The restricted maximal operators of partial sums with respect to bounded Vilenkin systems are investigated. We derive the maximal subspace of positive numbers, for which this operator is bounded from the Hardy space H p {H-{p}} to the Lebesgue space L p {L-{p}} for all 0 < p ≤ 1 {0<p\leq 1}. We also prove that the result is sharp in a particular sense. 

Let Lambda(q)(omega), q > 0, denote the Lorentz space equipped with the (quasi) norm parallel to f parallel to(Lambda q(omega)) := (integral(1)(0) (f*(t)omega(t))(q)dt/t)(1/q) for a function integral on [0,1] and with omega positive and equipped with some additional growth properties. A generalization of Boas theorem in the form of a two-sided inequality is obtained in the case of both general regular system Phi = {phi(k)}(k=1)(infinity) and generalized Lorentz Lambda(q) (omega) spaces.

This second edition provides a thorough introduction to contemporary convex function theory with many new results. A large variety of subjects are covered, from the one real variable case to some of the most advanced topics.  The new edition includes considerably more material emphasizing the rich applicability of convex analysis to concrete examples.  Chapters 4, 5, and 6 are entirely new, covering important topics such as the Hardy-Littlewood-Pólya-Schur theory of majorization, matrix convexity, and the Legendre-Fenchel-Moreau duality theory.

This book can serve as a reference and source of inspiration to researchers in several branches of mathematics and engineering, and it can also be used as a reference text for graduate courses on convex functions and applications.

Convex functions and their relatives are ubiquitous in a large variety of applications such as optimization theory, mass transportation, mathematical economics, and geometric inequalities related to isoperimetric problems. This chapter is devoted to a succinct presentation of their theory in the context of real normed linear spaces, but most of the illustrations will refer to the Euclidean space RN,">RN,RN, the matrix space MN(R)">MN(R)MN(R) of all N&#x00D7;N">N×NN×N-dimensional real matrices (endowed with the Hilbert–Schmidt norm or with the operator norm), and the Lebesgue spaces Lp(RN)">Lp(RN)Lp(RN) with p&#x2208;[1,&#x221E;]">p∈[1,∞]p∈[1,∞].

The study of convex functions of one real variable offers an excellent glimpse of the beauty and fascination of advanced mathematics. The reader will find here a large variety of results based on simple and intuitive arguments that have remarkable applications. At the same time they provide the starting point of deep generalizations in the setting of several variables, that will be discussed in the next chapters.

The natural domain for a convex function is a convex set. In this chapter we review some basic facts, necessary for a deep understanding of the concept of convexity in real linear spaces. For reader’s convenience, all results concerning the separation of convex sets in Banach spaces are stated in Section 2.2 with proofs covering only the particular (but important) case of Euclidean spaces. Full details in the general case are to be found in Appendix  B

This chapter is aimed to offer a glimpse on the majorization theory and the beautiful inequalities associated to it. Introduced by G. H. Hardy, J. E. Littlewood, and G. Pólya (Messenger Math. 58:145–152, (1929), [208]) in 1929, and popularized by their celebrated book on Inequalities (Hardy et al., Inequalities, Cambridge University Press, 1952, [209]), the relation of majorization has attracted along the time a big deal of attention not only from the mathematicians, but also from people working in various other fields such as statistics, economics, physics, signal processing, data mining, etc. Part of this research activity is summarized in the 900 pages of the recent book by A. W. Marshall, I. Olkin, and B. Arnold (Inequalities: theory of majorization and its applications. Springer, New York (2011), [305]).

In this chapter we investigate three subjects concerning the convexity of functions defined on a space of matrices (or just on a convex subset of it). The first one is devoted to the convex spectral functions, that is, to the convex functions F:Sym(n,R)&#x2192;R">F:Sym(n,R)→RF:Sym(n,R)→R whose values F(A) depend only on the spectrum of A. The main result concerns their description as superpositions f&#x2218;&#x039B;">f∘Λf∘Λ between convex functions f:Rn&#x2192;R">f:Rn→Rf:Rn→R invariant under permutations, and the eigenvalues map &#x039B;">ΛΛ.

Convex optimization is one of the main applications of the theory of convexity and Legendre–Fenchel duality is a basic tool, making more flexible the approach of many concrete problems. The diet problem, the transportation problem, and the optimal assignment problem are among the many problems that during the Second World War and immediately after led L. Kantorovich, T. C. Koopmans, F. L. Hitchcock, and G. B. Danzig to develop the mathematical theory of linear programming. Soon it was realized that most results extend to the framework of convex functions, which marked the birth of convex programming. Later on, W. Fenchel, R. T. Rockafellar, and J. J. Moreau laid the foundations of convex analysis.

In this paper extensions and refinements of Hermite-Hadamard and Fejer type inequalities are derived including monotonicity of some functions related to the Fejer inequality and extensions for functions, which are 1-quasiconvex and for function with bounded second derivative. We deal also with Fejer inequalities in cases that p, the weight function in Fejer inequality, is not symmetric but monotone on [a, b] .