# Some Generalizations of the Jensen-Type Inequalities with Applications
## 4 Applications
Taking the limit \(x_{1}\to x_{0}\) in (22), we obtain_  
\[\lim_{x_{1}\to x_{0}}[x_{0},x_{1},x_{2}]f=[x_{0},x_{0},x_{2}]f=\frac{f(x_{2})-f (x_{0})-f^{\prime}(x_{0})(x_{2}-x_{0})}{(x_{2}-x_{0})^{2}},\ \ x_{2}\
eq x_{0},\]  
_assuming that \(f^{\prime}\) exists. Further, taking the limits \(x_{i}\to x_{0}\), \(i=1,2\), in (22), we obtain_  
\[\lim_{x_{2}\to x_{0}}\lim_{x_{1}\to x_{0}}[x_{0},x_{1},x_{2}]f=[x_{0},x_{0},x_{ 0}]f=\frac{f^{\prime\prime}(x_{0})}{2}\]  
_assuming that \(f^{\prime\prime}\) exists._  
_A function \(f:I\to\mathbb{R}\) is convex if and only if for every choice of three mutually different points \(x_{0},x_{1},x_{2}\in I\) holds \([x_{0},x_{1},x_{2}]f\geq 0\)._  
Now, we can start with our results. We will take s certain family of functions, then apply our functionals to it, and in this way, we will construct \(n-\)exponentially convex and exponentially convex functions. Before we proceed, using the functionals \(A_{i}\)\((i=1,2,3)\) we will define three new functionals to assure that the functionals we use in this process are always non-negative, whenever they are defined. For continuous function \(g:[a,b]\to\mathbb{R}\) with \(Im(g)\subseteq[m,M]\subseteq(\alpha,\beta)\), a continuous function or a function of bounded variation \(\lambda:[a,b]\to\mathbb{R}\) with \(\lambda(a)\
eq\lambda(b)\), and continuous convex function \(\phi:[\alpha,\beta]\to\mathbb{R}\), we are now defining new functionals \(F_{i}\)\((i=1,2,3)\) by:  
\[F_{1}(g,\lambda,\phi) =-A_{1}(g,\lambda,\phi),\ \text{if for any }p\in\{1,2,3,4\}\ \text{for every }s\in[\alpha,\beta]\ \text{holds (\ref{eq:1})};\] \[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]0 \[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]1 \[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]2 \[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]3 \[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]4 \[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]5 \[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]6 \[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]7  
It is now \(F_{i}(g,\lambda,\phi)\geq 0\)\((i=1,2,3)\) always when these functionals are defined. **Theorem 8**. : _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 \(n-\)exponentially convex in the Jensen sense on \(I\). Then the functions \(u\mapsto F_{i}(g,\lambda,\phi_{u})\)\((i=1,2,3)\) are also \(n-\)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 \(n-\)exponentially convex on \(I\)._  
Proof. : Let us define the function \(\chi\) by  
\[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]8where \(\rho_{j},\rho_{k}\in\mathbb{R}\), \(u_{j},u_{k}\in I\), \(\phi_{\frac{u_{j}+u_{k}}{2}}\in\Gamma\), for \(1\leq j,k\leq n\). As the linear combination of continuous functions from the set \(\Gamma\), this function is also continuous on \([\alpha,\beta]\). The fact that the mapping \(u\mapsto[x_{0},x_{1},x_{2}]\phi_{u}\) is \(n-\)exponentially convex in the Jensen sense on \(I\) implies that  
\[\rho_{1}^{2}f(x_{1})+2\rho_{1}\rho_{2}f\left(\frac{x_{1}+x_{2}}{2}\right)+ \rho_{2}^{2}f(x_{2})\geq 0.\]9  
for every three mutually different points \(x_{0},x_{1},x_{2}\in[\alpha,\beta]\), and this means that also our function \(\chi\) is convex on \([\alpha,\beta]\).