Journal of East China Normal University(Natural Sc ›› 2018, Vol. 2018 ›› Issue (3): 46-54,120.doi: 10.3969/j.issn.1000-5641.2018.03.006

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AM(s)-Convex function and its Jensen-type inequality

SONG Zhen-yun1, HU Fu-gao2   

  1. 1. Dean's office, Hubei Polytechnic Institute, Xiaogan Hubei 432000, China;
    2. School of Mathematics and Statistics, Hubei Engineering University, Xiaogan Hubei 432000, China
  • Received:2017-03-17 Online:2018-05-25 Published:2018-05-29

Abstract: Based on the convexity and general convexity of a function, the authors study extending issues of a convex function. Firstly, the concept and sign of weighted r-th power s-mean of n positives are introduced; secondly, the AM(s)-convex function is defined by weighted r-th power s-mean; thirdly, the judgment theorem and operation properties of AM(s)-convex function are discussed; and finally, the Jensen-type inequality of the AM(s)-convex function is proved and an equivalent form is provided. The study shows that the AM(s)-convex function is a subset of general convex functions that includes many convex functions. Studying the AM(s)-convex function with the method of weighted r-th power s-mean is an effective way of extending and studying convex functions. This method explores a new approach to extending and studying convex functions.

Key words: weighted r-th power s-mean, AM(s)-convex function, judgment theorem, operation property, Jensen-type inequality

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