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Quadratic transformation inequalities for Gaussian hypergeometric function

In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grötzsch ring function.

Convexity and concavity of the complete elliptic integrals with respect to Lehmer mean

In this paper, the authors present necessary and sufficient conditions for the complete elliptic integrals of the first and second kind to be convex or concave with respect to the Lehmer mean. MSC: 33C05, 26E60.

Optimal bounds for two Sándor-type means in terms of power means

In the article, we prove that the double inequalities M α ( a , b ) < S Q A ( a , b ) < M β ( a , b ) and M λ ( a , b ) < S A Q ( a , b ) < M μ ( a , b ) hold for all a , b > 0 with a ≠ b if and only if α ≤ log 2 / [ 1 + log 2 − 2 log ( 1 + 2 ) ] = 1.5517 …  , β ≥ 5 / 3 , λ ≤ 4 log 2 / [ 4 + 2 log 2 − π ] = 1.2351 … and μ ≥ 4 / 3 , where S Q A ( a , b ) = A ( a , b ) e Q ( a , b...

Optimal evaluation of a Toader-type mean by power mean

In this paper, we present the best possible parameters p , q ∈ R such that the double inequality M p ( a , b ) < T [ A ( a , b ) , Q ( a , b ) ] < M q ( a , b ) holds for all a , b > 0 with a ≠ b , and we get sharp bounds for the complete elliptic integral E ( t ) = ∫ 0 π / 2 ( 1 − t 2 sin 2 θ ) 1 / 2 d θ of the second kind on the interval ( 0 , 2 / 2 ) , where T ( a , b ) = 2...

Monotonicity properties of a function involving the psi function with applications

In this paper, we present the best possible parameter a ∈ ( 1 / 15 , ∞ ) such that the functions ψ ′ ( x + 1 ) − L x ( x , a ) and ψ ″ ( x + 1 ) − L x x ( x , a ) are strictly increasing or decreasing with respect to x ∈ ( 0 , ∞ ) , where L ( x , a ) = 1 90 a 2 + 2 log ( x 2 + x + 3 a + 1 3 ) + 45 a 2 90 a 2 + 2 log ( x 2 + x + 15 a − 1 45 a ) and ψ ( x ) is the classical psi...

Sharp bounds for the Neuman mean in terms of the quadratic and second Seiffert means

In this paper, we prove that α=0 and β=3π−4log(2+3)(2π−4)log(2+3)=0.29758⋯ are the best possible constants such that the double inequality αQ(a,b)+(1−α)T(a,b)<SCA(a,b)<βQ(a,b)+(1−β)T(a,b) holds for all a,b>0 with a≠b, where Q(a,b)=(a2+b2)/2, SCA(a,b)=(a−b)3(a2+b2)+2ab2(a+b)sinh−1((a−b)3(a2+b2)+2ab(a+b)2) and T(a,b)=(a−b)/[2arctan((a−b)/(a+b))] are the quadratic, Neuman and second...

Optimal Bounds for Neuman-Sándor Mean in Terms of the Convex Combinations of Harmonic, Geometric, Quadratic, and Contraharmonic Means

, Hangzhou 310018, China Received 15 October 2012; Accepted 5 December 2012 Academic Editor: Julian López-Gómez Copyright © 2012 Tie-Hong Zhao et al. This is an open access article distributed under the

Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means

313000, China 4School of Automation, Southeast University, Nanjing 210096, China Received 19 January 2013; Accepted 1 February 2013 Academic Editor: Khalil Ezzinbi Copyright © 2013 Tie-Hong Zhao et al

Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means

313000, China 4School of Automation, Southeast University, Nanjing 210096, China Received 19 January 2013; Accepted 1 February 2013 Academic Editor: Khalil Ezzinbi Copyright © 2013 Tie-Hong Zhao et al

A Class of Logarithmically Completely Monotonic Functions Associated with a Gamma Function

We show that the function is strictly logarithmically completely monotonic on if and only if and is strictly logarithmically completely monotonic on if and only if .

Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions

Using the series-expansion of digamma functions and other techniques, some monotonicity and logarithmical concavity involving the ratio of gamma function are obtained, which is to give a partially affirmative answer to an open problem posed by B.-N.Guo and F.Qi. Several inequalities for the geometric means of natural numbers are established.