Searcharxiv⌕ Search

arXiv subjects

Man Kam Kwong

Publications and source records attributed to Man Kam Kwong.

15 recordsLinked to original sources
arXiv2023

Inequalities for trigonometric sums

We present several new inequalities for trigonometric sums. Among others, we show that the inequality $$ \sum_{k=1}^n (n-k+1)(n-k+2)k\sin(kx) > \frac{2}{9} \sin(x) \bigl( 1+2\cos(x) \bigr)^2 $$ holds for all $n\geq 1$ and $x\in (0, 2π/3)$. The constant factor $2/9$ is sharp. This refines the classical Szegö-Schweitzer inequality which states that the sine sum is positive for all $n\geq 1$ and $x\in (0,2 π/3)$. Moreover, as an application of one of our results, we obtain a two-parameter class of absolutely monotonic functions.

math.CA↗
arXiv2020

Inequalities for Taylor series involving the divisor function

Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved that $$ H(q) := T(q)- \frac{\log(1-q)}{\log(q)} $$ is strictly increasing in $ (0,1) $ while $$ F(q) := \frac{1-q}{q} \,H(q) $$ is strictly decreasing in $ (0,1) $. These results are then applied to obtain various inequalities, one of which states that the double-inequality $$ α\,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)} < T(q)< β\,\frac{q}{1-q}+\frac{\log(1-q)}{\log(q)}, \quad 0<q<1, $$ holds with the best possible constant factors $α=γ$ and $β=1$. Here, $γ$ denotes Euler's constant. This refines a result of Salem, who proved the inequalities with $α=1/2$ and $β=1$.

math.NT↗
arXiv2019

Sharp Bounds for the Arc Lemniscate Sine Function

The arc lemniscate sine function is given by $$ \mbox{arcsl}(x)=\int_0^x \frac{1}{\sqrt{1-t^4}}dt. $$ In 2017, Mahmoud and Agarwal presented bounds for $\mbox{arcsl}$ in terms of the Lerch zeta function $$ Φ(z,s,a)=\sum_{k=0}^\infty \frac {z^k}{(k+a)^s}. $$ They proved $$ \frac{1}{8} \, x \, Φ(x^4, 3/2, 1/4) < \mbox{arcsl}(x)< \frac{1}{4} \, x \, Φ(x^4,3/2,1/4)\qquad{(0<x<1)}. $$ We %use the monotone form of l'Hopital's rule to show that the factor $1/4$ can be replaced by $\mbox{arcsl}(1)/Φ(1,3/2,1/4)=0.12836...$. This constant is best possible.

math.CA↗
arXiv2017

A New Family of Nonnegative Sine Polynomials

We present a new family of sine polynomials that are nonnegative for all $x$ in $[0,π]$. We also characterize all nonnegative sine polynomials of degree 3 and all nonnegative cosine polynomials of degree 2. In the latest version, typos in (1.4) and (1.6) are corrected (with >= replaced by <=).

math.CA↗
arXiv2017

Technical Details of the Proof of the Sine Inequality \\[1.2ex] {\normalsize $\displaystyle \sum_{k=1}^{n-1}\left( \frac{n}{k} - \frac{k}{n} \right) ^β\sin(kx) \geq 0$

In a recent study, H. Alzer and the author showed that the sine polynomial $$ \sum_{k=1}^{n-1} \left( \frac{n}{k} - \frac{k}{n} \right) ^β\,\sin(kx) > 0 $$ is nonnegative for $ x\in[0,π] $, $ n\geq 2, \, β\geq β_1 := \frac{\log(2)}{\log(16/5)} . $ This result, among others, will be presented in a forthcoming article. The proof relies on quite a number of technical Lemmas and inequalities. We have decided to delegate all the tedious details of the proofs of these Lemmas in a separate article, namely, the current one. Some of the proofs require brute-force numerical computation, performed with the help of the computer software MAPLE. A few of the Lemmas included here are of independent interest.

math.CA↗
arXiv2016

On a Sine Polynomial of Turan

In 1935, P. Turán proved that $$ S_{n,a}(x)= \sum_{j=1}^n{n+a-j\choose n-j} \sin(jx)>0 \quad{(n,a\in\mathbf{N}; 0<x<π).} $$ We present various related inequalities. Among others, we show that the refinements $$ S_{2n-1,a}(x)\geq \sin(x) \quad\mbox{and} \quad{S_{2n,a}(x)\geq 2\sin(x)(1+\cos(x))} $$ are valid for all integers $n\geq 1$ and real numbers $a\geq 1$ and $x\in(0,π)$. Moreover, we apply our theorems on sine sums to obtain inequalities for the Chebyshev polynomials of the second kind.

math.CA↗
arXiv2016

Nonnegative Trigonometric Polynomials, Sturms Theorem, and Symbolic Computation

In this paper, we explain a procedure based on a classical result of Sturm that can be used to determine rigorously whether a given trigonometric polynomial is nonnegative in a certain interval or not. Many examples are given. This technique has been employed by the author in several recent works. The procedure often involves tedious computations that are time-consuming and error-prone. Fortunately, symbolic computation software is available to automate the procedure. In this paper, we give the details of its implementation in MAPLE 13. Some who are strongly attached to a more traditional theoretical research framework may find such details boring or even consider computer-assisted proofs suspicious. However, we emphasize again that the procedure is completely mathematically rigorous.

math.CA↗
arXiv2015

Nonnegative Trigonometric Polynomials and Sturms Theorem

In an earlier article [3], we presented an algorithm that can be used to rigorously check whether a specific cosine or sine polynomial is nonnegative in a given interval or not. The algorithm proves to be an indispensable tool in establishing some recent results on nonnegative trigonometric polynomials. See, for example, [2], [4] and [5]. It continues to play an essential role in several ongoing projects. The algorithm, however, cannot handle general trigonometric polynomials that involve both cosine and sine terms. Some ad hoc methods to deal with such polynomials have been suggested in [3], but none are, in general, satisfactory. This note supplements [3] by presenting an algorithm applicable to all general trigonometric polynomials. It is based on the classical Sturm Theorem, just like the earlier algorithm. A couple of the references in [3] are also updated.

math.CA↗
arXiv2015

Improved Vietoris Sine Inequalities for Non-Monotone, Non-Decaying Coefficients

Recently the author established an improvement of the classical Vietoris sine inequality to include sine polynomials with non-monotone coefficients. In this paper two further improvements are presented admitting sine polynomials with non-monotone and non-decaying coefficients. The extremal sums of the two results have the coefficient sequences {2a, a, 4/3, 1, 6/5, 1, 8/7, 1, ...}, where a = 0.78265..., and {3, 3/2, 7/3, 7/4, 11/5, 11/6, ...}.

math.CA↗
arXiv2015

On Hopital-style rules for monotonicity and oscillation

We point out the connection of the so-called Hôpital-style rules for monotonicity and oscillation to some well-known properties of concave/convex functions. From this standpoint, we are able to generalize the rules under no differentiability requirements and greatly extend their usability. The improved rules can handle situations in which the functions involved have non-zero initial values and when the derived functions are not necessarily monotone. This perspective is not new; it can be dated back to Hardy, Littlewood and Polya.

math.CA↗
arXiv2014

Periodic Solutions of 2D Isothermal Euler-Poisson Equations with Possible Applications to Spiral and Disk-like Galaxies

Compressible Euler-Poisson equations are the standard self-gravitating models for stellar dynamics in classical astrophysics. In this article, we construct periodic solutions to the isothermal ($γ=1$) Euler-Poisson equations in $R^{2}$ with possible applications to the formation of plate, spiral galaxies and the evolution of gas-rich, disk-like galaxies. The results complement Yuen's solutions without rotation (M.W. Yuen, Analytical Blowup Solutions to the 2-dimensional Isothermal Euler-Poisson Equations of Gaseous Stars, J. Math. Anal. Appl. 341(2008), 445--456.). Here, the periodic rotation prevents the blowup phenomena that occur in solutions without rotation. Based on our results, the corresponding $3$D rotational results for Goldreich and Weber's solutions are conjectured.

math-ph↗
arXiv2014

Sturm Theorem and a Refinement of Vietoris Inequality for Cosine Polynomials

In a recent work, the authors established a refinement of the well-known 1958 result of Vietoris on nonnegative cosine polynomials. In four places of the proof, use was made of the classical Sturm Theorem on determining the number of real roots of an algebraic polynomials in a given interval. Although absolutely rigorous, the Sturm procedure involves lengthy technical computations carried out with the help of the software MAPLE 13. This article provides such details which were omitted in the article presenting the recent work.

math.CA↗