A Sharp double inequality involving trigonometric functions and its applications

A Sharp Double Inequality for Trigonometric Functions and Its Applications

Abstract and Applied Analysis ◽

10.1155/2014/592085 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9 ◽

Cited By ~ 6

Author(s):

Zhen-Hang Yang ◽

Yu-Ming Chu ◽

Ying-Qing Song ◽

Yong-Min Li

Keyword(s):

Trigonometric Functions ◽

Double Inequality

We present the best possible parameterspandqsuch that the double inequality(2/3)cos2p(t/2)+1/31/p<sin t/t<(2/3)cos2q(t/2)+1/31/qholds for anyt∈(0,π/2). As applications, some new analytic inequalities are established.

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Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

Mathematical Problems in Engineering ◽

10.1155/2016/8901258 ◽

2016 ◽

Vol 2016 ◽

pp. 1-7 ◽

Cited By ~ 2

Author(s):

Wei-Mao Qian ◽

Yu-Ming Chu

Keyword(s):

Unique Solution ◽

Trigonometric Functions ◽

Logarithmic Mean ◽

Double Inequality ◽

Logarithmic Means ◽

Inverse Trigonometric Functions

We prove that the double inequalityLp(a,b)<U(a,b)<Lq(a,b)holds for alla,b>0witha≠bif and only ifp≤p0andq≥2and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, wherep0=0.5451⋯is the unique solution of the equation(p+1)1/p=2π/2on the interval(0,∞),U(a,b)=(a-b)/[2arctan⁡((a-b)/2ab)], andLp(a,b)=[(ap+1-bp+1)/((p+1)(a-b))]1/p (p≠-1,0),L-1(a,b)=(a-b)/(log⁡a-log⁡b)andL0(a,b)=(aa/bb)1/(a-b)/eare the Yang, andpth generalized logarithmic means ofaandb, respectively.

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Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions

Axioms ◽

10.3390/axioms10040308 ◽

2021 ◽

Vol 10 (4) ◽

pp. 308

Author(s):

Yogesh J. Bagul ◽

Ramkrishna M. Dhaigude ◽

Marko Kostić ◽

Christophe Chesneau

Keyword(s):

Exponential Type ◽

Bernoulli Numbers ◽

Cosine Function ◽

Sinc Function ◽

Series Expansions ◽

Trigonometric Functions ◽

Hyperbolic Functions ◽

Double Inequality ◽

Exponential Bounds ◽

Cosine Functions

Recent advances in mathematical inequalities suggest that bounds of polynomial-exponential-type are appropriate for evaluating key trigonometric functions. In this paper, we innovate in this sense by establishing new and sharp bounds of the form (1−αx2)eβx2 for the trigonometric sinc and cosine functions. Our main result for the sinc function is a double inequality holding on the interval (0, π), while our main result for the cosine function is a double inequality holding on the interval (0, π/2). Comparable sharp results for hyperbolic functions are also obtained. The proofs are based on series expansions, inequalities on the Bernoulli numbers, and the monotone form of the l’Hospital rule. Some comparable bounds of the literature are improved. Examples of application via integral techniques are given.

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A New Solution for Maxterm Problem in Trigonometric Functions by Simulated Annealing Algorithm

International Journal of Applied Evolutionary Computation ◽

10.4018/jaec.2013040102 ◽

2013 ◽

Vol 4 (2) ◽

pp. 20-28

Author(s):

Farhad Soleimanian Gharehchopogh ◽

Hadi Najafi ◽

Kourosh Farahkhah

Keyword(s):

Simulated Annealing ◽

Simulated Annealing Algorithm ◽

Trigonometric Functions ◽

Matlab Software ◽

Annealing Algorithm ◽

Do So

The present paper is an attempt to get total minimum of trigonometric Functions by Simulated Annealing. To do so the researchers ran Simulated Annealing. Sample trigonometric functions and showed the results through Matlab software. According the Simulated Annealing Solves the problem of getting stuck in a local Maxterm and one can always get the best result through the Algorithm.

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Analytical angular solutions for the atom–diatom interaction potential in a basis set of products of two spherical harmonics: two approaches

Journal of Mathematical Chemistry ◽

10.1007/s10910-021-01282-y ◽

2021 ◽

Author(s):

Mariusz Pawlak ◽

Marcin Stachowiak

Keyword(s):

Spherical Harmonics ◽

Interaction Potential ◽

Chem Phys ◽

Basis Set ◽

Matrix Elements ◽

Trigonometric Functions ◽

Variational Theory ◽

Molecular Collision ◽

The Matrix ◽

Analytical Expressions

AbstractWe present general analytical expressions for the matrix elements of the atom–diatom interaction potential, expanded in terms of Legendre polynomials, in a basis set of products of two spherical harmonics, especially significant to the recently developed adiabatic variational theory for cold molecular collision experiments [J. Chem. Phys. 143, 074114 (2015); J. Phys. Chem. A 121, 2194 (2017)]. We used two approaches in our studies. The first involves the evaluation of the integral containing trigonometric functions with arbitrary powers. The second approach is based on the theorem of addition of spherical harmonics.

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