Convexity Secrets

  • IICSE University


It was supposed to be a finger-friendly catalog of convexity types for functions where we could also find a history of the progresses with full reference list and etc. Trafford did not have the resource to make the finger-friendly index. The introduction is in very poor English and we regret having written it. With the fight support. we can certainly prepare a much better version of this material. Researchers were supposed to find plenty of material for innovative research and use the book as a tool to save time and effort. They should be able to simply copy the entire history of the convexity type from the book when producing their own material.
... We use the symbols from [8] here: 1 s < ≤ , used for the second type of s-convexity. Remark 1. ...
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In this note, we discuss the definition of the S1-convexity Phenomenon. We first make use of some results we have attained for s K 2 in the past, such as those contained in [1], to refine the definition of the phenomenon. We then observe that easy counterexamples to the claim s K 1 extends K0 are found. Finally, we make use of one theorem from [2] and a new theorem that appears to be a supplement to that one to infer that s K 1 does not properly extend K0 in both its original and its revised version.
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In this paper, we propose a refinement in the analytical definition of the s 2-convex classes of functions aiming to progress further in the direction of including s 2-convexity properly in the body of Real Analysis.
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