Abstract
The main purpose of the present paper is to introduce a new concept by calculating first order variation of multivariate function, and call it \(\alpha \)-multivariate partial derivatives, which is a generalization of the well-known \(\alpha \)-conformable derivatives. For the \(\alpha \)-multivariate partial derivatives, we prove Lagrange’s, Rolle’s and Cauchy’s mean value theorems in n-dimensional space. As applications, we establish some new Čebyšev–Grüss type inequalities for the \(\alpha \)-multivariate partial derivatives, which in special cases yield Milovanović, Pečarič and Fink’s, and Pachpatte’s inequalities.
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The author express his thanks to the two referees for their excellent suggestions. The author express also his thanks to Ms. Jinhua Ji for his valuable help.
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Research is supported by National Natural Science Foundation of China (11371334, 10971205).
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Zhao, CJ. Čebyšev–Grüss inequalities for \(\alpha \)-partial derivatives. RACSAM 115, 11 (2021). https://doi.org/10.1007/s13398-020-00948-7
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DOI: https://doi.org/10.1007/s13398-020-00948-7