Multiplicative Concavity of the Integral of Multiplicatively Concave Functions
© Y.-M. Chu and X.-M. Zhang. 2010
Received: 25 March 2010
Accepted: 7 June 2010
Published: 28 June 2010
For convenience of the readers, we first recall some definitions and notations as follows.
From Definitions 1.1 and 1.2, the following Theorem A is obvious.
is multiplicatively convex.
Equivalently, is a multiplicatively convex function if and only if is a convex function of . Modulo this characterization, the class of all multiplicatively convex functions was first considered by Motel , in a beautiful paper discussing the analogues of the notion of convex function in variables. However, the roots of the research in this area can be traced long before him. In a long time, the subject of multiplicative convexity seems to be even forgotten, which is a pity because of its richness. Recently, the multiplicative convexity has been the subject of intensive research. In particular, many remarkable inequalities were found via the approach of multiplicative convexity (see [2–18]).
The main purpose of this paper is to prove Theorem 1.6.
2. Lemmas and the Proof of Theorem 1.6
For the sake of readability, we first introduce and establish several lemmas which will be used to predigest the proof of Theorem 1.6.
Lemma 2.1 can be derived from Definitions 1.4 and 1.5.
If is a multiplicatively convex set, and is multiplicatively convex (or concave, resp.), then is convex (or concave, resp.) on . Conversely, if is a convex set, and is convex (or concave, resp.), then is multiplicatively convex (or concave, resp.) on .
Lemma 2.2 (see ).
Making use of Lemmas 2.1 and 2.2 we get the following Lemma 2.3.
Lemma 2.4 (see ).
We divide the proof into five cases.
Therefore, Lemma 2.7 follows from (2.16) and (2.17) together with Lemma 2.3.
Lemma 2.8 (see ).
From Definitions 1.1 and 1.2, Theorem A, and Lemma 2.8 we can get Lemma 2.9 immediately.
For each continuous multiplicatively convex (or concave, resp.) function , there exists a sequence of infinitely differentiable multiplicatively convex (or concave, resp.) functions , such that converges uniformly to on .
Proof of Theorem 1.6.
Since is a continuous multiplicatively concave function, from Lemma 2.9 we know that there exists a sequence of infinitely differentiable multiplicatively concave function , such that converges uniformly to on .
Therefore, Theorem 1.6 follows from Definition 1.5 and (2.18).
The research was supported by the Natural Science Foundation of China under Grant 60850005 and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.
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