# Some Generalizations of the Jensen-Type Inequalities with Applications
## 3 Mean-Value Theorems  
Let \(g:[a,b]\to\mathbb{R}\) be a continuous function, \(Im(g)\subseteq[m,M]\subseteq(\alpha,\beta)\), and suppose that \(\lambda:[a,b]\to\mathbb{R}\) is a continuous function or a function of bounded variation with \(\lambda(a)\
eq\lambda(b)\). The previous three theorems and inequalities (9), (16) and (19) offer us the possibility of formulating new Lagrange-type and Cauchy-type mean value theorems, and these mean value theorems can give us a possibility to derive some new means. Firstly, for easier notation and formulation of these results, for functions \(g\) and \(\lambda\) and for continuous convex function \(\phi:[\alpha,\beta]\to\mathbb{R}\), we will define three functionals by subtraction of the right-hand from the left-hand side of inequalities (9), (16) and (19):  
\[A_{1}(g,\lambda,\phi)=\frac{\int_{a}^{b}\phi(g(x))d\lambda(x)}{ \int_{a}^{b}d\lambda(x)}-\phi(\overline{g})-\frac{(M-\overline{g})(\overline {g}-m)}{M-m}\big{(}\phi^{\prime}(M)-\phi^{\prime}(m)\big{)},\]  
\[A_{2}(g,\lambda,\phi)= \frac{M-\overline{g}}{M-m}\phi(m)+\frac{\overline{g}-m}{M-m}\phi (M)-\frac{\int_{a}^{b}\phi(g(x))d\lambda(x)}{\int_{a}^{b}d\lambda(x)}\] \[-\frac{(M-\overline{g})(\overline{g}-m)}{M-m}\big{(}\phi^{\prime }(M)-\phi^{\prime}(m)\big{)},\]  
\[A_{3}(g,\lambda,\phi)= \frac{\int_{a}^{b}\phi(g(x))d\lambda(x)}{\int_{a}^{b}d\lambda(x)} -\phi(\overline{g})\] \[-\max\bigg{\{}\frac{M-\overline{g}}{M-m},\frac{\overline{g}-m}{M -m}\bigg{\}}\cdot\bigg{[}\phi(m)+\phi(M)-2\phi\bigg{(}\frac{m+M}{2}\bigg{)} \bigg{]},\]  
with a remark that for \(A_{1}(g,\lambda,\phi)\) and \(A_{3}(g,\lambda,\phi)\) the value of \(\overline{g}\) has to be in \([\alpha,\beta]\). From Theorems 1-3 we obtain that:  
* \(A_{1}(g,\lambda,\phi)\leq 0\) if for any \(p\in\{1,2,3,4\}\) we have that for every \(s\in[\alpha,\beta]\) holds (10), and \(A_{1}(g,\lambda,\phi)\geq 0\) if for any \(p\in\{1,2,3,4\}\) we have that for every \(s\in[\alpha,\beta]\) holds (10) with reversed inequality sign;
* \(A_{2}(g,\lambda,\phi)\leq 0\) if for any \(p\in\{1,2,3,4\}\) we have that for every \(s\in[\alpha,\beta]\) holds (17), and \(A_{2}(g,\lambda,\phi)\geq 0\) if for any \(p\in\{1,2,3,4\}\) we have that for every \(s\in[\alpha,\beta]\) holds (17) with reversed inequality sign;
* \(A_{3}(g,\lambda,\phi)\leq 0\) if for any \(p\in\{1,2,3,4\}\) we have that for every \(s\in[\alpha,\beta]\) holds (20), and \(A_{3}(g,\lambda,\phi)\geq 0\) if for any \(p\in\{1,2,3,4\}\) we have that for every \(s\in[\alpha,\beta]\) holds (20) with reversed inequality sign. For each of these functionals we will now derive the Lagrange-type mean value theorem and after that also Cauchy-type mean value theorem. Theorem 3.: _Let \(g:[a,b]\to\mathbb{R}\) be a continuous function, \(Im(g)\subseteq[m,M]\subseteq(\alpha,\beta)\), and let \(\phi:[\alpha,\beta]\to\mathbb{R}\) be such that \(\phi\in C^{2}([\alpha,\beta])\). Suppose that \(\lambda:[a,b]\to\mathbb{R}\) is a continuous function or a function of bounded variation where \(\lambda(a)\
eq\lambda(b)\) and let \(\overline{g}\in[\alpha,\beta]\)._  
_If for any \(p\in\{1,2,3,4\}\) for every \(s\in[\alpha,\beta]\) holds (10) or if for any \(p\in\{1,2,3,4\}\) for every \(s\in[\alpha,\beta]\) holds (10) with reversed inequality sign, then there exists \(\xi\in[\alpha,\beta]\) such that_  
\[A_{1}(g,\lambda,\phi)=\frac{1}{2}\phi^{\prime\prime}(\xi)A_{1}(g,\lambda,\phi_ {0}),\]  
_where \(\phi_{0}(t)=t^{2}\)._Proof. : Suppose that for some \(p\in\{1,2,3,4\}\) for every \(s\in[\alpha,\beta]\) holds inequality (10), or holds (10) with reversed inequality sign, i.e., that for that \(p\in\{1,2,3,4\}\) the term  
\[\frac{\int_{a}^{b}G_{p}(g(x),s)d\lambda(x)}{\int_{a}^{b}d\lambda(x)}-G_{p}( \overline{g},s)-\frac{(M-\overline{g})(\overline{g}-m)}{M-m}\big{(}(G_{p})^{ \prime}_{x}(M,s)-(G_{p})^{\prime}_{x}(m,s)\big{)}\]  
retains the same sign on the whole \([\alpha,\beta]\).