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Three families of two-parameter means constructed by trigonometric functions
In this paper, we establish three families of trigonometric functions with two parameters and prove their monotonicity and bivariate log-convexity. Based on them, three two-parameter families ofExpand
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Some properties of the divided difference of psi and polygamma functions
Abstract Let ψ n = ( − 1 ) n − 1 ψ ( n ) for n ≥ 0 , where ψ ( n ) stands for the psi and polygamma functions. For p , q ∈ R and ρ = min ⁡ ( p , q ) , let D [ x + p , x + q ; ψ n − 1 ] ≡ − ϕ n ( x )Expand
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Necessary and Sufficient Condition for Schur Convexity of the Two-Parameter Symmetric Homogeneous Means
An necessary and sufficient condition for Schur convexity of the two-parameter symmetric homogeneous means is given, which improves Witkowski's result. As an application, Schur convexity of theExpand
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On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind
Abstract In the article, we prove that the double inequalities 1 + ( 6 p − 7 ) r ′ p + ( 5 p − 6 ) r ′ π tanh − 1 ⁡ ( r ) 2 r K ( r ) 1 + ( 6 q − 7 ) r ′ q + ( 5 q − 6 ) r ′ π tanh − 1 ⁡ ( r ) 2 r ,Expand
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Sharp Gautschi inequality for parameter 0
In the article, we present the best possible parameters a,b on the interval (0,∞) such that the Gautschi double inequality [(xp +a) − x]/a < ex ∫ ∞ x e−t p dt < [(xp +b) − x]/b holds for all x > 0Expand
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On rational bounds for the gamma function
AbstractIn the article, we prove that the double inequality x2+p0x+p0<Γ(x+1)<x2+9/5x+9/5$$ \frac{x^{2}+p_{0}}{x+p_{0}}< \Gamma(x+1)< \frac{x^{2}+9/5}{x+9/5} $$ holds for all x∈(0,1)$x\in(0, 1)$,Expand
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On approximating the error function
*Correspondence: chuyuming2005@126.com 1School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of theExpand
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Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function
Abstract In this paper, we establish some lower and upper bounds for certain hyperbolic functions in terms of partial sums of their Taylor series. These allow us to present two completely monotonicExpand
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Monotonicity and sharp inequalities related to gamma function
In this paper, we investigate the monotonicity pattern of the function x → lnΓ(x+1) ln (x2 +a)− ln (x+a) on (0,1) for a 1 and resolve an open problem. From which we prove that the double inequality (Expand
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