Bulletin of the Malaysian Mathematical Sciences Society

In the present paper, we study the semilinear elliptic problem $$\displaystyle -\Delta u -\mu \frac{u}{|y|^{2}}=\frac{|u|^{2^{*}(s)-2}u}{|y|^{s}}+ f(x,u)$$ in bounded domain. Replacing the Ambrosetti–Rabinowitz condition by general superquadratic assumptions and the nonquadratic assumption, we establish the existence results of positive solutions.

In this paper, we prove that the equation $$x^2-(p^{2k+2}+1)y^2=-p^{2l+1}$$ , $$l \in \{0,1,\dots ,k\}, k \ge 0$$ , where p is an odd prime number, is not solvable in positive integers x and y. By combining that result with other known results on the existence of Diophantine quadruples, we are able to prove results on the extensibility of some $$D(-1)$$ -pairs to quadruples in the ring $${\mathbb {Z}}[\sqrt{-t}], t>0$$ .

The aim of a study of the presented paper is the differential subordination involving harmonic means of the expressions $$p(z)$$ , $$p(z) + zp'(z)$$ , and $$p(z) + \frac{zp'(z)}{p(z)}$$ when $$p$$ is an analytic function in the unit disk, such that $$p(0)=1, p(z)\not \equiv 1$$ . Several applications in the geometric functions theory are given.

Let G be a simple graph with n vertices and m edges, and let $$q_{n}(G)$$ be the least signless Laplacian eigenvalue of G. Recently, Guo, Chen and Yu proved that if $$n\ge 6$$ , then $$\begin{aligned} q_{n}(G)\ge \frac{2m}{n-2}-n+1, \end{aligned}$$ which confirms a conjecture proposed by de Lima, Oliveira, de Abreu and Nikiforov. In this note, we present a new proof for the above inequality, where we also completely characterize the case when the equality holds.

Sarmanov family of bivariate distributions, which was suggested by Sarmanov (Mathematical models in hydrology symposium, 1974) as a new mathematical model of hydrological processes, is considered one of the most flexible and efficient extended families of the traditional FGM family. Despite its manifest advantages, it was never investigated in the literature. In this paper, we revisit this family by revealing several new prominent statistical properties. The distribution theory of concomitants of order statistics from this family is investigated. Besides, some aspects of information measures, namely the Shannon entropy, inaccuracy measure, and Fisher information number, are theoretically and numerically studied. Two bivariate real-world data sets have been analyzed for illustrative purposes, and the performance is quite satisfactory.

In this paper, based on combined Type-II censored samples, an exponential form for the underlaying distributions is considered for estimating the unknown parameters using the maximum likelihood and Bayesian methods. Three loss functions are used to derive the Bayesian estimators. Also, the Bayesian prediction for the order statistics from a future sample is calculated. Moreover, the results of the exponential and Pareto distributions are presented as illustrative examples. Finally, for illustrating all the inferential methods discussed in this paper, Monte Carlo simulation and data analysis for the exponential distribution are presented.