Sharp upper bounds for the Hankel, generalized Zalcman and Krushkal coefficient functionals are investigated for the class of normalized analytic functions f defined in an open unit disk with the quantity zf′ / f lying in the evolute of a nephroid curve in the right-half plane. The similar analysis is also carried out when the expression zf′/f is replaced with . Moreover, the sharp radii constants concerning the class are also computed.

The aim of this paper is to study lattice-like coverings with congruent translation balls and the packings and coverings with a type of translation cylinders in Sol space related to the fundamental lattices. We introduce the notions of the densities of the considered problems and give upper estimate to ball coverings using the radii and the volumes of the circumscribed translation spheres of given translation tetrahedra. Moreover we determine the exact optimal packing and covering densities of a type of cylinder packings belonging to the fundamental lattices.

Based on the mixed cone-volume measure, we introduce in this paper a new ellipsoid, called the mixed-sine Lutwak-Yang-Zhang (LYZ) ellipsoid, which serves as a sine counterpart to the mixed LYZ ellipsoid. Several related volume inequalities for this ellipsoid are also established.

Graphs serve as a bridge between algebra and numerous mathematical fields, including combinatorics, number theory, and computer science. Let Z(R) represent the set of zero-divisors in a commutative ring R. The zero-divisor graph of R, denoted by Γ(R), is an undirected graph with vertex set Z(R)∗ = Z(R) \\ {0}, where two vertices are adjacent if their product is zero. The Steiner tree and Steiner radial number are fundamental concepts in network optimization, offering cost-efficient strategies for connecting multiple points with minimal total distance. In this study, we compute the m-eccentricity of all vertices and determine the Steiner radial number of the zero-divisor graph of ℤ n . Additionally, for some positive integer m ≥ 2, we establish the existence of non-isomorphic zero-divisor graphs with Steiner radial number m. This work not only enhances the understanding of zero-divisor graphs but also provides a foundation for future studies in algebraic graph theory and network optimization.

We characterize the gradient Ricci-Yamabe soliton of a doubly warped product manifold and provide a classification for We further establish the conditions for the doubly warped product manifold to be a gradient Ricci-Yamabe soliton and demonstrate this result for a physical model of the universe, namely, for the doubly warped spacetime. Finally, we obtain a proper example for a gradient Ricci-Yamabe soliton for a doubly warped spacetime.

A Catalan word of length n that avoids the pattern (≥,≥) is a sequence w = w 1…wn with w 1 = 0 and 0 ≤ wi ≤ wi −1 + 1 for all i, while ensuring that no subsequence satisfies wi ≥ wi +1 ≥ wi +2 for i = 2, … , n. These words are enumerated by the n-th Motzkin number. From such a word, we associate a n-column Motzkin polyomino (called a (≥,≥)-polyomino), where the i-th column contains wi + 1 bottom-aligned cells. In this paper, we derive generating functions for (≥,≥)-polyominoes based on their length, area, semiperimeter, last symbol value, and number of interior points. We provide asymptotic analyses and closed-form expressions for the total area, total semiperimeter, sum of the last symbol values, and total number of interior points across all (≥,≥)-polyominoes of a given length. Finally, we express all these results as linear combinations of trinomial coefficients.

This paper investigates a class of Kirchhoff-type double phase problems distinguished by the inclusion of two nonlocal source terms and formulated in the framework of variable exponent Sobolev spaces. By applying a variant of Bonanno’s critical point theorem [12], we establish the existence of at least three distinct solutions. These results not only enhance the current understanding of double phase equations exhibiting sublinear growth but also extend several recent developments by incorporating variable exponents and nonlocal effects into the analysis.

In this study, we present a problem which consists quantum difference equation with matrix value impulsive conditions and eigenparameter dependent boundary condition. Our examination is based on the concept of main spectral and scattering properties of this problem. We introduce Jost solution, Jost function and scattering function of the problem. Then, we find resolvent operator, Green function, eigenvalues and continuous spectrum of the operator generated by same problem.

In his second notebook, Ramanujan discovered the following identity for the special values of ζ (s) at the odd positive integers where α and β are positive numbers such that αβ = π 2 and m is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler’s famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan’s identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.