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.

中文翻译：

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β-偏导数的Opial-Traple型不等式

在论文中，我们引入一个新的偏导数的呼叫它β -偏导数为偏导数与极限定义的最自然的扩展β -衍生物，其遵循经典的性质，包括：连续性，线性度，产品规则，商法则，幂定律，链式定律和消失的导数，以实现恒定功能。作为应用，我们为β-偏导数建立了一些新的Opial-Traple型不等式。