Authors:Marek Niezgoda Pages: 1 - 20 Abstract: In this paper, by using Wu–Debnath’s method, we establish inequalities of Hammer–Bullen type for convex sequences and nondecreasing convex functions. In particular, we give an extension of a recent Farissi’s inequality. By assuming the symmetry of an involved sequence, we derive an inequality of Fejér–Hammer–Bullen type for convex sequences. In addition, we establish some companions of an Hermite–Hadamard like inequality by Latreuch and Belaïdi. In our approach we use matrix methods based on column stochastic matrices. PubDate: 2017-02-01 DOI: 10.1007/s00010-016-0444-9 Issue No:Vol. 91, No. 1 (2017)

Authors:Enrique Jordá; Antonio M. Peralta Pages: 99 - 114 Abstract: Let \({\Omega}\) be a compact Hausdorff space and let A be a C*-algebra. We prove that if every weak-2-local derivation on A is a linear derivation and every derivation on \({C(\Omega,A)}\) is inner, then every weak-2-local derivation \({\Delta:C(\Omega,A)\to C(\Omega,A)}\) is a (linear) derivation. As a consequence we derive that, for every complex Hilbert space H, every weak-2-local derivation \({\Delta : C(\Omega,B(H)) \to C(\Omega,B(H))}\) is a (linear) derivation. We actually show that the same conclusion remains true when B(H) is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if B denotes a compact C*-algebra (in particular, when \({B=K(H)}\) ), then every weak-2-local derivation on \({C(\Omega,B)}\) is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra M and every compact Hausdorff space \({\Omega}\) , every 2-local derivation on \({C(\Omega,M)}\) is a (linear) derivation. PubDate: 2017-02-01 DOI: 10.1007/s00010-016-0438-7 Issue No:Vol. 91, No. 1 (2017)

Authors:Ulrich Abel Pages: 115 - 120 Abstract: We present a generalization of Halphén’s formula on the higher order derivatives of \(f\left( 1/x\right) g\left( x\right) \) . Applications yield explicit representations of exponential partial Bell polynomials and Bessel polynomials. PubDate: 2017-02-01 DOI: 10.1007/s00010-016-0448-5 Issue No:Vol. 91, No. 1 (2017)

Authors:Monika Nowicka; Alfred Witkowski Pages: 121 - 128 Abstract: We establish a new refinement of the right-hand side of the Hermite–Hadamard inequality for simplices, based on the average values of a convex function over the faces of a simplex and over the values at their barycenters. PubDate: 2017-02-01 DOI: 10.1007/s00010-016-0433-z Issue No:Vol. 91, No. 1 (2017)

Authors:Feng Lü; Qi Han Pages: 129 - 136 Abstract: We study the non-existence of finite order meromorphic solutions to the Fermat-type difference equation \({f^3(z)+f^3(z+c)=1}\) over the complex plane C. An application to the unique range set problem is discussed. PubDate: 2017-02-01 DOI: 10.1007/s00010-016-0443-x Issue No:Vol. 91, No. 1 (2017)

Authors:Hamzeh Agahi; Milad Yadollahzadeh Pages: 153 - 159 Abstract: The concept of monotonic stochastic processes was introduced by Skowroński [Aequationes mathematicae 44 (1992) 249–258]. In this paper, we introduce the concept of comonotonic stochastic processes. As an application, we propose the well-known Chebyshev type inequality for two comonotonic stochastic processes via the generalized mean-square stochastic integral. PubDate: 2017-02-01 DOI: 10.1007/s00010-016-0442-y Issue No:Vol. 91, No. 1 (2017)

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

Authors:Andrzej Olbryś 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: 2017-03-10 DOI: 10.1007/s00010-017-0472-0

Authors:Pentti Haukkanen 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: 2017-03-07 DOI: 10.1007/s00010-017-0473-z

Authors:Mohamed Jleli; Donal O’Regan; Bessem Samet 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: 2017-02-28 DOI: 10.1007/s00010-017-0470-2

Authors:El-sayed El-hady; Janusz Brzdęk; Hamed Nassar 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: 2017-02-28 DOI: 10.1007/s00010-017-0471-1

Authors:Xiaowei Xu; Chen Li; Jianwei Zhu Abstract: Let n and s be integers such that \(1\le s<\frac{n}{2}\) , and let \(M_n(\mathbb {K})\) be the ring of all \(n\times n\) matrices over a field \(\mathbb {K}\) . Denote by \([\frac{n}{s}]\) the least integer m with \(m\ge \frac{n}{s}\) . In this short note, it is proved that if \(g:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) is a map such that \(g\left( \sum _{i=1}^{[\frac{n}{s}]}A_i\right) =\sum _{i=1}^{[\frac{n}{s}]}g(A_i)\) holds for any \([\frac{n}{s}]\) rank-s matrices \(A_1,\ldots ,A_{[\frac{n}{s}]}\in M_n(\mathbb {K})\) , then \(g(x)=f(x)+g(0)\) , \(x\in M_n(\mathbb {K})\) , for some additive map \(f:M_n(\mathbb {K})\rightarrow M_n(\mathbb {K})\) . Particularly, g is additive if \(char\mathbb {K}\not \mid \left( [\frac{n}{s}]-1\right) \) . PubDate: 2017-02-11 DOI: 10.1007/s00010-017-0469-8

Authors:Luis Barreira; Davor Dragičević; Claudia Valls Abstract: For the flow determined by a nonautonomous linear differential equation, we characterize the existence of a strong nonuniform exponential dichotomy in terms of the Fredholm property of a certain linear operator. We consider both cases of one-sided and two-sided exponential dichotomies. Moreover, we use the characterizations to establish the robustness of the notion of a strong nonuniform exponential dichotomy in a simple manner. PubDate: 2017-02-08 DOI: 10.1007/s00010-017-0468-9

Authors:Daniel Reem Abstract: This paper examines various aspects related to the Cauchy functional equation \(f(x+y)=f(x)+f(y)\) , a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation. PubDate: 2017-02-07 DOI: 10.1007/s00010-016-0463-6

Authors:S. S. Linchuk; Yu. S. Linchuk Abstract: We solve generalized the generalized Rubel equation on the space of analytic functions in domains. PubDate: 2017-02-04 DOI: 10.1007/s00010-017-0467-x

Authors:Boštjan Brešar; Sandi Klavžar; Douglas F. Rall; Kirsti Wash Abstract: The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(\Pi _1,\ldots ,\Pi _k\) , where \(\Pi _i\) , \(i\in [k]\) , is an i-packing. The following conjecture is posed and studied: if G is a subcubic graph, then \(\chi _{\rho }(S(G))\le 5\) , where S(G) is the subdivision of G. The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if G is a generalized prism of a cycle, then G is (1, 1, 2, 2)-colorable if and only if G is not the Petersen graph. The validity of the conjecture is further proved for graphs that can be obtained from generalized prisms in such a way that one of the two n-cycles in the edge set of a generalized prism is replaced by a union of cycles among which at most one is a 5-cycle. The packing chromatic number of graphs obtained by subdividing each of its edges a fixed number of times is also considered. PubDate: 2017-01-06 DOI: 10.1007/s00010-016-0461-8

Authors:Karol Baron; Janusz Morawiec Abstract: We study the problems of the existence, uniqueness and continuous dependence of Lipschitzian solutions \(\varphi \) of equations of the form $$\begin{aligned} \varphi (x)=\int _{\Omega }g(\omega )\varphi \big (f(x,\omega )\big )\mu (d\omega )+F(x), \end{aligned}$$ where \(\mu \) is a measure on a \(\sigma \) -algebra of subsets of \(\Omega \) and \(\int _{\Omega }g(\omega )\mu (d\omega )\!=\!1\) . PubDate: 2017-01-02 DOI: 10.1007/s00010-016-0455-6

Authors:Che Tat Ng; Hou Yu Zhao Abstract: Schauder’s fixed point theorem and the Banach contraction principle are used to study the polynomial-like iterative functional equation $$\begin{aligned} \lambda _1f(x)+\lambda _2f^2(x)+\cdots +\lambda _n f^n(x)=F(x). \end{aligned}$$ We give sufficient conditions for the existence, uniqueness, and stability of the periodic and continuous solutions. We examine the monotonicity, convexity, and differentiability of the solutions of the family \(2f(x)+\lambda f^2(x)=\sin (x)\) , ( \(\lambda \in [0,1]\) ). PubDate: 2017-01-02 DOI: 10.1007/s00010-016-0456-5

Authors:Lin Li Abstract: It is proved that every continuous iterative root of a piecewise monotone function with finite nonmonotonicity height, is an extension from a root on the characteristic interval. In this paper, a topological classification for those iterative roots is investigated. We first characterize all conjugacies between those piecewise monotone functions with finite nonmonotonicity height, and then determine a topological relation for their continuous iterative roots. PubDate: 2017-01-02 DOI: 10.1007/s00010-016-0457-4

Authors:Elqorachi Elhoucien Abstract: In this paper we study the solutions of the integral Van Vleck’s functional equation for the sine $$\begin{aligned} \int _{S}f(x\tau (y)t)d\mu (t)-\int _{S}f(xyt)d\mu (t) =2f(x)f(y),\; x,y\in S \end{aligned}$$ and the integral Kannappan’s functional equation $$\begin{aligned} \int _{S}f(xyt)d\mu (t)+\int _{S}f(x\tau (y)t)d\mu (t) =2f(x)f(y),\; x,y\in S, \end{aligned}$$ where S is a semigroup, \(\tau \) is an involution of S and \(\mu \) is a measure that is a linear combination of Dirac measures \((\delta _{z_{i}})_{i\in I}\) , such that for all \(i\in I\) , \(z_{i}\) is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of d’Alembert’s classic functional equation with involution. PubDate: 2016-12-05 DOI: 10.1007/s00010-016-0447-6