# Some Generalizations of the Jensen-Type Inequalities with Applications
## 4 Applications
The proofs for that can be obtained using_ Remark 10 _and adequate characterization of convexity._  
Now we will look at some families of functions that fulfill the assumptions of Theorem 8, Corollaries 1 and 2, and using them we will get some Cauchy-type means. **Example 1**. : _Let us define a family of functions_  
\[\Gamma_{1}=\{\psi_{u}:\mathbb{R}\rightarrow[0,+\infty):\;u\in\mathbb{R}\}\]  
_by_  
\[\psi_{u}(x)\;=\;\left\{ \begin{array}{ll}\frac{1}{u^{2}}\,e^{ux},&u\
eq 0 ;\\ \\ \frac{1}{2}\,x^{2},&u=0.\end{array} \right.\]  
_It is \(\frac{d^{2}}{dx^{2}}\psi_{u}(x)=e^{ux}>0\) for \(x\in\mathbb{R}\), and so for every \(u\in\mathbb{R}\) the function \(\psi_{u}\) is convex on \(\mathbb{R}\)._ Remark 8 _gives us that \(u\mapsto\frac{d^{2}}{dx^{2}}\psi_{u}(x)\) is exponentially convex, and from [20] we also have that \(u\mapsto[x_{0},x_{1},x_{2}]\psi_{u}\) is exponentially convex and therefore exponentially convex in the Jensen sense. That means that the family \(\Gamma_{1}\) of functions \(\psi_{u}\) fulfills the assumptions from_ Corollary 1_, and therefore we have that for \(i=1,2,3\) functions \(u\mapsto F_{i}(g,\lambda,\psi_{u})\) are exponentially convex in the Jensen sense. Although \(u\mapsto\psi_{u}\) is not continuous at \(u=0\), these functions are continuous, and it follows that they are exponentially convex._  
_Applying_ Corollary 2 _on \(\Gamma_{1}\), we obtain_  
\[\mu_{u,v}(g,F_{i},\Gamma_{1})\;=\;\left\{ \begin{array}{ll}\left(\frac{F_{i} (g,\lambda,\phi_{u})}{F_{i}(g,\lambda,\phi_{v})}\right)^{\frac{1}{u-v}},&u\
eq v ;\\ \exp\left(\frac{F_{i}(g,\lambda,id;\psi_{u})}{F_{i}(g,\lambda,\phi_{v})}-\frac {2}{u}\right),&u=v\
eq 0;\\ \exp\left(\frac{1}{3}\frac{F_{i}(g,\lambda,id;\psi_{0})}{F_{i}(g,\lambda,\phi_ {v})}\right),&u=v=0;\end{array} \right.\]  
_and from (24) we conclude that they are monotone in parameters \(u\) and \(v\)._Using the Cauchy-type theorem from Section 3, applied for \(\phi=\psi_{u}\in\Gamma_{1}\) and \(\psi=\psi_{v}\in\Gamma_{1}\), we get that for  
\[\mathcal{M}_{u,v}(g,F_{i},\Gamma_{1})\ =\ \log\mu_{u,v}(g,F_{i},\Gamma_{1})\ \ \ (i=1,2,3)\]  
holds  
\[\alpha\leq\mathcal{M}_{u,v}(g,F_{i},\Gamma_{1})\leq\beta,\ \ \text{for}\ i=1,2,3.\]  
If we set that \(Im(g)=[\alpha,\beta]\), we get that  
\[\alpha=\min_{t\in[a,b]}\{g(t)\}\leq\mathcal{M}_{u,v}(g,F_{i},\Gamma_{1})\leq \max_{t\in[a,b]}\{g(t)\}=\beta,\ \ \text{for}\ i=1,2,3,\]  
which means that then \(\mathcal{M}_{u,v}(g,F_{i},\Gamma_{1})\) are means of the function \(g\). From (24) we have that \(\mathcal{M}_{u,v}(g,F_{i},\Gamma_{1})\) are also monotone. **Example 2**. : _Let us define a family of functions_  
\[\Gamma_{2}=\{\phi_{u}:\mathbb{R}^{+}\rightarrow\mathbb{R}:u\in\mathbb{R}\}\]  
_by_  
\[\phi_{u}(x)\ =\ \left\{ \begin{array}{ll}\frac{x^{u}}{u(u-1)},&u\
eq 0,1;\\ -\log x,&u=0;\\ x\log x,&u=1.\end{array} \right.\]  
_We have that \(\frac{d^{2}}{dx^{2}}\phi_{u}(x)=x^{u-2}=e^{(u-2)\log x}>0\), and so for every \(u\in\mathbb{R}\) the function \(\phi_{u}\) is convex for \(x>0\). From Remark 8 we have that \(u\mapsto\frac{d^{2}}{dx^{2}}\phi_{u}(x)\) is exponentially convex, and from [20] we have that \(u\mapsto[x_{0},x_{1},x_{2}]\phi_{u}\) is exponentially convex and therefore exponentially convex in the Jensen sense. This means that \(\Gamma_{2}\) fulfills the assumptions from Corollary 1._  
_Now we will assume that \([\alpha,\beta]\) from_ Corollaries 1 _and 2 is a subset of \(\mathbb{R}^{+}\), and we obtain that_  
\[\mu_{u,v}(g,F_{i},\Gamma_{2})\ =\ \left\{ \begin{array}{ll}\left(\frac{F_{i}(g, \lambda,\phi_{u})}{F_{i}(g,\lambda,\phi_{v})}\right)^{\frac{1}{u-v}},&u\
eq v; \\ \exp\left(\frac{1-2u}{u(u-1)}-\frac{F_{i}(g,\lambda,\phi_{u})}{F_{i}(g,\lambda, \phi_{u})}\right),&u=v\
eq 0,1;\\ \exp\left(1-\frac{F_{i}(g,\lambda,\phi_{u})}{2F_{i}(g,\lambda,\phi_{u})}\right),&u=v=0;\\ \exp\left(\ -1-\frac{F_{i}(g,\lambda,\phi_{u}\phi_{1})}{2F_{i}(g,\lambda,\phi_{u})} \right),&u=v=1.\end{array}