scholarly journals Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
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.

scholarly journals The Optimal Upper and Lower Power Mean Bounds for a Convex Combination of the Arithmetic and Logarithmic Means

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Wei-Feng Xia ◽  
Yu-Ming Chu ◽  
Gen-Di Wang
Keyword(s):  
Convex Combination ◽  
Arithmetic Mean ◽  
Power Mean ◽  
Logarithmic Mean ◽  
Positive Real ◽  
Lower Power ◽  
Double Inequality ◽  
Logarithmic Means

Forp∈ℝ, the power meanMp(a,b)of orderp, logarithmic meanL(a,b), and arithmetic meanA(a,b)of two positive real valuesaandbare defined byMp(a,b)=((ap+bp)/2)1/p, forp≠0andMp(a,b)=ab, forp=0,L(a,b)=(b-a)/(log⁡b-log⁡a), fora≠bandL(a,b)=a, fora=bandA(a,b)=(a+b)/2, respectively. In this paper, we answer the question: forα∈(0,1), what are the greatest valuepand the least valueq, such that the double inequalityMp(a,b)≤αA(a,b)+(1-α)L(a,b)≤Mq(a,b)holds for alla,b>0?

scholarly journals The chord-length distribution of a polyhedron

2020 ◽  
Vol 76 (4) ◽  
pp. 474-488
Author(s):  
Salvino Ciccariello
Keyword(s):  
Distribution Function ◽  
Length Distribution ◽  
Chord Length ◽  
Square Root ◽  
Trigonometric Functions ◽  
Chord Length Distribution ◽  
Analytic Expressions ◽  
Analytic Expression ◽  
Positive Integers ◽  
Inverse Trigonometric Functions

The chord-length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

scholarly journals A Sharp Double Inequality for Trigonometric Functions and Its Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
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.

The Ambiguity of the Inverse Secant

2018 ◽  
Vol 111 (6) ◽  
pp. 470-475
Author(s):  
Thomas J. Bannon
Keyword(s):  
Trigonometric Functions ◽  
Definition Of ◽  
Inverse Trigonometric Functions

Defining inverse trigonometric functions involves choosing ranges for the functions. The choices made for the inverse sine, cosine, tangent, and cotangent functions follow generally accepted conventions. However, different authors make different choices when defining y = arcsec x and y = arccsc x for negative x. I first discovered that the definitions of these functions were not a settled convention when I found an alternate definition in Schaum's (Ayers and Mendelson 2012) and Anton's (1995) books. The more commonly used definition is simpler and results in a function more easily evaluated and for that reason is preferable when introducing the inverse trigonometric functions in an algebra or precalculus course. As we shall see, though, the alternate definition of the inverse secant function has many advantages when we move on to calculus. Since we have a choice in our definitions, we should choose what makes the most sense in context.

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