Second Note on the Definition of S1-Convexity
Author(s)
I. M. R. Pinheiro*
ABSTRACT
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.
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.
KEYWORDS
Analysis, Convexity, Definition, s-Convexity, Geometry, Shape, S-Convexity, s-Convex Function, S-Convex Function
Analysis, Convexity, Definition, s-Convexity, Geometry, Shape, S-Convexity, s-Convex Function, S-Convex Function
Cite this paper
Pinheiro, I. (2015) Second Note on the Definition of S1-Convexity. Advances in Pure Mathematics, 5, 127-130. doi: 10.4236/apm.2015.53015.
Pinheiro, I. (2015) Second Note on the Definition of S1-Convexity. Advances in Pure Mathematics, 5, 127-130. doi: 10.4236/apm.2015.53015.
References
[1] Pinheiro, M.R. (2013) Minima Domain Intervals and the S-Convexity, as Well as the Convexity, Phenomenon. Advances in Pure Mathematics, 3, 457-458. [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. [4] Dragomir, S.S. and Pearce, C.E.M. (2002) Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA, Monographs. Online at rgmia.vu.edu.au. [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.
[1] Pinheiro, M.R. (2013) Minima Domain Intervals and the S-Convexity, as Well as the Convexity, Phenomenon. Advances in Pure Mathematics, 3, 457-458. [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. [4] Dragomir, S.S. and Pearce, C.E.M. (2002) Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA, Monographs. Online at rgmia.vu.edu.au. [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.