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Ding.
Formal analysis: Yunshen Jiao, Lingyu Ding, Xiaofan Zhao.
Investigation: Yunshen Jiao, Xi Chen, Zirui Gao, Likai Gao.
Project administration: Yunshen Jiao, **Ming** **Chu**, Yuedan Wang.
Resources: Jiarui ... Kang.
Software: Yunshen Jiao, Lingyu Ding, Tieshan Wang.
Supervision: **Ming** **Chu**, Yuedan Wang.
Visualization: Lingyu Ding.
Writing ± original draft: Yunshen Jiao.
Writing ± review & editing: Lingyu

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 ) 0 , where erf ( x ) = 2 ∫ 0 x e − t 2 d t / π is the ...

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.

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.

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 if and only if a ≤ min { 1 , Γ − p ( 1 + 1 / p ) } and b ≥ max { 1 , Γ − p ( 1 + 1 / p ) } . MSC: 33B20, 26D07, 26D15.

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 0 and p > 1 , where I p ( x ) = e x p ∫ x ∞ e − t p d t is the incomplete gamma function. MSC: 33B20, 26D07, 26D15.

**Ming** **Chu** Kong 0 1
0 Laboratory for Mechanics of Materials and Nanostructures, EMPA, Swiss Federal Laboratories for Materials Testing and Research , Feuerwerkerstrasse 39, 3602 Thun , Switzerland
1

**Ming** **Chu** Kong 0 1
0 Laboratory for Mechanics of Materials and Nanostructures, EMPA, Swiss Federal Laboratories for Materials Testing and Research , Feuerwerkerstrasse 39, 3602 Thun , Switzerland
1

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 ) ) ] 1 ...

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.

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.

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.

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.

Conventional genome-wide association studies (GWAS) have been proven to be a successful strategy for identifying genetic variants associated with complex human traits. However, there is still a large heritability gap between GWAS and transitional family studies. The “missing heritability” has been suggested to be due to lack of studies focused on epistasis, also called gene–gene ...

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 hold for all x ∈ ( 0 , π / 2 ) . MSC: 26D05, 33B10.

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 ) / ( 2 ...

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.

In this paper, we present the best possible parameters p and q such that the double inequality M p ( 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.

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 ...

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 ) 0 with a ≠ b , where T ∗ ( a , b ) , A ( a , b ) , Q ( a , b ) and N Q A ( a , b ) are the Toader, arithmetic, quadratic, and Neuman means of a and b, respectively. MSC: 26E60.