# Some Generalizations of the Jensen-Type Inequalities with Applications
## 2 Main Results
In the case when \(\phi\in C^{2}([\alpha,\beta])\) we can use the representations (5)-(8), and we have  
\[\phi^{\prime}(x) =\phi^{\prime}(\beta)+\int_{a}^{\beta}(G_{1})^{\prime}_{x}(x,s) \phi^{\prime\prime}(s)ds, \tag{11}\] \[\phi^{\prime}(x) =\phi^{\prime}(\alpha)+\int_{a}^{\beta}(G_{2})^{\prime}_{x}(x,s) \phi^{\prime\prime}(s)ds,\] (12) \[\phi^{\prime}(x) =\phi^{\prime}(\alpha)+\int_{a}^{\beta}(G_{3})^{\prime}_{x}(x,s) \phi^{\prime\prime}(s)ds,\] (13) \[\phi(x)=\phi(\beta)+(x-\beta)\phi^{\prime}(\alpha)+\int_{\alpha}^{\beta}G_{2 }(x,s)\phi^{\prime\prime}(s)ds, \tag{6}\]0  
After some calculation, for every function \(G_{p}(p=1,2,3,4)\) we obtain the following relation  
\[\phi(x)=\phi(\beta)+(x-\beta)\phi^{\prime}(\alpha)+\int_{\alpha}^{\beta}G_{2 }(x,s)\phi^{\prime\prime}(s)ds, \tag{6}\]1 \[\phi(x)=\phi(\beta)+(x-\beta)\phi^{\prime}(\alpha)+\int_{\alpha}^{\beta}G_{2 }(x,s)\phi^{\prime\prime}(s)ds, \tag{6}\]2  
If \(\phi\) is convex function, then for every \(s\in[\alpha,\beta]\) we have that \(\phi^{\prime\prime}(s)\geq 0\). Further, if for every \(s\in[\alpha,\beta]\) holds (10), then the term in the square brackets in (15) is less then or equal to zero. That means that the left hand side of (15) also has to be less then or equal to zero, which means that for every continuous convex function \(\phi:[\alpha,\beta]\to\mathbb{R}\), such that \(\phi\in C^{2}([\alpha,\beta])\), holds inequality (9). After all, also notice that the existence of the second derivative of the function \(\phi\) is not necessary, because it is possible to uniformly approximate a continuous convex function by convex polynomials (see also [4], p. 172). We will skip the proof of the last sentence of our theorem since it can be conducted in the same way. **Remark 1**. : _Note that the condition \([m,M]\subseteq(\alpha,\beta)\) assures that the one sided derivatives are finite. If \(\phi^{\prime}_{+}(\alpha)\) and \(\phi^{\prime}_{-}(\beta)\) are finite, then we can also allow that \(Im(g)\) can be the whole interval \([\alpha,\beta]\)._  
**Remark 2**. : _The previous theorem considers the case when the function \(\phi\) is convex. Concluding in a similar way, we can also get the results that hold for concave function. Under the conditions of the previous theorem, we have the following._  
_For every continuous concave function \(\phi:[\alpha,\beta]\to\mathbb{R}\) holds (9) if and only if for every \(s\in[\alpha,\beta]\) holds (10) with reversed inequality sign._Also, for every continuous concave function \(\phi:[\alpha,\beta]\to\mathbb{R}\) holds (9) with reversed inequality sign if and only if for every \(s\in[\alpha,\beta]\) holds (10). Our next theorem states a similar result for the converse Jensen inequality, in the literature also known as the Lah-Ribaric or Edmundson-Lah-Ribaric inequality. Theorem 4.2: _Let \(g:[a,b]\to\mathbb{R}\) be a continuous function and let \(Im(g)\subseteq[m,M]\subseteq(\alpha,\beta)\). Suppose that \(\lambda:[a,b]\to\mathbb{R}\) is a continuous function or a function of bounded variation where \(\lambda(a)\
eq\lambda(b)\). For every continuous convex function \(\phi:[\alpha,\beta]\to\mathbb{R}\) holds_  
\[\phi(x)=\phi(\beta)+(x-\beta)\phi^{\prime}(\alpha)+\int_{\alpha}^{\beta}G_{2 }(x,s)\phi^{\prime\prime}(s)ds, \tag{6}\]3 \[\phi(x)=\phi(\beta)+(x-\beta)\phi^{\prime}(\alpha)+\int_{\alpha}^{\beta}G_{2 }(x,s)\phi^{\prime\prime}(s)ds, \tag{6}\]4  
_if and only if for every \(s\in[\alpha,\beta]\) holds_  
\[\phi(x)=\phi(\beta)+(x-\beta)\phi^{\prime}(\alpha)+\int_{\alpha}^{\beta}G_{2 }(x,s)\phi^{\prime\prime}(s)ds, \tag{6}\]5 \[\phi(x)=\phi(\beta)+(x-\beta)\phi^{\prime}(\alpha)+\int_{\alpha}^{\beta}G_{2 }(x,s)\phi^{\prime\prime}(s)ds, \tag{6}\]6  
_where the functions \(G_{p}\)\((p=1,2,3,4)\) are defined in (1)-(4)._  
_Moreover, this equivalence holds also in the case if we reverse the inequality sign in both (16) and (17)._  
Proof. : We can carry out this proof in the same way as the previous one. Let for every continuous convex function \(\phi:[\alpha,\beta]\to\mathbb{R}\) inequality (16) hold.