scholarly journals A Sharp Double Inequality between Harmonic and Identric Means

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
Yu-Ming Chu ◽  
Miao-Kun Wang ◽  
Zi-Kui Wang
Keyword(s):  
Double Inequality

We find the greatest valuepand the least valueqin(0,1/2)such that the double inequalityH(pa+(1-p)b,pb+(1-p)a)<I(a,b)<H(qa+(1-q)b,qb+(1-q)a)holds for alla,b>0witha≠b. Here,H(a,b), andI(a,b)denote the harmonic and identric means of two positive numbersaandb, respectively.

scholarly journals A refinement of a double inequality for the gamma function

2012 ◽  
Vol 80 (3-4) ◽  
pp. 333-342 ◽  
Author(s):  
JIAO-LIAN ZHAO ◽  
BAI-NI GUO ◽  
FENG QI
Keyword(s):  
Gamma Function ◽  
Double Inequality

scholarly journals Optimal Bounds for Seiffert Mean in terms of One-Parameter Means

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Hua-Nan Hu ◽  
Guo-Yan Tu ◽  
Yu-Ming Chu
Keyword(s):  
Double Inequality ◽  
Seiffert Mean ◽  
Optimal Bounds

The authors present the greatest valuer1and the least valuer2such that the double inequalityJr1(a, b)<T(a, b)<Jr2(a, b)holds for alla, b>0witha≠b, whereT(a, b)andJp(a, b)denote the Seiffert andpth one-parameter means of two positive numbersaandb, respectively.

scholarly journals Sharp inequalities for the harmonic numbers

2012 ◽  
Vol 28 (2) ◽  
pp. 223-229
Author(s):  
CHAO-PING CHEN ◽  
Keyword(s):  
Harmonic Number ◽  
Harmonic Numbers ◽  
Double Inequality

Let Hn be the nth harmonic number, and let γ be the Euler-Mascheroni constant. We prove that for all integers n ≥ 1, the double-inequality ... holds with the best possible constants ... We also establish inequality for the Euler-Mascheroni constant.

A sharp trigonometric double inequality

2021 ◽  
Vol 98 (1-2) ◽  
pp. 231-242
Author(s):  
Yongbeom Kim ◽  
Tuo Yeong Lee ◽  
Vengat S. ◽  
Hui Xiang Sim ◽  
Jay Kin Heng Tai
Keyword(s):  
Double Inequality

scholarly journals A Solution to Qi’s Conjecture on a Double Inequality for a Function Involving the Tri- and Tetra-Gamma Functions

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1098 ◽  
Author(s):  
Ladislav Matejíčka
Keyword(s):  
Open Problem ◽  
Complete Monotonicity ◽  
Double Inequality ◽  
Gamma Functions

In the paper, the author gives a solution to a conjecture on a double inequality for a function involving the tri- and tetra-gamma functions, which was first posed in Remark 6 of the paper “Complete monotonicity of a function involving the tri- and tetragamma functions” (2015) and repeated in the seventh open problem of the paper “On complete monotonicity for several classes of functions related to ratios of gamma functions” (2019).

scholarly journals Inequalities for Some Integrals Involving Modified Lommel Functions of the First Kind

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Robert E. Gaunt
Keyword(s):  
Hypergeometric Function ◽  
Generalized Hypergeometric Function ◽  
Double Inequality ◽  
Lommel Functions ◽  
Modified Struve Function ◽  
Lommel Function

AbstractIn this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $$t_{\mu ,\nu }(x)$$tμ,ν(x). In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function $$t_{\mu ,\nu }(x)$$tμ,ν(x), for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.

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.

scholarly journals Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Yu-Ming Chu ◽  
Shou-Wei Hou
Keyword(s):  
Sharp Bounds ◽  
Double Inequality ◽  
Seiffert Mean ◽  
Contraharmonic Mean

We find the greatest valueαand the least valueβin(1/2,1)such that the double inequalityC(αa+(1-α)b,αb+(1-α)a)<T(a,b)<Cβa+1-βb,βb+(1-βa)holds for alla,b>0witha≠b. Here,T(a,b)=(a-b)/[2 arctan((a-b)/(a+b))]andCa,b=(a2+b2)/(a+b)are the Seiffert and contraharmonic means ofaandb, respectively.

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