Authors:Paweł Pasteczka Pages: 7 - 24 Abstract: In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013–2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow–Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance. PubDate: 2018-02-01 DOI: 10.1007/s00010-017-0513-8 Issue No:Vol. 92, No. 1 (2018)

Authors:Hamid Reza Moradi; Mohsen Erfanian Omidvar; Muhammad Adil Khan; Kazimierz Nikodem Pages: 25 - 37 Abstract: In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp\left( A \right) \subset \left( 1,\infty \right) \) , then $$\begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned}$$ and if \(Sp\left( A \right) \subset \left( 0,1 \right) \) , then $$\begin{aligned} \left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right) ,\quad 0<r<1 \end{aligned}$$ for each positive operator A and \(x\in \mathcal {H}\) with \(\left\ x \right\ =1\) . PubDate: 2018-02-01 DOI: 10.1007/s00010-017-0496-5 Issue No:Vol. 92, No. 1 (2018)

Authors:Josef Šlapal Pages: 75 - 90 Abstract: We introduce and discuss a connectedness induced by n-ary relations ( \(n>1\) an integer) on their underlying sets. In particular, we focus on certain n-ary relations with the induced connectedness allowing for a definition of digital Jordan curves. For every integer \(n>1\) , we introduce one such n-ary relation on the digital plane \({\mathbb {Z}}^2\) and prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For \(n=2\) , such a structure coincides with the (specialization order of the) Khalimsky topology and, for \(n>2\) , it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology. PubDate: 2018-02-01 DOI: 10.1007/s00010-017-0508-5 Issue No:Vol. 92, No. 1 (2018)

Authors:Jixia Yuan; Xiaomin Tang Pages: 91 - 109 Abstract: In this paper, we characterize super-biderivations of classical simple Lie superalgebras over the complex field \(\mathbb {C}\) . Furthermore, we prove that all super-biderivations of classical simple Lie superalgebras are inner super-biderivations. As an application, the super-biderivations of a general linear Lie superalgebra are studied. We find that there exist non-inner and non-super-skewsymmetric super-biderivations. Finally, using the results on biderivations we characterize linear super commuting maps on the classical simple Lie superalgebras and general linear Lie superalgebras. PubDate: 2018-02-01 DOI: 10.1007/s00010-017-0503-x Issue No:Vol. 92, No. 1 (2018)

Authors:Dawid Kotrys; Kazimierz Nikodem Pages: 111 - 122 Abstract: Characterizations of pairs of stochastic processes that can be separated by Jensen and by affine stochastic processes are presented. As a consequence, some stability results of the Hyers–Ulam-type are obtained. PubDate: 2018-02-01 DOI: 10.1007/s00010-017-0528-1 Issue No:Vol. 92, No. 1 (2018)

Authors:M. Kelbert; I. Stuhl; Y. Suhov Pages: 165 - 200 Abstract: Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes ‘weights’ of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting. We focus on properties of the optimal portfolios and discuss a number of simple examples extending the well-known Kelly betting scheme. An important restriction is that the investment does not exceed the current capital value and allows the trader to cover the worst possible losses. The paper deals with a class of discrete-time models. A continuous-time extension is a topic of an ongoing study. PubDate: 2018-02-01 DOI: 10.1007/s00010-017-0515-6 Issue No:Vol. 92, No. 1 (2018)

Authors:Bruce Ebanks Abstract: We continue the study of additive functions \(f_k:R\rightarrow F \;(1\le k\le n)\) linked by an equation of the form \(\sum _{k=1}^n p_k(x)f_k(q_k(x))=0\) , where the \(p_k\) and \(q_k\) are polynomials, R is an integral domain of characteristic 0, and F is the fraction field of R. A method is presented for solving all such equations. We also consider the special case \(\sum _{k=1}^n x^{m_k}f_k(x^{j_k})=0\) in which the \(p_k\) and \(q_k\) are monomials. In this case we show that if there is no duplication, i.e. if \((m_k,j_k)\ne (m_p,j_p)\) for \(k\ne p\) , then each \(f_k\) is the sum of a linear function and a derivation of order at most \(n-1\) . Furthermore, if this functional equation is not homogeneous then the maximal orders of the derivations are reduced in a specified way. PubDate: 2018-02-20 DOI: 10.1007/s00010-017-0537-0

Authors:Mohammad Reza Oboudi Abstract: Let \(k\ge 1\) and \(n_1,\ldots ,n_k\ge 1\) be some integers. Let \(S(n_1,\ldots ,n_k)\) be a tree T such that T has a vertex v of degree k and \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1},\ldots ,P_{n_k}\) , that is \(T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}\) so that every neighbor of v in T has degree one or two. The tree \(S(n_1,\ldots ,n_k)\) is called starlike tree, a tree with exactly one vertex of degree greater than two, if \(k\ge 3\) . In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if \(k\ge 4\) and \(n_1,\ldots ,n_k\ge 2\) , then \(\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}\) , where \(\lambda _1(T)\) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval \((-2,2)\) . PubDate: 2018-01-24 DOI: 10.1007/s00010-017-0533-4

Authors:Janusz Brzdęk; Zbigniew Leśniak; Renata Malejki Abstract: We study a generalization of the Fréchet functional equation, stemming from a characterization of inner product spaces. We show, in particular, that under some weak additional assumptions each solution of such an equation is additive. We also obtain a theorem on the Ulam type stability of the equation. In its proof we use a fixed point result to show the existence of an exact solution of the equation that is close to a given approximate solution. PubDate: 2018-01-24 DOI: 10.1007/s00010-017-0536-1

Authors:Anna Bahyrycz; Harald Fripertinger; Jens Schwaiger Abstract: In Baak et al. (J Math Anal Appl 314(1):150–161, 2006) the authors considered the functional equation $$\begin{aligned} r f\left( \frac{1}{r}\,\sum _{j=1}^{d}x_j\right)+ & {} \sum _{i(j)\in \{0,1\} \atop \sum _{1\le j\le d} i(j)=\ell }r f\left( \frac{1}{r}\,\sum _{j=1}^d (-1)^{i(j)}x_j\right) \\= & {} \left( {d-1\atopwithdelims ()\ell }-{d-1\atopwithdelims ()\ell -1} +1\right) \sum _{j=1}^{d} f(x_j) \end{aligned}$$ where \(d,\ell \in \mathbb {N}\) , \(1<\ell <d/2\) and \(r\in \mathbb {Q}{\setminus }\{0\}\) . The authors determined all odd solutions \(f:X\rightarrow Y\) for vector spaces X, Y over \(\mathbb {Q}\) . In Oubbi (Can Math Bull 60:173–183, 2017) the author considered the same equation but now for arbitrary real \(r\not =0\) and real vector spaces X, Y. Generalizing similar results from (J Math Anal Appl 314(1):150–161, 2006) he additionally investigates certain stability questions for the equation above, but as for that equation itself for odd approximate solutions only. The present paper deals with the general solution of the equation and the corresponding stability inequality. In particular it is shown that under certain circumstances non-odd solutions of the equation exist. PubDate: 2018-01-24 DOI: 10.1007/s00010-017-0534-3

Authors:Marcin Balcerowski Abstract: We show connections between generalized versions of some conditional Cauchy equation and its basic form. As a consequence we obtain the solution of the generalized equations in some classes of regular functions. PubDate: 2018-01-23 DOI: 10.1007/s00010-017-0510-y

Authors:Benjamin Hackl; Clemens Heuberger; Sara Kropf; Helmut Prodinger Abstract: Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only the leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results include exact expressions and asymptotic expansions for the expected value and the variance as well as central limit theorems. PubDate: 2018-01-23 DOI: 10.1007/s00010-017-0529-0

Authors:Gyula Maksa Abstract: In this paper, we give the solution of a problem formulated in Kominek and Sikorska (Aequationes Math 90:107–121, 2016) in connection with the functional equation $$\begin{aligned} f(xy)-f(x)-f(y)=g(x+y)-g(x)g(y). \end{aligned}$$ Our result can also be interpreted in the way that, under some additional condition, the logarithmic and the exponential Cauchy equations are strongly alien. PubDate: 2018-01-20 DOI: 10.1007/s00010-017-0535-2

Authors:G. Dolinar; B. Kuzma; N. Stopar Abstract: We classify maps which preserve orthogonality on the Cayley projective plane over octonions. In addition, we also classify orthogonality preserving maps on finite dimensional projective spaces over reals, complexes, or quaternions. Unlike similar results which extend Uhlhorns’s theorem we assume neither injectivity/surjectivity nor that orthogonality is preserved in both directions. PubDate: 2018-01-20 DOI: 10.1007/s00010-017-0532-5

Authors:M. H. Ahmadi Gandomani; M. J. Mehdipour Abstract: Let G be a locally compact abelian group, \(\omega \) be a weighted function on \({\mathbb {R}}^+\) , and let \(\mathfrak {D}\) be the Banach algebra \(L_0^\infty (G)^*\) or \(L_0^\infty (\omega )^*\) . In this paper, we investigate generalized derivations on the noncommutative Banach algebra \(\mathfrak {D}\) . We characterize \(\textsf {k}\) -(skew) centralizing generalized derivations of \(\mathfrak {D}\) and show that the zero map is the only \(\textsf {k}\) -skew commuting generalized derivation of \(\mathfrak {D}\) . We also investigate the Singer–Wermer conjecture for generalized derivations of \(\mathfrak {D}\) and prove that the Singer–Wermer conjecture holds for a generalized derivation of \(\mathfrak {D}\) if and only if it is a derivation; or equivalently, it is nilpotent. Finally, we investigate the orthogonality of generalized derivations of \(L_0^\infty (\omega )^*\) and give several necessary and sufficient conditions for orthogonal generalized derivations of \(L_0^\infty (\omega )^*\) . PubDate: 2018-01-15 DOI: 10.1007/s00010-017-0531-6

Authors:Z. Guerrero-Zarazua; J. Jerónimo-Castro Abstract: Consider a long, convex, homogenous cylinder with horizontal axis and with a planar convex body K as transversal section. Suppose the cylinder is immersed in water and let \(K_w\) be the wet part of K. In this paper we study some properties of the locus of the centroid of \(K_w\) and prove an analogous result to Klamkin–Flanders’ theorem when the locus is a circle. We also study properties of bodies floating at equilibrium when either the origin or the centroid of the body is pinned at the water line. In some sense this is the floating body problem for a density varying continuously. Finally, in the last section we give an isoperimetric type inequality for the perimeter of the centroid body (defined by C. M. Petty in Pacific J Math 11:1535–1547, 1961) of convex bodies in the plane. PubDate: 2018-01-12 DOI: 10.1007/s00010-017-0525-4

Authors:Martin Obst Abstract: Measuring angles in the Euclidean plane is a well-known topic, but for general normed planes there exists a variety of different concepts. These can be of a special kind, e.g. also preserving special orthogonality types. But these concepts are no angle measures in the sense of measure theory since they are not additive. This motivates us to define a new angle measure for normed planes that is in fact a measure in the sense of measure theory. Furthermore, we look at related types of rotation and reflection. PubDate: 2017-12-27 DOI: 10.1007/s00010-017-0526-3

Authors:Irina Asekritova; Yuri Karlovich; Natan Kruglyak Abstract: For \(p\in [1,\infty ]\) , we establish criteria for the one-sided invertibility of binomial discrete difference operators \({{\mathcal {A}}}=aI-bV\) on the space \(l^p=l^p(\mathbb {Z})\) , where \(a,b\in l^\infty \) , I is the identity operator and the isometric shift operator V is given on functions \(f\in l^p\) by \((Vf)(n)=f(n+1)\) for all \(n\in \mathbb {Z}\) . Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators \(A=aI-bU_\alpha \) on the Lebesgue space \(L^p(\mathbb {R}_+)\) for every \(p\in [1,\infty ]\) , where \(a,b\in L^\infty (\mathbb {R}_+)\) , \(\alpha \) is an orientation-preserving bi-Lipschitz homeomorphism of \([0,+\infty ]\) onto itself with only two fixed points 0 and \(\infty \) , and \(U_\alpha \) is the isometric weighted shift operator on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )\) . Applications of binomial discrete operators to interpolation theory are given. PubDate: 2017-12-19 DOI: 10.1007/s00010-017-0522-7

Authors:Károly Bezdek Abstract: Let \(\mathbb {M}^d\) denote the d-dimensional Euclidean, hyperbolic, or spherical space. The r-dual set of a given set in \(\mathbb {M}^d\) is the intersection of balls of radii r centered at the points of the a given set. In this paper we prove that for any set of given volume in \(\mathbb {M}^d\) the volume of the r-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser–Poulsen Conjecture states that if the centers of a family of N congruent balls in Euclidean d-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers, that is, when the pairwise distances of the two sets are separated by some positive real number. We prove a special case of the Kneser–Poulsen conjecture namely, we prove the conjecture for uniform contractions (with sufficiently large N) in \(\mathbb {M}^d\) . PubDate: 2017-11-23 DOI: 10.1007/s00010-017-0516-5

Authors:Mohammad Hossein Ghaffari; Zohreh Mostaghim Abstract: The Paul Erdős and András Gyárfás conjecture states that every graph of minimum degree at least 3 contains a simple cycle whose length is a power of two. In this paper, we prove that the conjecture holds for Cayley graphs on generalized quaternion groups, dihedral groups, semidihedral groups and groups of order \(p^3\) . PubDate: 2017-11-21 DOI: 10.1007/s00010-017-0518-3