A lower bound on the sinc function and its application.

Yue Hu,Cristinel Mortici

doi:10.1155/2014/571218

Abstract

A lower bound on the sinc function is given. Application for the sequence {b n}n=1 ∞ which related to Carleman inequality is given as well.

Highlights

The sinc function is defined to be{ sin (x) sinc (x) = {{1 x x ≠ 0, x = 0. (1)This function plays a key role in many areas of mathematics and its applications [1–6].The following result that provides a lower bound for the sinc is well known as Jordan inequality [7]: sinc (x) ⩾ 2 π, x ∈ [0, π 2 ]
To the best of the authors’ knowledge, few results have been obtained on fractional lower bound for the sinc function
It is the first aim of the present paper to establish the following fractional lower bound for the sinc function

Summary

Introduction

{ sin (x) sinc (x) = {{1 x x ≠ 0, x = 0 This function plays a key role in many areas of mathematics and its applications [1–6]. The following result that provides a lower bound for the sinc is well known as Jordan inequality [7]: sinc (x). We noticed that the lower bound in (3) is the fractional function. To the best of the authors’ knowledge, few results have been obtained on fractional lower bound for the sinc function. It is the first aim of the present paper to establish the following fractional lower bound for the sinc function. In [37], Yang proved that for any positive integer m, the following Carleman type inequality holds:.

The Proof of Theorem 1

The Proof of Theorem 2