The kinetics of the E-Z-E isomerisation and liquid-crystalline properties of selected azobenzene derivatives investigated by the prism of the ester group inversion

Two new groups of azobenzene ester derivatives were synthesised: alkyl 4-[4-(nonyloxy)phenyl]diazenyl]benzoates and 4-[4-(nonyloxy)phenyl]diazenyl]phenyl alkanoates. All 35 presented homologues are mesogenic. Moreover, some of the above-mentioned compounds exhibit rich liquid-crystalline polymorphism likewise tetramorphism. During this investigation by the use of polarising optical microscopy, differential scanning calorimetry and X-Ray studies, six types of mesophases were detected: nematic, smectics (A, C, I, F) and G. Furthermore, due to the presence of the photosensitive azo moiety, the E–Z isomerisation reaction is possible. This process, which is initiated by the UV irradiation, causes significant changes in the UV-Vis absorption spectra of investigated compounds. However, the photoisomerisation is a reversible process and in the dark the thermal relaxation of Z isomer takes place. Based on the achieved data, the kinetic constants of the isomerisation and relaxation processes were calculated. It shows that conversion of the ester bond makes some changes in the optical properties. The shift of about 7 nm of the absorbance maximum was observed. Surprisingly, the inversion of the ester group has significant influence on the liquid-crystalline polymorphism replacing one mesophase (for benzoates) into four (for alkanoates).     

Linear maps preserving group majorization

A necessary and sufficient condition for a linear map to preserve group majorizations is given. The condition is applied to prove some preservation results.     

Fejér and Hermite–Hadamard type results for H-invex functions with applications

Positivity - We investigate the class of H-invex functions including, e.g., the subclasses of convex, c-strongly convex, $$ \varphi $$ -uniformly convex and superquadratic functions. For H-invex...     

Commutators and accretive operators

In this paper, singular values of commutators of Hilbert space operators are estimated. To this aim the accretivity of a transform of the operators is applied. Some recent results of Kittaneh [F. Kittaneh, Singular value inequalities for commutators of Hilbert space operators, Linear Algebra Appl. 430 (2009) 2362-2367] are extended.     

Generalized convex functions on normal decomposition systems

Aequationes mathematicae - In this paper, the class of weakly $$ \psi $$ -uniformly convex functions is introduced and studied. Such functions are defined on a convex subcone D of a linear space V...     

Bifractional inequalities and convex cones

In this paper we give a systematic study of a class of linear inequalities related to convex cones in linear spaces. In particular, Chebyshev and Andersson type inequalities are discussed. Some classical and new inequalities are derived from the results.     

Majorization and relative concavity

In this paper, some results established in [H.-N. Shi, Refinement and generalization of a class of inequalities for symmetric functions, Math. Practice Theory 29 (4) (1999) pp. 81-84] are extended from the classical majorization preordering to group-induced cone orderings. To this end the notion of relative concavity introduced in [C.P. Niculescu, F. Popovici, The extension of majorization inequalities within the framework of relative convexity, J. Inequal. Pure Appl. Math. 7 (1) (2005) (Article 27)] is used. In addition, some Ky Fan’s inequalities are discussed.     

Remarks on convex functions and separable sequences,II

In this paper we generalize results on convex sequences and convex functions due to Wu and Debnath [S. Wu, L. Debnath, Inequalities for convex sequences and their applications, Comput. Math. Appl. 54 (2007) 525–534] by using some results on similarly separable sequences established in Niezgoda [M. Niezgoda, Remarks on convex functions and separable sequences, Discrete Math. 308 (2008) 1765–1773].     

An extension of a result of Hwang and Pyo

In this paper, we extend a majorization result of Hwang and Pyo [LAA 332-334 (2001) pp. 15-21] from the ordinary majorization ordering to the class of group induced cone orderings induced by non-effective groups. The case of effective groups is also investigated. A particular attention is paid to positive operators.     

Some remarks on the Chebyshev functional

Identities are provided for the Chebyshev functional defined on a real linear space equipped with a bilinear form. In addition, some inequalities for the functional are discussed.