# Some Generalizations of the Jensen-Type Inequalities with Applications
## 4 Applications
\right.\]  
_As in the previous example, we have that functions \(u\mapsto F_{i}(g,\lambda,\phi_{u})\) (\(i=1,2,3\)) are exponentially convex, and that \(\mu_{u,v}(g,F_{i},\Gamma_{2})\) are monotone._  
_Using the Cauchy-type theorem from Section 3, applied for \(\phi=\phi_{u}\in\Gamma_{2}\) and \(\psi=\phi_{v}\in\Gamma_{2}\), we have that there exist_  
\[\xi_{i}\in[\alpha,\beta]\ \ \ (i=1,2,3),\]  
_such that_  
\[\xi_{i}^{u-v}=\frac{F_{i}(g,\lambda,\phi_{u})}{F_{i}(g,\lambda,\phi_{v})}.\]  
_As \(\xi\mapsto\xi^{u-v}\) is invertible for \(u\
eq v\), in that case we obtain_  
\[\alpha\leq\left(\frac{F_{i}(g,\lambda,\phi_{u})}{F_{i}(g,\lambda,\phi_{v})} \right)^{\frac{1}{u-v}}\leq\beta,\ \ \text{where}\ i=1,2,3.\]_As before, if we set \(Im(g)=[\alpha,\beta]\), we get_  
\[\alpha=\min_{t\in[a,b]}\left\{g(t)\right\}\leq\left(\frac{F_{i}(g,\lambda,\phi_{u })}{F_{i}(g,\lambda,\phi_{v})}\right)^{\frac{1}{2-v}}\leq\max_{t\in[a,b]}\left\{ g(t)\right\}=\beta \tag{27}\]  
_and that shows us that then \(\mu_{u,v}(g,F_{i},\Gamma_{2})\) are means of function \(g\)._  
_We can also add here one additional parameter, we will denote it by \(r\). In that case, for \(r\
eq 0\) we use the substitution \(g\to g^{r}\), \(u\rightarrow\frac{u}{r}\) and \(v\rightarrow\frac{v}{r}\) in (27), and we obtain_  
\[\min_{t\in[a,b]}\left\{(g(t))^{r}\right\}\leq\left(\frac{F_{i}(g^{r},\lambda, \phi_{u})}{F_{i}(g^{r},\lambda,\phi_{v})}\right)^{\frac{r}{u-v}}\leq\max_{t\in [a,b]}\left\{(g(t))^{r}\right\},\ \ \text{for}\ i=1,2,3.\]  
_We can define new generalized mean by_  
\[\mu_{u,v,r}(g,F_{i},\Gamma_{2})=\left\{ \begin{array}{ll}\left(\mu_{u,v,r} \left(g^{r},F_{i},\Gamma_{2}\right)\right)^{\frac{1}{r}},&r\
eq 0;\\ \mu_{u,v}(\log g,F_{i},\Gamma_{2}),&r=0.\end{array} \right.\]  
_Such means are also monotone. Namely, for \(u,v,y,z\in\mathbb{R}\), \(r\
eq 0\) such that \(u\leq y\), \(v\leq z\), we have_  
\[\mu_{u,v,r}(g,F_{i},\Gamma_{2})\leq\mu_{y,zr}(g,F_{i},\Gamma_{2})\,\ \ \text{where}\ \ \ i=1,2,3,\]  
_as_  
\[\mu_{\frac{u}{r},\frac{v}{r}}(g^{r},F_{i},\Gamma_{2})=\left(\frac{F_{i}(g^{r}, \lambda,\phi_{\frac{u}{r}})}{F_{i}(g^{r},\lambda,\phi_{\frac{v}{r}})}\right)^ {\frac{r}{u-v}}\leq\left(\frac{F_{i}(g^{r},\lambda,\phi_{\frac{v}{r}})}{F_{i} (g^{r},\lambda,\phi_{\frac{v}{r}})}\right)^{\frac{r}{r-z}}=\mu_{\frac{v}{r}, \frac{z}{r}}(g^{r},F_{i},\Gamma_{2}),\]  
_for \(u,v,y,z\in\mathbb{R}\), \(r\
eq 0\), such that \(\frac{u}{r}\leq\frac{v}{r},\frac{v}{r}\leq\frac{z}{r}\), and because \(\mu_{u,v}(g,F_{i},\Gamma_{2})\) for \(i=1,2,3\) are monotone in both parameters. The result when \(r=0\) can be derived by taking the limit \(r\to 0\)._  
**Example 3**. : _Let us define a family of functions_  
\[\Gamma_{3}=\left\{\theta_{u}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}:\,u\in \mathbb{R}^{+}\right\}\]  
_by_  
\[\theta_{u}(x)=\frac{e^{-x\sqrt{u}}}{u}.\]  
_We have that \(\frac{d^{2}}{dx^{2}}\theta_{u}(x)=e^{-x\sqrt{u}}>0\), and so for every \(u\in\mathbb{R}\) the function \(\theta_{u}\) is convex for \(x>0\)._ Remark 8_ gives us that \(u\mapsto\frac{d^{2}}{dx^{2}}\theta_{u}(x)\) is exponentially convex, and from [20] we then also have that \(u\mapsto[x_{0},x_{1},x_{2}]\theta_{u}\) is exponentially convex. This means that \(\Gamma_{3}\) fulfills the assumptions of Corollary 1._  
_If we set that \([\alpha,\beta]\subset\mathbb{R}^{+}\), we get_  
\[\mu_{u,v}(g,F_{i},\Gamma_{3})\ =\ \left\{ \begin{array}{ll}\left(\frac{F_{i}(g, \lambda,\theta_{u})}{F_{i}(g,\lambda,\theta_{v})}\right)^{\frac{1}{u-v}},&u\
eq v ;\\ \exp\Big{(}-\frac{F_{i}(g,\lambda,\theta_{u},\theta_{u})}{2\sqrt{u}F_{i}(g, \lambda,\theta_{u})}-\frac{1}{u}\Big{)},&u=v.\end{array}