Barkat Ali Bhayo, Li Yin
In this paper we prove the conjecture posed by Klén et al. in \cite{kvz}, and give optimal inequalities for generalized trigonometric and hyperbolic functions.
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In this paper we present an overview of results for discrete trigonometric and hyperbolic systems. These systems are discrete analogues of trigonometric and hyperbolic linear Hamiltonian systems. We show results which can be viewed as discrete n-dimensional extensions of scalar continuous trigonometric and hyperbolic formulas
We prove various new trigonometric and hyperbolic inequalities of Jordan, Wilker, Huygens or Cusa-Huygens type. Connections with bivariate means, as well as monotonicity and convexity properties are pointed out, too.
In this paper, we present series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions. By using the obtained results, we establish some inequalities for trigonometric and hyperbolic functions.
We study the power mean inequality of the generalized trigonometric and hyperbolic functions with two parameters. The generalized $p$-trigonometric and $(p, q)$-trigonometric functions were introduced by P. Lindqvist and S. Takeuchi, respectively.
The aim of this paper is to prove new trigonometric and hyperbolic inequalities, which constitute among others refinements or analogs of famous Cusa-Huygens, Wu-Srivastava, and related inequalities. In most cases, the obtained results are sharp.
The generalized trigonometric functions occur as an eigenfunction of the Dirichlet problem for the one-dimensional $p-$Laplacian. The generalized hyperbolic functions are defined similarly. Some classical inequalities for trigonometric and hyperbolic functions, such as Mitrinović-Adamović inequality, Lazarević's inequality, Huygens-type inequalities, Wilker-type inequalities, and Cuza-Huygens-type inequalities, are generalized to the case of generalized functions.
Various miscellaneous functional inequalities are deduced for the so-called generalized inverse trigonometric and hyperbolic functions. For instance, functional inequalities for sums, difference and quotient of generalized inverse trigonometric and hyperbolic functions are given, as well as some Grünbaum inequalities with the aid of the classical Bernoulli inequality. Moreover, by means of certain already derived bounds, bilateral bounding inequalities are obtained for the generalized hypergeometric ${}_3F_2$ Clausen function.
This article is the collection of the six research papers, recently written by the authors. In these papers authors refine the inequalities of trigonometric and hyperbolic functions such as Adamovic-Mitrinovic inequality, Cusa-Huygens inequality, Lazarevic inequality, Huygens inequality, Jordan's inequality, Carlson's inequality, Wilker's inequality, Redheffer's inequality, Wilker-Anglesio inequality, Becker-Stark inequality, Kober's inequality, Shafer's inequality and Shafer-Fink's inequality. The relation between trigonometric and hyperbolic functions has been built too. In the last paper, t…
Generalized trigonometric functions and generalized hyperbolic functions can be converted to each other by the duality formulas previously discovered by the authors. In this paper, we apply the duality formulas to prove dual pairs of Wilker-type inequalities, Huygens-type inequalities, and (relaxed) Cusa-Huygens-type inequalities for the generalized functions. In addition, multiple- and double-angle formulas not previously obtained are also given.
The well known table of Gradshteyn and Ryzhik contains indefinite and definite integrals of both elementary and special functions. We give proofs of several entries containing integrands with some combination of hyperbolic and trigonometric functions. In fact, we occasionally present an extension of such entries or else give alternative evaluations. We develop connections with special cases of special functions including the Hurwitz zeta function. Before concluding we mention new integrals coming from the investigation of certain elliptic functions.
We propose two new definitions of the exponential function on time scales. The first definition is based on the Cayley transformation while the second one is a natural extension of exact discretizations. Our eponential functions map the imaginary axis into the unit circle. Therefore, it is possible to define hyperbolic and trigonometric functions on time scales in a standard way. The resulting functions preserve most of the qualitative properties of the corresponding continuous functions. In particular, Pythagorean trigonometric identities hold exactly on any time scale. Dynamic equations sati…
We prove that the inequalities $\sin_{p,q}(\sqrt{rs})\geq \sqrt{\sin_{p,q}(r)\sin_{p,q}(s)}$ and $\sinh_{p,q}(\sqrt{r^*s^*}) \leq \sqrt{\sinh_{p,q}(r^*)\sinh_{p,q}(s^*)}$ hold for all $p,q\in(1,\infty)$, $r,s\in(0,\int_{0}^{1}(1-t^q)^{-1/p}dt)$ and $r^*,s^*\in(0,\int_{0}^{\infty}(1+t^q)^{-1/p}dt)$, where $\sin_{p,q}$ and $\sinh_{p,q}$ are the generalized trigonometric and hyperbolic sine functions, respectively. As a consequence of the results, we prove a conjecture due to Bhayo and Vuorinen [J. Approx. Theory, 164(2012)].
Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic) and of type E_6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level de…
We prove that for $p\in (0,1]$, the double inequality% \begin{equation*} \tfrac{1}{3p^{2}}\cos px+1-\tfrac{1}{3p^{2}}<\frac{\sin x}{x}<\tfrac{1}{% 3q^{2}}\cos qx+1-\tfrac{1}{3q^{2}} \end{equation*}% holds for $x\in (0,π/2)$ if and only if $0<p\leq p_{0}\approx 0.77086$ and $\sqrt{15}/5=p_{1}\leq q\leq 1$. While its hyperbolic version holds for $% x>0$ if and only if $0<p\leq p_{1}=\sqrt{15}/5$ and $q\geq 1$. As applications, some more accurate estimates for certain mathematical constants are derived, and some new and sharp inequalities for Schwab-Borchardt mean\ and logarithmic means are estab…
Principal Branches of Inverse Trigonometric and Inverse Hyperbolic Functions
本文为三个关键平方根函数及反三角、反双曲函数开发主分支，利用复分析将定义转化为具体表达式，旨在扩大 DLMF 结果适用范围并降低用户的复变量负担。
In this paper we prove two conjectures stated by Chao-Ping Chen in [Int. Trans. Spec. Funct. 23:12 (2012), 865--873], using a method for proving inequalities of mixed trigonometric polynomial functions.
In this paper, the versions of trigonometric functions of certain known inequalities for hyperbolic ones are proved, and then corresponding inequalities for means are presented.
It is shown that generalized trigonometric functions and generalized hyperbolic functions can be transformed from each other. As an application of this transformation, a number of properties for one immediately lead to the corresponding properties for the other. In this way, Mitrinović-Adamović-type inequalities, multiple-angle formulas, and double-angle formulas for both can be produced.
This paper considers various integrals where the integrand includes the log gamma function (or its derivative, the digamma function) multiplied by a trigonometric or hyperbolic function. Some apparently new integrals and series are evaluated.
In this paper, authors generalize logarithmic mean $L$, Neuman-Sándor $M$, two Seiffert means $P$ and $T$ as an application of generalized trigonometric and hyperbolic functions. Moreover, several two-sided inequalities involving these generalized means are established.
A look at generalized trigonometric functions as functions of their two parameters and further new properties
本文研究含两个参数的广义三角函数与双曲函数关于参数的单调性和凸性，给出部分函数的（对数）凸性/凹性结果及凸性充要条件，并发现新的超几何表示和四个显式积分求值。
The sums of three trigonometric series with logarithmic coefficients are derived by extending an approach first utilized by Lerch. By applying Frullani's theorem to two of these series, two non-trivial integrals involving hyperbolic functions are evaluated in terms of the digamma function.
In this work, we generalize the D'Aurizio-Sándor inequalities (\cite{D'Aurizio,Sandor}) using an elementary approach. In particular, our approach provides an alternative proof of the D'Aurizio-Sándor inequalities. Moreover, as an immediate consequence of the generalized D'Aurizio-Sándor inequalities, we establish the D'Aurizio-Sándor-type inequalities for hyperbolic functions.
In this paper we study the inverse of the eigenfunction $\sin_p$ of the one-dimensional $p$-Laplace operator and its dependence on the parameter $p$, and we present a Turán type inequality for this function. Similar inequalities are given also for other generalized inverse trigonometric and hyperbolic functions. In particular, we deduce a Turán type inequality for a series considered by Ramanujan, involving the digamma function.
The generalized $p$-trigonometric and ($p,q$)-trigonometric functions were introduced by P. Lindqvist and S. Takeuchi, respectively. We prove some inequalities and present a few conjectures for the ($p,q$)-functions.
We study the convexity properties of the generalized trigonometric functions considered as functions of parameter. We show that $p\to\sin_p(y)$ and $p\to\cos_p(y)$ are log-concave on the appropriate intervals while $p\to\tan_p(y)$ is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle the major part of a conjecture put forward in a recent paper by Baricz, Bhayo and Vuorinen.
We develop a new point of view to introduce families of functions, which can be identified as generalization of the ordinary trigonometric or hyperbolic functions. They are defined using a procedure based on umbral methods, inspired to the Bessel Calculus of Bochner, Cholewinsky and Haimo. We propose further extensions of the method and of the relevant concepts as well and obtain new families of integral transform allowing the framing of the previous concepts within the context of generalized Borel tranforms.
Applying the theory of elliptic functions we establish two Jacobi theta function identities. From these identities we confirm two q-trigonometric identities conjectured by Gosper. As an application, we give a new and simple proof of a Pi_{q}-identity of Gosper.
This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as (sin x)/x and x/(sinh x) are proved.
In math.QA/0309252, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.
Our aim is to extend some trigonometric inequalities to Bessel functions. Moreover, we extend the hyperbolic analogue of these trigonometric inequalities. As an application of these results we present a generalization of Cusa-type inequality to modified Bessel function. Our main motivation to write this paper is a recent publication of Chen and Sándor, which we wish to complement.
We prove that for all p>1/2 there exists a constant $γ_p>0$ such that, for any symmetric measurable set of positive measure $E\subset \TT$ and for any $γ<γ_p$, there is an idempotent trigonometrical polynomial f satisfying $\int_E |f|^p > γ\int_{\TT} |f|^p$. This disproves a conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence of $γ_p>0$ for p>1 and conjectured that it does not exists for p=1. Furthermore, we prove that one can take $γ_p=1$ when p>1 is not an even integer, and that polynomials f can be chosen with arbitrarily large gaps when $p\neq 2$. This shows str…
In the paper, the authors prove that the generalized sine function $\sin_{p,q}(x)$ and the generalized hyperbolic sine function $\sinh_{p,q}(x)$ are geometrically concave and geometrically convex, respectively. Consequently, the authors verify a conjecture posed in the paper "B. A. Bhayo and M. Vuorinen, On generalized trigonometric functions with two parameters, J. Approx. Theory 164 (2012), no.~10, 1415\nobreakdash--1426; Available online at \url{http://dx.doi.org/10.1016/j.jat.2012.06.003}".
In the article, some Huygens and Wilker type inequalities involving trigonometric and hyperbolic functions are refined and sharpened.
Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials
研究双曲交叉三角多项式一致范数的采样离散化，构造显式赋范集并证明随机采样达到最优指数，同时给出赋范集大小的下界，排除二次界。
Table in Gradshteyn and Ryzhik: Derivation of definite integrals of a Hyperbolic Function
本文利用围道积分和特殊函数的解析延拓，推导 Gradshteyn 与 Ryzhik 表中的双曲函数定积分及相关无穷和，并以对数函数乘三角函数的积分为例，得到 Catalan 常数和 π 等结果。
Remez-type inequalities for the hyperbolic cross polynomials
本文研究了具有超球面谐波的三角多项式的Remez型不等式，探讨了Remez与Nikolskii不等式之间的关系及其应用。
From Ingham to Nazarov's inequality: a survey on some trigonometric inequalities
本文综述了调和与非调和三角多项式在 L^1 和 L^2 范数下的若干不等式，重点介绍 Ingham 不等式、Littlewood 猜想的证明及 Nazarov 的非调和对应结果，并补充了小区间上的定量估计与更强条件下的下界。
Trigonometric polynomials with frequencies in the set of squares
本文研究频率限制在平方数集合中的三角多项式，证明了当频率范围长度为 N^{γ0-ε} 时其 L4 范数可由 L2 范数控制，推进了 Cilleruelo-Cordoba 猜想，并揭示了该猜想与 Ruzsa 因数分布猜想之间的联系。
Norm bounds on Fourier series with polynomial spectra and constrained coefficients
本文针对谱为多项式的三角多项式，通过系数约束建立了L⁴范数上界，补充了前人成果并解决了特定猜想，确定了对应范数增长的精确阶。
Generalised definitions of exponential, trigonometric sine and cosine and hyperbolic sine and cosine functions are given. In the lowest order, these functions correspond to ordinary exponential, trigonometric sine etc. Some of the properties of the generalised functions are discussed. Importance of these functions and their possible applications are also considered.
Bicycle tracks with hyperbolic monodromy -- results and conjectures
本文研究闭平面曲线自行车单值性为双曲的条件，利用“双曲展开”方法给出新的充要条件，并基于计算机实验对严格凸闭曲线提出两个猜想。
We prove a multivariable elliptic extension of Jackson's summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.
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