# Search: authors:"Ming Chu"

63 papers found.
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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.

#### Sharp bounds for a special quasi-arithmetic mean in terms of arithmetic and geometric means with two parameters

In the article, we present the best possible parameters λ = λ ( p ) $\lambda=\lambda (p)$ and μ = μ ( p ) $\mu=\mu(p)$ on the interval [ 0 , 1 / 2 ] $[0, 1/2]$ such that the double inequality G p [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] A 1 − p ( a , b ) < E ( a , b ) < G p [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] A 1 − p ( a , b ) \begin{aligned}& G^{p}\bigl[\lambda a+(1...

#### On approximating the error function

In the article, we present the necessary and sufficient condition for the parameter p on the interval ( 7 / 5 , ∞ ) such that the function x → erf ( x ) / B p ( x ) is strictly increasing (decreasing) on ( 0 , ∞ ) , and find the best possible parameters p, q on the interval ( 7 / 5 , ∞ ) such that the double inequality B p ( x ) < erf ( x ) < B q ( x ) holds for all x > 0 , where...

#### Necessary and sufficient conditions for the two parameter generalized Wilker-type inequalities

In the article, we provide the necessary and sufficient conditions for the parameters α and β such that the generalized Wilker-type inequality 2 β α + 2 β ( sin x x ) α + α α + 2 β ( tan x x ) β − 1 > ( < ) 0 holds for all x ∈ ( 0 , π / 2 ) . MSC: 26D05, 33B10.

#### Monotonicity and absolute monotonicity for the two-parameter hyperbolic and trigonometric functions with applications

In this paper, we present the monotonicity and absolute monotonicity properties for the two-parameter hyperbolic and trigonometric functions. As applications, we find several complete monotonicity properties for the functions involving the gamma function and provide the bounds for the error function. MSC: 33B10, 33B15, 33B20, 26A48, 26D07.

#### Monotonicity of the incomplete gamma function with applications

In the article, we discuss the monotonicity properties of the function x → ( 1 − e − a x p ) 1 / p / ∫ 0 x e − t p d t for a , p > 0 with p ≠ 1 on ( 0 , ∞ ) and prove that the double inequality Γ ( 1 + 1 / p ) ( 1 − e − a x p ) 1 / p < ∫ 0 x e − t p d t < Γ ( 1 + 1 / p ) ( 1 − e − b x p ) 1 / p holds for all x > 0 if and only if a ≤ min { 1 , Γ − p ( 1 + 1 / p ) } and b ≥ max { 1...

#### Monotonicity and inequalities involving the incomplete gamma function

In the article, we deal with the monotonicity of the function x → [ ( x p + a ) 1 / p − x ] / I p ( x ) on the interval ( 0 , ∞ ) for p > 1 and a > 0 , and present the necessary and sufficient condition such that the double inequality [ ( x p + a ) 1 / p − x ] / a < I p ( x ) < [ ( x p + b ) 1 / p − x ] / b for all x > 0 and p > 1 , where I p ( x ) = e x p ∫ x ∞ e − t p d t is...

#### Monotonicity of the ratio for the complete elliptic integral and Stolarsky mean

In the article, we prove that the function r ↦ E ( r ) / S 9 / 2 − p , p ( 1 , r ′ ) is strictly increasing on ( 0 , 1 ) for p ≤ 7 / 4 and strictly decreasing on ( 0 , 1 ) for p ∈ [ 2 , 9 / 4 ] , where r ′ = 1 − r 2 , E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ( t ) d t is the complete elliptic integral of the second kind, and S p , q ( a , b ) = [ q ( a p − b p ) / ( p ( a q − b q...

#### On approximating the modified Bessel function of the first kind and Toader-Qi mean

In the article, we present several sharp bounds for the modified Bessel function of the first kind I 0 ( t ) = ∑ n = 0 ∞ t 2 n 2 2 n ( n ! ) 2 and the Toader-Qi mean T Q ( a , b ) = 2 π ∫ 0 π / 2 a cos 2 θ b sin 2 θ d θ for all t > 0 and a , b > 0 with a ≠ b . MSC: 33C10, 26E60.

#### Lazarević and Cusa type inequalities for hyperbolic functions with two parameters and their applications

In the article, we establish several Lazarević and Cusa type inequalities involving the hyperbolic sine and cosine functions with two parameters. As applications, we find some new bounds for certain bivariate means. MSC: 26D05, 26D07, 33B10, 26E60.

#### 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.

#### Improvements of the bounds for Ramanujan constant function

In the article, we establish several inequalities for the Ramanujan constant function R ( x ) = − 2 γ − ψ ( x ) − ψ ( 1 − x ) on the interval ( 0 , 1 / 2 ] , where ψ ( x ) is the classical psi function and γ = 0.577215 ⋯ is the Euler-Mascheroni constant. MSC: 33B15, 26D07.

#### Generalized Wilker-type inequalities with two parameters

In the article, we present certain p , q ∈ R such that the Wilker-type inequalities 2 q p + 2 q ( sin x x ) p + p p + 2 q ( tan x x ) q > ( < ) 1 and ( π 2 ) p ( sin x x ) p + [ 1 − ( π 2 ) p ] ( tan x x ) q > ( < ) 1 hold for all x ∈ ( 0 , π / 2 ) . MSC: 26D05, 33B10.

#### Inequalities for certain means in two arguments

In this paper, we present the sharp bounds of the ratios U ( a , b ) / L 4 ( a , b ) , P 2 ( a , b ) / U ( a , b ) , N S ( a , b ) / P 2 ( a , b ) and B ( a , b ) / N S ( a , b ) for all a , b > 0 with a ≠ b , where L 4 ( a , b ) = [ ( b 4 − a 4 ) / ( 4 ( log b − log a ) ) ] 1 / 4 , U ( a , b ) = ( b − a ) / [ 2 arctan ( ( b − a ) / 2 a b ) ] , P 2 ( a , b ) = [ ( b 2 − a 2...

#### On approximating Mills ratio

In the article, we present several sharp bounds for the Mills ratio R ( x ) = e x 2 / 2 ∫ x ∞ e − t 2 / 2 d t ( x > 0 ) in terms of the functions I a ( x ) = a / [ x 2 + 2 a + ( a − 1 ) x ] and J ( x ) = a / [ x 2 + 2 a 2 / π + 2 a x / π ] with parameter a > 0 . MSC: 60E15, 26A48, 26D15.

#### Optimal power mean bounds for the second Yang mean

In this paper, we present the best possible parameters p and q such that the double inequality M p ( a , b ) < V ( a , b ) < M q ( a , b ) holds for all a , b > 0 with a ≠ b , where M r ( a , b ) = [ ( a r + b r ) / 2 ] 1 / r ( r ≠ 0 ) and M 0 ( a , b ) = a b is the rth power mean and V ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / 2 a b ) ] is the second Yang mean. MSC: 26E60.

#### Sharp bounds for cyclic sums of the ratio of the exradius to the sides of a triangle

In this paper, sharp bounds for cyclic sums of the ratio of the exradius to the sides of a triangle are established depending on the circumradius and inradius of the triangle. The best possible parameters for the expressions of bounds are derived. Moreover, an alternative bound for the ratio of the exradius to the sides of triangle, expressed by trigonometric functions, is also...

#### Sharp bounds for Toader mean in terms of arithmetic, quadratic, and Neuman means

In this paper, we present the best possible parameters α , β ∈ R and λ , μ ∈ ( 1 / 2 , 1 ) such that the double inequalities α N A Q ( a , b ) + ( 1 − α ) A ( a , b ) < T ∗ ( a , b ) < β N A Q ( a , b ) + ( 1 − β ) A ( a , b ) , Q [ λ a + ( 1 − λ ) b , λ b + ( 1 − λ ) a ] < T ∗ ( a , b ) < Q [ μ a + ( 1 − μ ) b , μ b + ( 1 − μ ) a ] hold for all a , b > 0 with a ≠ b , where T...

#### Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

In the article, we present the best possible parameters α 1 , α 2 , β 1 , β 2 ∈ ( 0 , 1 ) and α 3 , α 4 , β 3 , β 4 ∈ ( 0 , 1 / 2 ) such that the double inequalities α 1 A ( a , b ) + ( 1 − α 1 ) H ( a , b ) < X ( a , b ) < β 1 A ( a , b ) + ( 1 − β 1 ) H ( a , b ) , α 2 A ( a , b ) + ( 1 − α 2 ) G ( a , b ) < X ( a , b ) < β 2 A ( a , b ) + ( 1 − β 2 ) G ( a , b ) , H [ α 3 a...

#### 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...