On a paper of Erdös and Szekeres

Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products\(\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}\) integers. In the discussion below, {a₁,..., a_n} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.     

On a question of H. Kraljević

In this paper, assuming a certain set-theoretic hypothesis, a positive answer is given to a question of H. Kraljevi, namely it is shown that there exists a Lebesgue measurable subsetA of the real line such that the set {c R: A + cA contains an interval} is nonmeasurable. Here the setA + cA = {a + ca: a, a A}. Two other results about sets of the formA + cA are presented.     

On a conjecture of the Randić index and the minimum degree of graphs

The Randi? index $R\left(G\right)$ of a graph $G$ is defined by $R\left(G\right)={\sum }_{uv}\frac{1}{\sqrt{d\left(u\right)d\left(v\right)}}$, where $d\left(u\right)$ is the degree of a vertex $u$ and the summation extends over all edges $uv$ of $G$. Delorme et al. (2002)  [6] put forward a conjecture concerning the minimum Randi? index among all$n$-vertex connected graphs with the minimum degree at least $k$. In this work, we show that the conjecture is true given the graph contains $k$ vertices of degree $n?1$. Further, it is true among $k$-trees.     

Corrections and complements to my paper “On a class of operator monotone functions of several variables”

Mathematical Notes -     

On a Result of Pandey and Bhargava

Let S denote the class of functions f(z) which are analytic in the unit disc D={z:|z|<1} and normalized by the conditions f(0)=0=f′(0)-1. Let K(μ)and S (m,M ) be the subclasses of S and satisfying the conditions     

Properties of a Subclass of Avakumović Functions and Their Generalized Inverses

We study properties of a subclass of ORV functions introduced by Avakumovi and provide their applications for the strong law of large numbers for renewal processes.     

On quasi-contraction mappings of Ćirić and Fisher type via ω-distance

Kada, Suzuki, and Takahashi introduced and studies the concept of ω- distance in fixed point theory. In this paper, we generalize and unify ?iri?’ and Fisher fixed points results for quasi-contractions on metric space to ω-distance on complete metric spaces. We also extend some results of Kada, Suzuki and Takahashi, and Suzuki. Our methods of proofs are new and even simpler than the corresponding methods in metric spaces.     

On Sums of Two Squares of Primes and a Cube of a Prime

In this paper we are concerned with the exceptional set of the sum of two squares of primesand a cube of a prime P;+p;+p;.Noting that竹三1 or 3(mod 6)is a necessary conditionfor the solvability of the equation n=P}+P;+P;(see【1]),we define E(N)=Card{n:n≤N,礼∈三and n≠P;+Pi+P;for any Pi,1≤i s 3), (1)where三={n:n三1 or 3(mod 6)). This and similar problems have been studied by a number of authors.In 1937,Davenportand Heilbronn[2J proved that if后2 3 is an odd integer then almost all posi…     

On Sums of a Prime, and a Square of Prime, and a κ-power of Prime

In 1923, Hardy and Littlewood[1] conjectured that each integer n can be written asp+m12+ m22 = n,and Linnik[2,3] proved that this conjecture is true. But if these mi with i = 1,2 are restricted to primes Pi, the corresponding result is out of reach at present. We consider the following Diophantine equation     

On a problem of Erdős and Moser

A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\), \(B\in A^{(r-1)}\) is in \(\mathcal H\). Erd?s and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.     

On a Question of Erdős and Ulam

Ulam asked in 1945 if there is an everywhere dense rational set, i.e., 1 a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős’ conjecture for algebraic curves by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.