Second Note on the Definition of S1-Convexity
I. M. R. Pinheiro*
RGMIA, AMS, PROz., Melbourne, Australia.
DOI: 10.4236/apm.2015.53015   PDF   HTML   XML   4,021 Downloads   4,507 Views   Citations


In this note, we discuss the definition of the S1-convexity Phenomenon. We first make use of some results we have attained for   in the past, such as those contained in [1], to refine the definition of the phenomenon. We then observe that easy counter-examples to the claim 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  does not properly extend K0 in both its original and its revised version.

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Pinheiro, I. (2015) Second Note on the Definition of S1-Convexity. Advances in Pure Mathematics, 5, 127-130. doi: 10.4236/apm.2015.53015.

Conflicts of Interest

The authors declare no conflicts of interest.


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[2] Pinheiro, M.R. (2014) First Note on the Definition of s1-Convexity. Advances in Pure Mathematics, 4, 674-679.
[3] Hudzik, H. and Maligranda, L. (1994) Some Remarks on s-Convex Functions. Aequationes Mathematicae, 48, 100-111.
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[5] Dragomir, S.S. and Fitzpatrick, S. (1999) The Hadamard’s Inequality for S-Convex Functions in the Second Sense. Demonstratio Mathematica, 32, 687-696.
[6] Pinheiro, M.R. (2007) Exploring the Concept of s-Convexity. Aequationes Mathematicae, 74, 201-209.
[7] Pinheiro, M.R. (2004) Exploring the Concept of s-Convexity. Proceedings of the 6th WSEAS Int. Conf. on Mathematics and Computers in Physics (MCP '04).
[8] Pinheiro, M.R. (2008) Convexity Secrets. Trafford Canada.
[9] Pinheiro, M.R. (2014) Third Note on the Shape of S-Convexity. International Journal of Pure and Applied Mathematics, 93, 729-739.

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