63 papers found.

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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.

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...

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...

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 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...

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...

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...

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.

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.

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...

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 ) < 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.

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 ) < 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...

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...

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...