arXiv subjects

Narmin N. Aliyeva

Publications and source records attributed to Narmin N. Aliyeva.

4 recordsLinked to original sources

arXiv2026

Equivalence of the minimality conditions for the root functions of Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

We study the minimality of the system of root functions associated with a Sturm--Liouville problem whose boundary condition depends linearly on the eigenparameter. Two different criteria for minimality were previously obtained using independent approaches. In this paper, we establish the equivalence of these criteria and provide a unified characterization of the exceptional cases in which the removal of certain associated functions fails to preserve minimality. The theoretical results are illustrated by several examples involving multiple eigenvalues, demonstrating the consistency of the two approaches and clarifying the structure of the corresponding root function systems.

math.CA↗

arXiv2026

Optimal bounds for the ratio of differences of quadratic, arithmetic, and harmonic means

We determine the optimal constants in inequalities comparing the differences of the quadratic, arithmetic, and harmonic means of $n$ nonnegative real numbers. Specifically, we prove that for $n\ge3$ the sharp double inequality \[ \frac{1}{\sqrt{n}}\le \frac{A_n-H_n}{Q_n-H_n}\le \sqrt{\frac{n-1}{n}} \] holds true. This extends earlier results by T. Mitev, which established the sharp bounds only for the cases $n=3,4,$ and $5$. Our approach is based on a variant of the classical optimization method of Cauchy and Maclaurin, in which a quadratic symmetric function is optimized under simultaneous constraints on the arithmetic and harmonic means.

math.CA↗

arXiv2026

Minimality of the root functions of Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

We consider a Sturm--Liouville problem in which the spectral parameter appears linearly in one of the boundary conditions. The study focuses on the root functions of the problem, including eigenfunctions and associated functions corresponding to multiple eigenvalues. By employing the characteristic function of the boundary value problem, explicit representations are obtained for the biorthogonal system and for several special associated functions that play a crucial role in the spectral analysis. These representations allow previously established criteria for the basis and minimality properties of the system of root functions to be reformulated directly in terms of the characteristic function and its derivatives at the eigenvalues. As a consequence, the investigation of particular boundary value problems becomes considerably simpler. Several illustrative examples are analyzed to demonstrate the effectiveness of the proposed approach and to show its agreement with known results in the literature.

math.CA↗

arXiv2026

Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

This paper studies a Sturm--Liouville boundary value problem in which one of the boundary conditions depends linearly on the spectral parameter. The differential equation is considered on the interval $(0,1)$ with a classical boundary condition at one endpoint and an eigenparameter--dependent boundary condition at the other. Explicit formulas for the inner products and norms of the root functions are obtained. These relations make it possible to analyze the structure of the system of root functions and the corresponding biorthogonal system. Using these results, the minimality of the system of root functions in $L_2(0,1)$ is established. Furthermore, the basis properties of the system of root functions in the spaces $L_p(0,1)$, $1<p<\infty$, are investigated. Necessary and sufficient conditions under which the system forms a basis are derived. Special attention is given to the cases of multiple eigenvalues and the case when the eigenvalue coincides with the critical value $-d/c$. The obtained results reveal a symmetry between different spectral cases and provide a simpler approach that avoids the use of the exit space $L_2(0,1) \oplus \mathbb{C}$. Several examples are presented to illustrate the theoretical results.

math.CA↗