# Some Generalizations of the Jensen-Type Inequalities with Applications
## 4 Applications
This implies  
\[(x_{3}-x_{2})f(x_{1})+(x_{1}-x_{3})f(x_{2})+(x_{2}-x_{1})f(x_{3})\geq 0\;.\]0  
and therefore  
\[(x_{3}-x_{2})f(x_{1})+(x_{1}-x_{3})f(x_{2})+(x_{2}-x_{1})f(x_{3})\geq 0\;.\]1  
which means that \(u\mapsto F_{i}(g,\lambda,\phi_{u})\ (i=1,2,3)\) are \(n-\)exponentially convex in the Jensen sense on \(I\). If \(u\mapsto F_{i}(g,\lambda,\phi_{u})\ (i=1,2,3)\) is additionally also continuous on \(I\), then it is \(n\)-exponentially convex by definition. As an immediate consequence of this theorem, we have the following corollary. Corollary 1: _Let \(\Gamma=\{\phi_{u}:u\in I\}\) be a family of functions \(\phi_{u}:[\alpha,\beta]\to\mathbb{R}\) where \(\phi_{u}\in C([\alpha,\beta])\), such that for every three mutually different points \(x_{0},x_{1},x_{2}\in[\alpha,\beta]\) the function \(u\mapsto[x_{0},x_{1},x_{2}]\phi_{u}\) is exponentially convex in the [PERSON] sense on \(I\). Then the functions \(u\mapsto F_{i}(g,\lambda,\phi_{u})\ (i=1,2,3)\) are also exponentially convex in the [PERSON] sense on \(I\). If the function \(u\mapsto F_{i}(g,\lambda,\phi_{u})\ (i=1,2,3)\) is also continuous on \(I\), then it is exponentially convex on \(I\)._  
We also have the following corollary. Corollary 2: _Let \(\Gamma=\{\phi_{u}:u\in I\}\) be a family of functions \(\phi_{u}:[\alpha,\beta]\to\mathbb{R}\) where \(\phi_{u}\in C([\alpha,\beta])\), such that for every three mutually different points \(x_{0},x_{1},x_{2}\in[\alpha,\beta]\) the function \(u\mapsto[x_{0},x_{1},x_{2}]\phi_{u}\) is \(2-\)exponentially convex in the Jensen sense on \(I\)._  
1. _If the function_ \(u\mapsto F_{i}(g,\lambda,\phi_{u})\ (i=1,2,3)\) _is continuous on_ \(I\)_, then it is_ \(2-\)_exponentially convex on_ \(I\)_. Further, if_ \(u\mapsto F_{i}(g,\lambda,\phi_{u})\ (i=1,2,3)\) _is also strictly positive, then it is also log-convex on_ \(I\)_, and it holds_ \[(x_{3}-x_{2})f(x_{1})+(x_{1}-x_{3})f(x_{2})+(x_{2}-x_{1})f(x_{3})\geq 0\;.\]2 (23) _for_ \(u_{1},u_{2},u_{3}\in I\) _such that_ \(u_{1}<u_{2}<u_{3}\)_._
2. _If the function_ \(u\mapsto F_{i}(g,\lambda,\phi_{u})\ (i=1,2,3)\) _is strictly positive and differentiable on_ \(I\)_, then it holds_ \[(x_{3}-x_{2})f(x_{1})+(x_{1}-x_{3})f(x_{2})+(x_{2}-x_{1})f(x_{3})\geq 0\;.\]3 (24) _for every_ \(u,v,y,z\in I\) _such that_ \(u\leq y\) _and_ \(v\leq z\)_, where_ \[\mu_{u,v}(g,F_{i},\Gamma)=\ \left\{ \begin{array}{ll}\bigg{(}\frac{F_{i}(g, \lambda,\phi_{u})}{F_{i}(g,\lambda,\phi_{v})}\bigg{)}^{\frac{1}{n-\beta}},&u \
eq v,\\ \\ \exp\bigg{(}\frac{\frac{d}{du}F_{i}(g,\lambda,\phi_{u})}{F_{i}(g,\lambda,\phi_ {u})}\bigg{)},&u=v,\end{array} \right.\] (25) _for_ \(\phi_{u},\phi_{v}\in\Gamma\)Proof. : \((i)\) From Theorem 8 we get that the first sentence is valid, and the log-convexity follows from Remark 9. We still have to prove (25). We have that the function \(u\mapsto F_{i}(g,\lambda,\phi_{u})\) is strictly positive, and we can apply Lemma 1 on the function \(f(x)=\log F_{i}(g,\lambda,\phi_{x})\). We have that  
\[(u_{3}-u_{2})F_{i}(g,\lambda,\phi_{u_{1}})+(u_{1}-u_{3})F_{i}(g,\lambda,\phi_{u _{2}})+(u_{2}-u_{1})F_{i}(g,\lambda,\phi_{u_{3}})\geq 0\]  
for \(u_{1},u_{2},u_{3}\in I\) (\(u_{1}<u_{2}<u_{3}\)) and therefore inequality (23) holds. \((ii)\) From \((i)\) we have that the function \(u\mapsto F_{i}(g,\lambda,\phi_{u})\) (\(i=1,2,3\)) is log-convex on \(I\), and that means that the function \(u\mapsto\log F_{i}(g,\lambda,\phi_{u})\) (\(i=1,2,3\)) is convex on \(I\). Applying Lemma 2 we get  
\[\frac{\log F_{i}(g,\lambda,\phi_{u})-\log F_{i}(g,\lambda,\phi_{v})}{u-v} \leq\frac{\log F_{i}(g,\lambda,\phi_{y})-\log F_{i}(g,\lambda,\phi_{z})}{y-z} \tag{26}\]  
where \(u\leq y,\;\;v\leq z,\;\;u\
eq v,\;\;y\
eq z\). This implies  
\[\mu_{u,v}(g,F_{i},\Gamma)\leq\mu_{y,z}(g,F_{i},\Gamma).\]  
We get the cases \(u=v\) and \(y=z\) as limit cases from (26). **Remark 11**. : _When two or all of the points \(x_{0},x_{1},x_{2}\in[\alpha,\beta]\) are equal, the results from_ Theorem 8_,_ Corollaries 1_and 2 are also valid.