Abstract
In this paper, we first show that the first Seiffert mean P is concave whereas the second Seiffert mean T and the Neuman-Sandor mean NS are convex. As applications, we establish the sub-stabilizability/super-stabilizability of certain bivariate means. Open problems are derived as well.
Highlights
A mean m is a binary map from (, ∞) × (, ∞) into (, ∞) satisfying min(x, y) ≤ m(x, y) ≤ max(x, y) for all x, y >
Symmetric means are defined in the usual way; see, for instance, [ ]
We identify a mean m with its value at (x, y) by setting m = m(x, y) for simplicity
Summary
A (bivariate) mean m is a binary map from ( , ∞) × ( , ∞) into ( , ∞) satisfying min(x, y) ≤ m(x, y) ≤ max(x, y) for all x, y >. The super-stabilizability of m is defined in an analogous manner (by reversing the previous mean inequalities). Section contains some applications of the previous results to the sub/super stabilizability of certain bivariate means. If m is symmetric homogeneous, m is partially convex Concave) if and only if the real function x −→ m(x, ) is convex For a mean of class C (i.e., twice differentiable on ( , ∞) × ( , ∞) with continuous second derivative), we can give a direct proof of the previous lemma without refiring to the general result of [ ] stated for general positively homogeneous functions. Such a mean is known as the power (binomial) mean of order p
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callSimilar Papers
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
