Tie-Hong Zhao

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

We present the best possible lower and upper bounds for the Neuman-Sandor mean in terms of the convex combinations of either the harmonic and quadratic means or the geometric and quadratic means or the harmonic and contraharmonic means.

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.

Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

monotonic on � 0, ∞� if � α, β� ∈{ � α, β� :1 / √ α ≤ β ≤ 1 ,α / 1 }∪{ � α, β� :0 <β ≤ 1 ,ϕ 1� α, β� ≥ 0 ,ϕ 2� α, β� ≥ 0} andfα,β� x�� −1 is strictly logarithmically completely monotonic on � 0, ∞� if � α, β� ∈ {� α, β� :0 <α ≤ 1/2,0 <β ≤ 1 }∪{ � α, β� :1 ≤ β ≤ 1/ √ α ≤ √ 2 ,α / 1 }∪{ � α, β� :1 /2 ≤ α< 1 ,β ≥ 1/� 1−α� } ,w hereϕ1� α, β ��� α 2 � α−1� β 2 �� 2α 2 −3α� 1� β−α and ϕ2� α, β ��� α−1� β 2 �� 2α 2 −5α� 2� β−1.

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

We prove that the double inequalities hold for all with if and only if , , , and , where , , , and are the identric, Neuman-Sandor, quadratic, and contraharmonic means of and , respectively.

Generators for the Euclidean Picard modular groups

The goal of this article is to show that five explicitly given transformations, a rotation, two screw Heisenberg rotations, a vertical translation and an involution generate the Euclidean Picard modular groups with coefficient in the Euclidean ring of integers of a quadratic imaginary number field. We also obtain the relations of the isotropy subgroup by analysis of the combinatorics of the fundamental domain in Heisenberg group.

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 .

Sharp bounds for neuman-sándor mean in terms of the convex combination of quadratic and first Seiffert means

Abstract In this article, we prove that the double inequality α P ( a , b ) + 1 ( 1 − α ) Q ( a , b ) M ( a , b ) β P ( a , b ) + ( 1 − β ) Q ( a , b ) holds for any a,b > 0 with a ≠ b if and only if α ≥ 1/2 and β ≤ [ π ( 2 log ( 1 + 2 ) − 1 ) ] / [ ( 2 π − 2 ) log ( 1 + 2 ) ] = 03595… , where M(a,b), Q(a,b), and P(a,b) are the Neuman-Sandor, quadratic, and first Seiffert means of a and b, respectively.

A minimal volume arithmetic cusped complex hyperbolic orbifold

The sister of Eisenstein–Picard modular group is described explicitly in [ 10 ], whose quotient is a noncompact arithmetic complex hyperbolic 2-orbifold of minimal volume (see [ 16 ]). We give a construction of a fundamental domain for this group. A presentation of that lattice can be obtained from that construction, which relates to one of Mostows lattices.

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.

Optimal bounds for arithmetic-geometric and Toader means in terms of generalized logarithmic mean

In this paper, we find the greatest values α1,α2