A Mean Value Inequality for Euler's Beta Function

Alzer, Horst; Paris, R. B.

doi:10.33232/bims.0078.25.30

Irish Math. Soc. Bull.

2016

DOI: 10.33232/bims.0078.25.30

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A Mean Value Inequality for Euler's Beta Function

Horst Alzer¹,

R. B. Paris²

Abstract: Let B(x, y) be Euler's beta function. We prove that the inequalitieshold for all x, y > 0 with x = y. The given constant bounds are best possible. This result is extended to the case when the beta function in the numerator has arguments given by the weighted geometric and arithmetic means.

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Approximations and inequalities for the exponential beta function

Dragomir

Khosrowshahi

2019

J Inequal Appl

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In this paper, motivated by interest in simulating the expenditure patterns of construction projects, we introduce the mathematical concept of Exponential-Beta function by $$ F ( \alpha ,\beta ) := \int _{0}^{1}\exp \bigl[ x^{\alpha } ( 1-x ) ^{\beta } \bigr]\,dx, $$ F ( α , β ) : = ∫ 0 1 exp [ x α ( 1 − x ) β ] d x , where α, β are positive numbers. Taylor’s-type approximations, several analytic inequalities, and global convexity properties are established.

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Approximations and inequalities for the exponential beta function

Dragomir

Khosrowshahi

2019

J Inequal Appl

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A Mean Value Inequality for Euler's Beta Function

Abstract: Let B(x, y) be Euler's beta function. We prove that the inequalitieshold for all x, y > 0 with x = y. The given constant bounds are best possible. This result is extended to the case when the beta function in the numerator has arguments given by the weighted geometric and arithmetic means.

Cited by 1 publication

References 10 publications

Approximations and inequalities for the exponential beta function

Approximations and inequalities for the exponential beta function

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