On Opial-Traple type inequalities for <em>β</em>-partial derivatives

Chang-Jian Zhao; Wing-Sum Cheung; Chang-Jian Zhao; Wing-Sum Cheung

doi:10.3934/math.2020366

AIMS Mathematics

2020, Volume 5, Issue 6: 5716-5723. doi: 10.3934/math.2020366

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Research article

On Opial-Traple type inequalities for β-partial derivatives

Chang-Jian Zhao ^{1
,
,},
Wing-Sum Cheung ²

1.
Department of Mathematics, China Jiliang University, Hangzhou 310018, P. R. China
2.
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong

Received: 30 April 2020 Accepted: 30 June 2020 Published: 09 July 2020
MSC : 26A33, 26D125

In the paper, we introduce a new partial derivative call it β-partial derivatives as the most natural extensions of the limit definitions of the partial derivative and the β-derivative, which obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple type inequalities for the β-partial derivatives.
- β-derivative,
- partial derivative,
- β-partial derivative,
- Cauchy-Schwarz inequality
Citation: Chang-Jian Zhao, Wing-Sum Cheung. On Opial-Traple type inequalities for β-partial derivatives[J]. AIMS Mathematics, 2020, 5(6): 5716-5723. doi: 10.3934/math.2020366

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Abstract

In the paper, we introduce a new partial derivative call it β-partial derivatives as the most natural extensions of the limit definitions of the partial derivative and the β-derivative, which obeys classical properties including: continuity, linearity, product rule, quotient rule, power rule, chain rule and vanishing derivatives for constant functions. As applications, we establish some new Opial-Traple type inequalities for the β-partial derivatives.

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