Superadditivity and monotonicity of the Jensen-type functionals. New methods for improving the Jensen-type inequalities in real and in operator cases.

*(English)*Zbl 1358.26001
Monographs in Inequalities 11. Zagreb: Element (ISBN 978-953-197-599-5/pbk). xi, 367 p. (2016).

The book contains eleven chapters: 1) Basic notation and fundamental results, 2) On Jessen’s and McShane’s functionals, 3) Jensen-type functionals under the Steffensen’s conditions. Petrović type functionals, 4) Some further improvements. Levinson’s functional, 5) Different approaches to superadditivity, 6) Jensen-type functionals for the operators on a Hilbert space, 7) Improvements of some matrix and operator inequalities via the Jensen functional, 8) The converse Jensen inequality: variants, improvements and generalizations, 9) Further improvements and generalizations of the Jessen-Mercer inequality, 10) New improved forms of the Hermite-Hadamard-type inequalities, 11) On the refinements of the Jensen operator inequality.

The authors deduce the Jensen inequality considering the Jensen-type functionals which occur subtracting the left-hand side from the right-hand side of the considered inequality. Let us describe this idea on one particular functional – on Jessen’s functional. In the second chapter, the following theorem for Jessen’s functional \[ J(\Phi, f, p; A)=A(p\Phi(f))-A(p)\Phi\left( \frac{A(pf)}{A(p)} \right) \] is proved:

Let \(A:L\rightarrow \mathbb{R} \) be a positive linear functional. Suppose \(f\in L\) and \(p,q \in L^+\). If \(\Phi :I \rightarrow \mathbb{R} \), \(I\subseteq \mathbb{R}\), is a continuous and convex function, then \[ J(\Phi, f, p+q; A) \geq J(\Phi, f, p; A) +J(\Phi, f, q; A), \] that is, \(J(\Phi, f, \cdot ; A)\) is superadditive on \(L^+\). Moreover, if \(p,q \in L^+\) are such that \( p\geq q\), then \[ J(\Phi, f, p; A) \geq J(\Phi, f, q; A)\geq 0, \] that is, \(J(\Phi, f, \cdot ; A)\) is increasing on \(L^+\).

An immediate consequence of the above result is a corollary which provides lower and upper bounds for Jessen’s functional which are expressed by means of the non-weight functional of the same type.

Let function \(f\) and functional \(A\) be as in the above theorem. Suppose \(p\in L^+\) attains its minimal and maximal value on \(E\). If \(\Phi :I \rightarrow \mathbb{R} \), \(I\subseteq \mathbb{R}\), is a continuous and convex function, then \[ \left[ \min_{x\in E} p(x) \right] J(\Phi, f, 1 ; A) \leq J(\Phi, f, p ; A) \leq \left[ \max_{x\in E} p(x) \right] J(\Phi, f, 1 ; A), \] where \(J(\Phi, f, 1 ; A) = A(\Phi(f)\cdot 1)-A(1) \Phi\left( \frac{A(f)}{A(1)}\right)\).

In the same chapter, similar results for McShane’s functional are given. Also, applications to weight generalized means and to Hölder’s inequality are obtained. In a similar manner, the third chapter deals with Jensen-Steffensen, Jensen-Mercer and Petrović-type functionals in their discrete and integral forms. In the fourth chapter, further refinements of Jessen’s inequality are discussed together with applications to the Levinson functional. The fifth chapter is devoted to investigation of quasilinearity of the functional \((h \circ v)\cdot \left( \Phi \circ \frac gv \right)\) where \(\Phi\) is a monotone \(h\)-concave (\(h\)-convex) function and \(v\) and \(g\) are functionals with certain super(sub)additive properties. General results are applied to some special functionals generated with several inequalities such as the Jensen, the Jensen-Mercer, the Beckenbach, the Chebyshev and the Milne inequality.

In the sixth chapter, one finds transition from the domain of real analysis to the domain of the functional analysis. So, arguments of the Jensen-type functionals are bounded self-adjoint operators on a Hilbert space. Also, Jensen’s integral operator inequality with a correspondingly defined functional is studied as well as the multidimensional Jensen’s functional for operators with some applications to connections, solidarities and multidimensional weight geometric means. In the following chapter, several refinements of Heinz norm inequalities are derived by virtue of convexity of Heinz means and with the help of Jensen’s functional. Also, some improved weak majorization relations and eigenvalue inequalities for matrix versions of Jensen’s inequality are derived.

In the eighth and ninth chapters due to monotonicity property of certain functionals, the authors give various variants of the converse Jensen inequality motivated by the Lah-Ribarič and the Giaccardi-Petrović inequality. Several families of \(n\)-exponentially convex and exponentially convex functions are constructed. Improvements and generalizations of the Jessen-Mercer’s inequality are given. The tenth chapter is devoted to improvements of various forms of the Hermite-Hadamard inequality such as Fejer, Lupaş, Brenner-Alzer, Beesack-Pečarić inequality. These improvements are given in terms of positive linear functionals and obtained by means of the monotonicity property of correspondingly functionals. In the last chapter, several refinements of Jensen’s operator inequality are presented, for \(n\)-tuples of self-adjoint operators, unital \(n\)-tuples of positive linear mappings and real-valued continuous convex functions with the condition on the spectra of the operators.

The authors deduce the Jensen inequality considering the Jensen-type functionals which occur subtracting the left-hand side from the right-hand side of the considered inequality. Let us describe this idea on one particular functional – on Jessen’s functional. In the second chapter, the following theorem for Jessen’s functional \[ J(\Phi, f, p; A)=A(p\Phi(f))-A(p)\Phi\left( \frac{A(pf)}{A(p)} \right) \] is proved:

Let \(A:L\rightarrow \mathbb{R} \) be a positive linear functional. Suppose \(f\in L\) and \(p,q \in L^+\). If \(\Phi :I \rightarrow \mathbb{R} \), \(I\subseteq \mathbb{R}\), is a continuous and convex function, then \[ J(\Phi, f, p+q; A) \geq J(\Phi, f, p; A) +J(\Phi, f, q; A), \] that is, \(J(\Phi, f, \cdot ; A)\) is superadditive on \(L^+\). Moreover, if \(p,q \in L^+\) are such that \( p\geq q\), then \[ J(\Phi, f, p; A) \geq J(\Phi, f, q; A)\geq 0, \] that is, \(J(\Phi, f, \cdot ; A)\) is increasing on \(L^+\).

An immediate consequence of the above result is a corollary which provides lower and upper bounds for Jessen’s functional which are expressed by means of the non-weight functional of the same type.

Let function \(f\) and functional \(A\) be as in the above theorem. Suppose \(p\in L^+\) attains its minimal and maximal value on \(E\). If \(\Phi :I \rightarrow \mathbb{R} \), \(I\subseteq \mathbb{R}\), is a continuous and convex function, then \[ \left[ \min_{x\in E} p(x) \right] J(\Phi, f, 1 ; A) \leq J(\Phi, f, p ; A) \leq \left[ \max_{x\in E} p(x) \right] J(\Phi, f, 1 ; A), \] where \(J(\Phi, f, 1 ; A) = A(\Phi(f)\cdot 1)-A(1) \Phi\left( \frac{A(f)}{A(1)}\right)\).

In the same chapter, similar results for McShane’s functional are given. Also, applications to weight generalized means and to Hölder’s inequality are obtained. In a similar manner, the third chapter deals with Jensen-Steffensen, Jensen-Mercer and Petrović-type functionals in their discrete and integral forms. In the fourth chapter, further refinements of Jessen’s inequality are discussed together with applications to the Levinson functional. The fifth chapter is devoted to investigation of quasilinearity of the functional \((h \circ v)\cdot \left( \Phi \circ \frac gv \right)\) where \(\Phi\) is a monotone \(h\)-concave (\(h\)-convex) function and \(v\) and \(g\) are functionals with certain super(sub)additive properties. General results are applied to some special functionals generated with several inequalities such as the Jensen, the Jensen-Mercer, the Beckenbach, the Chebyshev and the Milne inequality.

In the sixth chapter, one finds transition from the domain of real analysis to the domain of the functional analysis. So, arguments of the Jensen-type functionals are bounded self-adjoint operators on a Hilbert space. Also, Jensen’s integral operator inequality with a correspondingly defined functional is studied as well as the multidimensional Jensen’s functional for operators with some applications to connections, solidarities and multidimensional weight geometric means. In the following chapter, several refinements of Heinz norm inequalities are derived by virtue of convexity of Heinz means and with the help of Jensen’s functional. Also, some improved weak majorization relations and eigenvalue inequalities for matrix versions of Jensen’s inequality are derived.

In the eighth and ninth chapters due to monotonicity property of certain functionals, the authors give various variants of the converse Jensen inequality motivated by the Lah-Ribarič and the Giaccardi-Petrović inequality. Several families of \(n\)-exponentially convex and exponentially convex functions are constructed. Improvements and generalizations of the Jessen-Mercer’s inequality are given. The tenth chapter is devoted to improvements of various forms of the Hermite-Hadamard inequality such as Fejer, Lupaş, Brenner-Alzer, Beesack-Pečarić inequality. These improvements are given in terms of positive linear functionals and obtained by means of the monotonicity property of correspondingly functionals. In the last chapter, several refinements of Jensen’s operator inequality are presented, for \(n\)-tuples of self-adjoint operators, unital \(n\)-tuples of positive linear mappings and real-valued continuous convex functions with the condition on the spectra of the operators.

Reviewer: Sanja Varošanec (Zagreb)

##### MSC:

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

26B25 | Convexity of real functions of several variables, generalizations |

26D07 | Inequalities involving other types of functions |

26D10 | Inequalities involving derivatives and differential and integral operators |