More About Hermite-Hadamard Inequalities, Cauchy's Means, and Superquadracity
© S. Abramovich et al. 2010
Received: 26 March 2010
Accepted: 4 August 2010
Published: 16 August 2010
New results associated with Hermite-Hadamard inequalities for superquadratic functions are given. A set of Cauchy's type means is derived from these Hermite-Hadamard-type inequalities, and its log-convexity and monotonicity are proved.
is holding for any convex function, that is, well known in the literature as the Hermite-Hadamard inequality (see [1, page 137]). In many areas of analysis applications of Hermite-Hadamard inequality appear for different classes of functions with and without weights; see for convex functions, for example, [2, 3]. Also some useful mappings are defined connected to this inequality see in [4–6]. Here we focus on a class of functions which are superquadratic and analogs and refinements of (1.1) are applied to obtain results useful in analysis.
Now we present definitions, theorems, and results that we use in this paper.
The following definition is given in .
The followings theorem is given in  and is used in our main results:
Proposition 1.2 (see [9, Proposition ]).
The next two sections are about mean value theorems, positive semidefiniteness, exponential convexity, log-convexity, Cauchy means, and their monotonicity, that are associated with Hermite-Hadamard inequalities for superquadratic functions.
2. Mean Value Theorems
In  we have the following Lemma.
Lemma 2.2 (see [12, Lemma ]).
By combining the above two inequalities we get that there exists such that (2.4) holds. Moreover if (for example) is bounded from above we have that (2.5) is valid. Also (2.4) holds when is not bounded.
is a new mean.
By combining the above two inequalities we get that there exist such that (2.23) holds. Moreover if (for example) is bounded from above we have that (2.24) is valid. Also (2.23) holds when is not bounded.
is a new mean.
3. Positive Semidefiniteness, Exponential Convexity, and Log-Convexity
Lemma 3.1 (see [12, Lemma ]).
One has the following
Similar to Theorem 3.2 we get the following.
The proof is the same as the proof of Theorem 3.2.
When log f is convex we see that (also see )
This research was partially funded by Higher Education Commission, Pakistan. The research of the third author was supported by the Croatian Ministry of Science, Education and Sports under the Research grant no. 117-1170889-0888.
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