of the operators $$ \hat M_{n,\alpha } (f,x) $$ for the case 0 < α < 1. In the end we propose the q-analogue of MKZK operators.

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Composition of Arithmetical functions with generalization of perfect and related numbers

75%

Shukla D. P., Yadav S.

Commentationes Mathematicae

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2012

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tom 52

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nr 2

EN

In this paper we have studied the deficient and abundent numbers connected with the composition of $\varphi$, $\varphi^*$, $\sigma$, $\sigma^*$ and $\psi$ arithmetical functions, where $\varphi$ is Euler totient, $\varphi^*$ is unitary totient, $\sigma$ is sum of divisor, $\sigma^*$ is unitary sum of divisor and $\psi$ is Dedekind's function. In 1988, J. Sandor conjectured that $\psi(\varphi(m)) \geq m$, for all $m$, all odd $m$ and proved that this conjecture is equivalent to $\psi(\varphi(m)) \geq \frac{m}{2}$, we have studied this equivalent conjecture. Further, a necessary and sufficient conditions of primitivity for unitary r-deficient numbers and unitary totient r-deficient numbers have been obtained. We have discussed the generalization of perfect numbers for an arithmetical function $E_\alpha$.

Ograniczanie wyników

3 Commentationes Mathematicae

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

1 Open Mathematics

2 Sulaiman W. T.

1 Gupta V.

1 Hachani M. A.

1 Shukla D. P.

1 Yadav S.

1 Zeng X.-M.

1 2017

1 2012

1 2009

1 2007

1 2006

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

On new multiple extension of Hilbert’s integral inequality

On new multiple extension of Hilbert’s integral inequality (2)

Approximation by the Bézier variant of the MKZ-Kantorovich operators in the case α < 1

Composition of Arithmetical functions with generalization of perfect and related numbers