# Some Generalizations of the Jensen-Type Inequalities with Applications
## 4 Applications  
In order to round off this paper, we would like to present some applications. When we speak about the mean-value theorems it is somehow most natural to derive some means. In order to get the Cauchy-type means with certain nice properties, we will use the method from the paper [20], which will firstly help us to define some new exponentially convex functions, and then to define the new means. At the very beginning of this section, before stating our results, we have to recall some of the very basic definitions and facts about exponential convexity. Throughout this section, with \(I\) we will denote an open interval in \(\mathbb{R}\). **Definition 1**. : _A function \(f:I\to\mathbb{R}\) is \(n-\)exponentially convex in the [PERSON] sense on \(I\) if_  
\[\sum_{i,j=1}^{n}\rho_{i}\rho_{j}f\left(\frac{x_{i}+x_{j}}{2}\right)\geq 0\]  
_for all \(\rho_{i}\in\mathbb{R}\) and \(x_{i}\in I\), \(i=1,\ldots,n\). A function \(f:I\to\mathbb{R}\) is \(n-\)exponentially convex if it is \(n-\)exponentially convex in the [PERSON] sense and continuous on \(I\)._  
**Remark 7**. : _Note that \(1-\)exponentially convex functions in the [PERSON] sense are non-negative functions, as we can see from the definition. Further, \(n-\)exponentially convex functions in the [PERSON] sense are \(k-\)exponentially convex in the [PERSON] sense for every \(k\in\mathbb{N}\), \(k\leq n\)._  
**Definition 2**. : _A function \(f:I\to\mathbb{R}\) is exponentially convex in the [PERSON] sense on 1, if it is \(n-\)exponentially convex in the [PERSON] sense for all \(n\in\mathbb{N}\). A function \(f:I\to\mathbb{R}\) is exponentially convex if it is exponentially convex in the [PERSON] sense and continuous._  
**Remark 8**. : _Here we mention also some examples of exponentially convex functions from [20], as we will need them in the process of constructing our means:_  
1. \(f:I\to\mathbb{R}\) _defined by_ \(f(x)=ce^{rx}\)_, where_ \(c\geq 0\) _and_ \(r\in\mathbb{R}\)_._
2. \(f:\mathbb{R}^{+}\to\mathbb{R}\) _defined by_ \(f(x)=x^{-r}\)_, where_ \(r>0\)_._
3. \(f:\mathbb{R}^{+}\to\mathbb{R}^{+}\) _defined by_ \(f(x)=e^{-r/\sqrt{x}}\)_, where_ \(r>0\)_._  
**Remark 9**. : _A positive function \(f:I\to\mathbb{R}^{+}\) is log-convex in the Jensen sense on 1 if and only if it is \(2-\)exponentially convex in the Jensen sense on 1, i.e., if and only if for every \(\rho_{1},\rho_{2}\in\mathbb{R}\) and \(x_{1},x_{2}\in I\) holds_  
\[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]  
_If such function is also continuous on \(I\), it follows that it is log-convex on 1._  
We also recall the following two lemmas from [4], p. 2. **Lemma 1**. : _If \(x_{1},x_{2},x_{3}\in I\) are such that \(x_{1}<x_{2}<x_{3}\), then the function \(f:I\to\mathbb{R}\) is convex if and only if_  
\[(x_{3}-x_{2})f(x_{1})+(x_{1}-x_{3})f(x_{2})+(x_{2}-x_{1})f(x_{3})\geq 0\;.\]  
**Lemma 2**. : _If \(f:I\to\mathbb{R}\) is a convex function and \(x_{1},x_{2},y_{1},y_{2}\in I\) are such that \(x_{1}\;\leq\;y_{1},\;x_{2}\;\leq\;y_{2},\;x_{1}\
eq x_{2},\;y_{1}\
eq y_{2}\), then_  
\[\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}\leq\frac{f(y_{2})-f(y_{1})}{y_{2}-y_{1}}\;.\]  
We have to recall also the definition of the divided difference of the second order. **Definition 3**. : _The divided difference of the second order of a function \(f:I\to\mathbb{R}\) at mutually different points \(x_{0},x_{1},x_{2}\in I\) is defined recursively by_  
\[[x_{i}]f = f(x_{i}),\qquad i=0,1,2,\] \[[x_{i},x_{i+1}]f = \frac{f(x_{i+1})-f(x_{i})}{x_{i+1}-x_{i}},\qquad i=0,1,\] \[[x_{0},x_{1},x_{2}]f = \frac{[x_{1},x_{2}]f-[x_{0},x_{1}]f}{x_{2}-x_{0}}. \tag{22}\]  
**Remark 10**. : _The value \([x_{0},x_{1},x_{2}]f\) is independent of the order of the points \(x_{0},x_{1}\) and \(x_{2}\). Also, we can extended this definition to include the case when some or all of the points are equal ([4], p. 14).