A New Proof of the Stronger Second Mean Value Theorem for Integrals
Received:November 09, 2020  Revised:January 03, 2021
Key Word: second mean value theorem for integrals   Riemann integrable   Lebesgue integrable  
Fund ProjectL:Supported by Natural Science Basic Research Program of Shaanxi (Program No.2021JM-487), the Special Scientific Research Program of the Education Department of Shaanxi Province (Grant No.18JK0161) and the Scientific Research Foundation of Shaanxi University of Technology (Grant No.SLGQD1807).
Author NameAffiliation
Peng ZHENG School of Mathematics and Computer Science, Shaanxi University of Technology, Shaanxi 723001, P. R. China 
Siquan SHI Division of Physics, Engineering, Mathematics, and Computer Science, Delaware State University, Dover 19901, U. S. A. 
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Abstract:
      R. Witula et al obtained a stronger version of the second mean value theorem for integral with some restrictions. In this paper, the stronger version theorem is proved without any restriction. The result is first restricted to the Riemann integrable functions and can be easily generalized to $L^p$ integrable functions by using the well-known result that continuous functions are dense in the Banach space $L^p[a,b]$ for any $p \geq 1$.
Citation:
DOI:10.3770/j.issn:2095-2651.2021.03.004
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