48 Result(s)
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Article
Complete monotonicity of the remainder in an asymptotic series related to the psi function
Let p, q ∈ ℝ with p − q ≽ 0, \(\sigma = {1 \over 2}(p + q - 1)\) σ = ...
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Convexity of ratios of the modified Bessel functions of the first kind with applications
Let \(I_{\nu }\left( x\right) \) I ν ...
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The signs rule for the Laplace integrals with applications
Let the function \(f\left( t\right) \) f t
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Absolute monotonicity of the accuracy of Ramanujan approximations for perimeter of an ellipse
In this paper, we prove that the absolute errors of 11 mean value approximations for the perimeter of an ellipse are all absolutely monotonic with respect to the eccentricity of this ellipse. More interestingl...
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Convexity and Monotonicity Involving the Complete Elliptic Integral of the First Kind
Let \(\mathcal {K}\left( r\right) \) K r ...
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Several Absolutely Monotonic Functions Related to the Complete Elliptic Integral of the First Kind
Let \({\mathcal {K}}\left( r\right) \) K r ...
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New properties of the divided difference of psi and polygamma functions
Let \(\psi _{n}=\left( -1\right) ^{n-1}\psi ^{\left( n\right) }\) ...
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Open AccessEditorial Expression of Concern: On approximating Mills ratio
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Sharp bounds for the Toader mean in terms of arithmetic and geometric means
For \(a,b>0\) a , b > ...
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A sharp lower bound for the complete elliptic integrals of the first kind
Let \({\mathcal {K}}\left( r\right) \) K r ...
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A positive answer to Bhatia—Li conjecture on the monotonicity for a new mean in its parameter
The Bhatia—Li mean \(\mathcal {B}_{p}\left( x,y\right) \) B p x , y of positive numbers x and y is defined as $$\begin{aligned} \frac{1}{\mathcal {B}_{p}\left( x,y\right) }=\frac{p}{B\left( 1/p,1/p\rig...
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Monotonicity, convexity, and complete monotonicity of two functions related to the gamma function
In this paper, we prove that for \(a\ge 31/98\)a≥31/98, the function $$\begin{aligned} f_{a}\left( x\right) =\ln \Gamma \left( x+\frac{1}{2}\right) -x\ln x+x-\frac{ 1}{2}\ln \left( 2\pi \right) +\dfrac{x}{24}\dfr...
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On Burnside type approximation for the gamma function
In the paper, by means of the monotonicity rule for the ratio of two Laplace transform we determine the best \(a,b>0\) ...
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Open AccessWindschitl type approximation formulas for the gamma function
In this paper, we present four new Windschitl type approximation formulas for the gamma function. By some unique ideas and techniques, we prove that four functions combined with the gamma function and Windschi...
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Open AccessMonotonicity of the ratio of modified Bessel functions of the first kind with applications
Let W v ...
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Open AccessAn accurate approximation formula for gamma function
In this paper, we present a very accurate approximation for the gamma function: Γ ...
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Open AccessSharp Smith’s bounds for the gamma function
Among various approximation formulas for the gamma function, Smith showed that Γ ...
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Open AccessMonotonicity and inequalities for the gamma function
In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the function ...
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Article
A double inequality for an integral mean in terms of the exponential and logarithmic means
In the paper, the authors aim to present a double inequality for the integral mean $$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }a^{...
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Open AccessComplete monotonicity involving some ratios of gamma functions
In this paper, by using the properties of an auxiliary function, we mainly present the necessary and sufficient conditions for various ratios constructed by gamma functions to be respectively completely and lo...
