This book presents the new fascinating area of continuous inequalities. It was recently discovered that several of the classical inequalities can be generalized and given in a more general continuous/family form. The book states, proves and discusses a number of classical inequalities in such continuous/family forms. Moreover, since many of the classical inequalities hold also in a refined form, it is shown that such refinements can be proven in the more general continuous/family frame.

Written in a pedagogical and reader-friendly way, the book gives clear explanations and examples on how this technique works. The presented interplay between classical theory of inequalities and these newer continuous/family forms, including some corresponding open questions, will appeal to a broad audience of mathematicians and serve as a source of inspiration for further research.

This third edition presents an expanded and updated treatment of convex analysis methods, incorporating many new results that have emerged in recent years. These additions are essential for grasping the practical applications of convex function theory in solving contemporary real-world problems.

To reflect these advancements, the material has been meticulously reorganized, with a greater emphasis on topics relevant to current research. Additionally, great care has been taken to ensure that the text remains accessible to a broad audience, including both students and researchers focused on the application of mathematics.

Ideal for undergraduate courses, graduate seminars, or as a comprehensive reference, this book is an indispensable resource for those seeking to understand the extensive potential of convex function theory.

There is a lot of information available concerning Hardy-Hilbert type inequalities in one or more dimensions. In this paper we introduce the development of such inequalities on homogeneous groups. Moreover, we point out a unification of several of the Hardy-Hilbert type inequalities in the classical case to a general kernel case. Finally, we generalize these results to the homogeneous group case.

The Peetre K-functional is a key object in the development of the real method of interpolation. In this paper we point out a less known relation to wavelet theory and its applications to approximation theory and engineering applications. As a new basis for further development of these studies we present some known properties in the form appropriate for further applications and then derive new information and prove some new results concerning the K-functional and its close relation to (almost) quasi-monotone functions, various indices and interpolation theory. In particular, we extend and unify some known function parameter generalizations of the standard real interpolation spaces (A₀, A₁) _ᶿ,q.

In this paper we discuss an analogy of the Carleson-Hunt theorem with respect to Vilenkin systems. In particular, we investigate the almost everywhere convergence of Vilenkin-Fourier series of f∈Lp(Gm) for p>1 in case the Vilenkin system is bounded. Moreover, we state an analogy of the Kolmogorov theorem for p=1 and construct a function f∈L1(Gm) such that the partial sums with respect to Vilenkin systems diverge everywhere. 

Classical inequalities in a measure space are usually described by involving finitely many functions f1,…,fN, N=2,3,…. In this paper we give an introduction and several examples when such inequalities can be given with infinitely many functions fs involved, where index s can even be taken from another measure space. Such a development was also inspired when developing an interpolation between families (continuously many) of Banach spaces. 

A weighted multidimensional Cochran-Lee inequality is characterized in the frame of Hardy-type inequalities with parameters 0<p≤q<∞. Moreover, for the case p=q and power weights, even the sharp constant is derived, thus generalizing the original Cochran-Lee inequality to a multidimensional setting. As an application, new inequalities are pointed out. Finally, some related Hardy-type inequalities on homogeneous groups are presented.

In this paper we prove by the concrete construction that for any set E of measure zero there exists a function f∈Lp(G)(1≤p<∞) such that the Féjer means with respect to Wals system diverge on this set. The key is to use new constructions of Walsh polynomials, which was introduced in Persson et al. (Martingale Hardy Spaces and Summability of One-dimensional Vilenkin-Fourier Series. Birkhäuser/Springer, 2022). In fact, the theorem we prove follows from the general result of Karagulyan (Mat. Sb. 202(1):11–36, 2011), but we provide an alternative approach and the constructed function in our proof has a simpler explicit representation. 

The current status concerning Hardy-type inequalities with sharp constants is presented in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure dx with the Haar measure dx/x. We also present some new two-sided Hardy-type inequalities for monotone functions, where not only the two constants are sharp but also where the involved function spaces are optimal. As applications, a number of both well-known and new Hardy-type inequalities are pointed out. These results are used to present some new information concerning sharpness in the relation between different quasi-norms in Lorentz spaces.

We describe some chosen ideas and results for more than 100 years prehistory and history of the remarkable development concerning Hardy-type inequalities. In particular, we present a newer convexity approach, which we believe could partly have changed this development if Hardy had discovered it. In order to emphasize the current very active interest in this subject, we finalize by presenting some examples of the recent results, which we believe have potential not only to be of interest for a broad audience from a historical perspective, but also to be useful in various applications.