The reverse Holder inequality for an elementary function

For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\in\mathbb R$ is defined by \smallskip\centerline{$\displaystyle\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),\qquad\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$} Assume that $0<A<B$, $0<\theta<1$ and consider the step function$g_{A<B,\theta}=B\cdot\chi_{[0,\theta)}+A\cdot\chi_{[\theta,1]}$, where $\chi_E$ is the characteristic function of the set $E$. Let $-\infty<p<q<+\infty$. The main result of this work consists in finding the term \smallskip\centerline{$\displaystyleC_{p<q,A<B}=\max\limits_{0\le\theta\le1}\frac{\|\,g_{A<B,\theta}\,\|_q}{\|\,g_{A<B,\theta}\,\|_p}.$} \smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $\theta_{p<q,A<B}$ with respect to $\beta=B/A\in(1,+\infty)$.The cases $p=0$ or $q=0$ are considered separately. The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].

Reduced Sombor index of bicyclic graphs

The concept of Sombor indices (SO) of a graph was recently introduced by Gutman and the reduced Sombor index [Formula: see text] of a graph [Formula: see text] is defined by [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text]. In this paper, we study the extremal properties of the reduced Sombor index and characterize the bicyclic graphs with extremal [Formula: see text]-value.

Radon Inversion Problem for Holomorphic Functions on Circular, Strictly Convex Domains

In this paper we study the so-called Radon inversion problem in bounded, circular, strictly convex domains with $${\mathcal {C}}^2$$ C 2 boundary. We show that given $$p>0$$ p > 0 and a strictly positive, continuous function $$\Phi $$ Φ on $$\partial \Omega $$ ∂ Ω , by use of homogeneous polynomials it is possible to construct a holomorphic function $$f \in {\mathcal {O}}(\Omega )$$ f ∈ O ( Ω ) such that $$\displaystyle \smallint _0^1 |f(zt)|^pdt = \Phi (z)$$ ∫ 0 1 | f ( z t ) | p d t = Φ ( z ) for all $$z \in \partial \Omega $$ z ∈ ∂ Ω . In our approach we make use of so-called lacunary K-summing polynomials (see definition below) that allow us to construct solutions with in some sense extremal properties.

Cactus Graphs with Maximal Multiplicative Sum Zagreb Index

A connected graph G is said to be a cactus if any two cycles have at most one vertex in common. The multiplicative sum Zagreb index of a graph G is the product of the sum of the degrees of adjacent vertices in G. In this paper, we introduce several graph transformations that are useful tools for the study of the extremal properties of the multiplicative sum Zagreb index. Using these transformations and symmetric structural representations of some cactus graphs, we determine the graphs having maximal multiplicative sum Zagreb index for cactus graphs with the prescribed number of pendant vertices (cut edges). Furthermore, the graphs with maximal multiplicative sum Zagreb index are characterized among all cactus graphs of the given order.

On product-one sequences over dihedral groups

Let [Formula: see text] be a finite group. A sequence over [Formula: see text] means a finite sequence of terms from [Formula: see text], where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over [Formula: see text] (with the concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This paper provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

The value of the high, low and close in the estimation of Brownian motion

The conditional density of Brownian motion is considered given the max, $$B(t|\max )$$ B ( t | max ) , as well as those with additional information: $$B(t|close, \max )$$ B ( t | c l o s e , max ) , $$B(t|close, \max , \min )$$ B ( t | c l o s e , max , min ) where the close is the final value: $$B(t=1)=c$$ B ( t = 1 ) = c and $$t \in [0,1]$$ t ∈ [ 0 , 1 ] . The conditional expectation and conditional variance of Brownian motion are evaluated subject to one or more of the statistics: the close (final value), the high (maximum), the low (minimum). Computational results displaying both the expectation and variance in time are presented and compared with the theoretical values. We tabulate the time averaged variance of Brownian motion conditional on knowing various extremal properties of the motion. The final table shows that knowing the high is more useful than knowing the final value among other results. Knowing the open, high, low and close reduces the time averaged variance to $$42\%$$ 42 % of the value of knowing only the open and close (Brownian bridge).

On the forgotten topological index of graphs

The forgotten topological index of a graph [Formula: see text] is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] and [Formula: see text] are the degrees of the vertices [Formula: see text] and [Formula: see text] in [Formula: see text], respectively. In this paper, we characterize the extremal properties of the F-index (forgotten topological index). We first introduce some graph transformations which increase or decrease this index. Furthermore, we will determine the extremal acyclic, unicyclic and bicyclic graphs with minimum and maximum of the F-index by a unified method, respectively. Recently, Akhter et al. [S. Akhter, M. Imran and M. R. Farahani, Extremal unicyclic and bicyclic graphs with respect to the F-index, AKCE Int. J. Graphs Comb. 14 (2017) 80–91] characterized the extremal graph of unicyclic and bicyclic graphs with minimum of the F-index. We will provide a shorter proof.