Authors:Ákos G. Horváth Pages: 401 - 418 Abstract: In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the n-dimensional volume of a regular simplex and the \((n-1)\) -dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for \(n=3\) . PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0465-4 Issue No:Vol. 91, No. 3 (2017)

Authors:Andrzej Olbryś Pages: 429 - 444 Abstract: In the present paper we introduce a notion of the \(\mathbb {K}\) -Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the \(\mathbb {K}\) -Riemann integral and the convexity notion is replaced by \(\mathbb {K}\) -convexity. PubDate: 2017-06-01 DOI: 10.1007/s00010-017-0472-0 Issue No:Vol. 91, No. 3 (2017)

Authors:El-sayed El-hady; Janusz Brzdęk; Hamed Nassar Pages: 445 - 477 Abstract: It is a survey on functional equations of a certain type, for functions in two complex variables, which often arise in queueing models. They share a common pattern despite their apparently different forms. In particular, they invariably characterize the probability generating function of the bivariate distribution characterizing a two-queue system and their forms depend on the composition of the underlying system. Unfortunately, there is no general methodology of solving them, but rather various ad-hoc techniques depending on the nature of a particular equation; most of the techniques involve advanced complex analysis tools. Also, the known solutions to particular cases of this type of equations are in general of quite involved forms and therefore it is very difficult to apply them practically. So, it is clear that the issues connected with finding useful descriptions of solutions to these equations create a huge area of research with numerous open problems. The aim of this article is to stimulate a methodical study of this area. To this end we provide a survey of the queueing literature with such two-place functional equations. We also present several observations obtained while preparing it. We hope that in this way we will make it easier to take some steps forward on the road towards a (more or less) general solving theory for this interesting class of equations. PubDate: 2017-06-01 DOI: 10.1007/s00010-017-0471-1 Issue No:Vol. 91, No. 3 (2017)

Authors:Mohamed Jleli; Donal O’Regan; Bessem Samet Pages: 479 - 490 Abstract: In this paper, some fractional integral inequalities involving m-convex functions are established. The presented results are generalizations of the obtained inequalities in Dragomir and Toader (Babeş-Bolyai Math 38:21–28, 1993). PubDate: 2017-06-01 DOI: 10.1007/s00010-017-0470-2 Issue No:Vol. 91, No. 3 (2017)

Authors:Teerapong Suksumran; Keng Wiboonton Pages: 491 - 503 Abstract: Möbius addition is defined on the complex open unit disk by $$\begin{aligned} a\oplus _M b = \dfrac{a+b}{1+\bar{a}b} \end{aligned}$$ and Möbius’s exponential equation takes the form \(L(a\oplus _M b) = L(a)L(b)\) , where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Möbius’s exponential equation is connected to Cauchy’s exponential equation. Möbius’s exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Möbius addition, considered as a nonassociative group-like structure. This suggests studying Schur’s lemma in a more general setting. PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0452-9 Issue No:Vol. 91, No. 3 (2017)

Authors:Tibor Kiss; Zsolt Páles Pages: 505 - 525 Abstract: It is well-known that if a real valued function acting on a convex set satisfies the n-variable Jensen inequality, for some natural number \(n\ge 2\) , then, for all \(k\in \{1,\dots , n\}\) , it fulfills the k-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the (M, N)-convexity property of functions and also for Hölder–Minkowski type inequalities. PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0459-2 Issue No:Vol. 91, No. 3 (2017)

Authors:Pentti Haukkanen Pages: 527 - 536 Abstract: It is well known that Euler’s totient function \(\phi \) satisfies the arithmetical equation \( \phi (mn)\phi ((m, n))=\phi (m)\phi (n)(m, n) \) for all positive integers m and n, where (m, n) denotes the greatest common divisor of m and n. In this paper we consider this equation in a more general setting by characterizing the arithmetical functions f with \(f(1)\ne 0\) which satisfy the arithmetical equation \( f(mn)f((m,n)) = f(m)f(n)g((m, n)) \) for all positive integers m, n with \(m,n \in A(mn)\) , where A is a regular convolution and g is an A-multiplicative function. Euler’s totient function \(\phi _A\) with respect to A is an example satisfying this equation. PubDate: 2017-06-01 DOI: 10.1007/s00010-017-0473-z Issue No:Vol. 91, No. 3 (2017)

Authors:S. S. Linchuk; Yu. S. Linchuk Pages: 537 - 545 Abstract: We solve generalized the generalized Rubel equation on the space of analytic functions in domains. PubDate: 2017-06-01 DOI: 10.1007/s00010-017-0467-x Issue No:Vol. 91, No. 3 (2017)

Authors:Daniel Eremita Pages: 563 - 578 Abstract: Let \(T_n(R)\) be the upper triangular matrix ring over a unital ring R. Suppose that \(B:T_n(R)\times T_n(R) \rightarrow T_n(R)\) is a biadditive map such that \(B(X,X)X = XB(X,X)\) for all \(X \in T_n(R)\) . We consider the problem of describing the form of the map \(X\mapsto B(X,X)\) . PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0462-7 Issue No:Vol. 91, No. 3 (2017)

Abstract: In this paper, we investigate the Hermite–Hadamard type inequality for the class of some h-convex stochastic processes, which is an extension of the Hermite–Hadamard inequality given by Barráez et al. (Math. Æterna 5:571–581, 2015). We also provide the estimates of both sides of the Hermite–Hadamard type inequality for h-convex stochastic processes, where h is any non-negative function with \(h(t)+h(1-t)\le 1\) for \(0\le t\le 1\) . PubDate: 2017-05-25

Abstract: In this paper we pose some questions about superderivations on \({\mathbb {Z}}_{2}\) -graded rings. Then we consider the quaternion rings and upper triangular matrix rings with special \({\mathbb {Z}}_{2}\) -gradings and we check the answer to these questions about them. PubDate: 2017-05-25

Abstract: The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in the case of homogeneous linear functional equations. The foundations of the theory can be found in Kiss and Varga (Aequat Math 88(1):151–162, 2014) and Kiss and Laczkovich (Aequat Math 89(2):301–328, 2015). We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to Koclȩga-Kulpa and Szostok (Ann Math Sylesianae 22:27–40, 2008), see also Koclȩga-Kulpa and Szostok (Georgian Math J 16:725–736, 2009; Acta Math Hung 130(4):340–348, 2011). They are motivated by quadrature rules of approximate integration. PubDate: 2017-05-25

Abstract: As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of [5]. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of \({\mathbb C}\) and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [8], see also [7, 9]. PubDate: 2017-05-24

Authors:Muaadh Almahalebi; Abdellatif Chahbi Abstract: In this paper, we present hyperstability results of Jensen functional equations in ultrametric Banach spaces. PubDate: 2017-05-19 DOI: 10.1007/s00010-017-0487-6

Authors:Marek Cezary Zdun Abstract: Let I be an interval. We consider the non-monotonic convex self-mappings \(f:I\rightarrow I\) such that \(f^2\) is convex. They have the property that all iterates \(f^n\) are convex. In the class of these mappings we study three families of functions possessing convex iterative roots. A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. A mapping f is said to be dyadically convex if for every \(n\ge 2\) there exists a convex iterative root \(f^{1/2^n}\) of order \(2^n\) and the sequence \(\{f^{1/2^n}\}\) satisfies the condition of compatibility, that is \( f^{1/2^n}\circ f^{1/2^n}= f^{1/2^{n-1}}.\) A function f is said to be flowly convex if it possesses a convex semi-flow of f, that is a family of convex functions \(\{f^t,t>0\}\) such that \(f^t\circ f^s=f^{t+s}, \ \ t,s >0\) and \(f^1=f\) . We show the relations among these three types of convexity and we determine all convex iterative roots of non-monotonic functions. PubDate: 2017-05-15 DOI: 10.1007/s00010-017-0483-x

Authors:Piotr Maćkowiak Abstract: In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014). PubDate: 2017-05-10 DOI: 10.1007/s00010-017-0491-x

Authors:Michael Schwarzenberger Abstract: In this note we determine the unique solution to the functional equation \(f(x + y) ( x- y) = \left( f(x) -f (y) \right) (x + y)\) . We require no additional assumptions on the function \(f\) . Moreover we solve this functional equation if \(f\) is only defined on a finite interval. The interest in this type of functional equation is motivated by the study of symmetrizing measures for (the generator of) a Lévy-driven Ornstein–Uhlenbeck process. PubDate: 2017-05-08 DOI: 10.1007/s00010-017-0484-9

Authors:Dijana Mosić Abstract: We study the generalization of Jacobson’s lemma for the group inverse, Drazin inverse, generalized Drazin inverse and pseudo Drazin inverse of \(1-bd\) (or \(1-ac\) ) in a ring when \(1-ac\) (or \(1-bd\) ) has a corresponding inverse, \(acd=dbd\) and \(bdb=bac\) (or \(dba=aca\) ). Thus, we recover some recent results. PubDate: 2017-04-04 DOI: 10.1007/s00010-017-0476-9

Authors:Jacek Chudziak; Zdeněk Kočan Abstract: We determine continuous solutions of the Goła̧b–Schinzel functional equation on cylinders. PubDate: 2017-03-27 DOI: 10.1007/s00010-017-0475-x

Authors:P. A. García-Sánchez; D. Llena; A. Moscariello Abstract: This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these results, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized. PubDate: 2017-03-14 DOI: 10.1007/s00010-017-0474-y