This paper introduces the sequence spaces \mathrm{?_p} ( \mathrm{?_q} ) and \mathrm{?_?} ( \mathrm{?_q} ) for q ? (0, 1), generated by an infinite matrix \mathrm{?_q} derived from the q-Euler totient function \varphi_q . We establish key properties of \varphi_q and comprehensively analyze the new sequence spaces, detailing their Schauder bases, ?-, ?-, and ?-duals, matrix class characterizations, and geometric properties.
In the paper, with aid of the Maclaurin power series expansions of the hyperbolic sine and cosine functions, and in light of stratification method invented and applied by Malesevic and his coauthors, the authors establish several inequalities for bounding the sums of the hyperbolic functions sinhc(2x), tanhc x, and their reciprocals. These inequalities are extensions of the Huygens-type inequalities.
In this paper, we provide several new Mitrinovic-Adamovic-type inequalities and pose some open problems.
The adjugate matrix adj(G, x) of a simple graph G having adjacency matrix A is the adjugate of xI A. In this paper, we prove two new results on adj(G, x), one of which is a more general version of Jacobi's theorem that is in terms of the characteristic polynomials of G and those of the vertex-deleted subgraphs G u, G v and G u v. These two results on adj(G, x) are used to present several necessary and sufficient conditions for G to keep its original characteristic polynomial after performing an edge rotation.
This paper is devoted to Bernstein-type growth estimates for polynomials that do not vanish within a disk of positive radius. We derive several new supnorm bounds for the growth of higher-order derivatives of such polynomials in the complex plane. These results encompass generalizations of the well-known inequality of Ankeny and Rivlin (Pacific J. Math., 5: 849-852, 1955), along with extensions of significant inequalities in approximation theory, notably those obtained by Jain (Turk. J. Math., 31: 89-94, 2007).
In this paper, we present monotonicity rules for the ratio of two power series x (sic) Sigma(infinity)(k=0) a(k)x(k)/ Sigma(infinity)(k =0) b(k)x(k) under the assumption that the sequence {a(k)/b(k)}(k >= 0) changes monotonicity twice. We also prove that the number of monotonicity changes of this function does not exceed that of the sequence {a(k)/b(k)}(k >= 0). As an application, we establish a necessary and sufficient condition for the monotonicity of a function involving the confluent hypergeometric function of the first kind, x (sic) e(-x) /x+1 M(a, b, cx), where a, b, c > 0.
In this study, factorial series involving generalizations of the harmonic numbers are investigated. New expressions for the Dirichlet series (also called Euler sums) associated with hyperharmonic and skew-harmonic numbers are obtained. In addition, the relationships between the inverse factorial series of a given sequence and the inverse factorial series of the binomial, Stirling and Lah transformations of that sequence are investigated. Furthermore, closed-form formulas are derived for inverse factorial series whose coefficients are given by the p-Stirling numbers.
Dyck-type lattice paths, consisting of up and down diagonal steps on the integer lattice, can be either unconfined - if the only restriction is to remain in the non-negative half-plane - or confined, if, in addition, they are not allowed to pass above a fixed horizontal line. In this paper, we establish a formula for the number of unconfined Dyck-type lattice paths of a given length, and also investigate the enumeration of confined Dyck-type paths, deriving recurrence relations involving Vieta-Fibonacci polynomials.
We present two new fixed point theorems on proximity spaces using p-distance in this paper and compare them with the earlier results. Then, the existence and uniqueness of a fractional boundary value problem (FBVP) with Riemann-Liouville fractional derivatives have been established using one of these fixed point theorems.
In this manuscript, we focus on discrete dynamical systems and propose some important results according to their affine periodic solutions. We reconsider two landmark results, namely Massera's theorem and Floquet's theorem, for linear discrete dynamical systems with respect to affine-periodicity. We adapt affine periodicity in Floquet's theory in order to obtain an affine-periodic Floquet's decomposition. We also examine the existence of (Q, T)-affine periodic solutions for certain kinds of difference systems.
In the paper, the authors establish several inequalities for bounding the sums of sinc(2x), tanc x, and their reciprocals. This is achieved by using the Maclaurin power series expansions of the functions sin x and cos x, as well as the stratification method developed by Malesevic and his coauthors. These results provide a significant and nontrivial extension of the classical Huygens-type inequalities.
The ABC spectral radius of a graph G, denoted by rho(G), has attracted more and more attentions. In this paper, we prove that rho(G) <= root triangle + (2m n + 1)/triangle 2, where G is a connected graph with n vertices, m edges, and maximum degree triangle. As applications, we reproduce the upper bounds of ABC spectral radius for connected graphs, trees, unicyclic graphs, and bicyclic graphs. In addition, we determine the unique tree with second largest ABC spectral radius.
In this paper, we present new Shafer-Fink type inequalities for the arc sine and arc tangent functions. We develop Shafer's inequality for the arc hyperbolic tangent function to produce sharp two-sided inequalities, and present a new lower bound for the arc hyperbolic tangent function. Also, we present new sharp inequalities for the arc lemniscate functions.
This study investigates necessary and sufficient conditions to existence and uniqueness results for nonlinear system of mixed Psi-Riemann-Liouville and Phi-Caputo fractional derivatives for the sum of three component functions with coefficients of continuous functions. We use the properties of mixed monotone fixed points theorem to derive these results, which are supported by Banach fixed point theorems. The novelty of our work is that, in this paper, we study coupled system of mixed Psi-Riemann-Liouville and Phi-Caputo fractional derivatives involving novel results of mixed monotone fixed points theorem
In this paper, we introduce a new type degenerate Simsek numbers and their generating function, which are different from degenerate Simsek number studied so far. We establish the explicit formula, recurrence relation and other identities for these numbers. We also derive several interesting expressions and relations between these numbers and certain other special numbers in the literature. In addition, several numerical examples and graphical illustrations are provided to illustrate the behavior of the introduced numbers.
We present analysis of some new means introduced by M. Ra & imath;& uml;ssouli and A. Rezgui. We establish comparison relations and results on (K, N)-sub/superstabilizability. Assuming that means involved have asymptotic expansions, we present the complete asymptotic expansion of the resultant mean-map. As an application of the obtained asymptotic expansions and the asymptotic inequality between M and R(Bp, M, Bq), we show how to find the optimal parameters p and q for which M is (Bp, Bq)-sub/super-stabilizable.
In this paper will be presented certain infinite dimensional generalizations of two inequalities introduced by G. V. Milovanovic and I. Z. Milovanovic [A generalization of a problem given by D. S. Mitrinovic, Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. Fiz. 602?633, 129?132.], and some its applications.
A signed line graph of a simply signed graph extends the notion of a generalized line graph defined in the framework of ordinary graphs. In this paper, the Krausz theorem on covering characterization of line graphs and the Whitney theorem on isomorphism are extended to the context of signed line graphs.
For the hypergraphs among the set of the connected linear 3-uniform hypergraphs on n vertices without the Berge?Cl, we present two upper bounds for their spectral radius and ?-spectral radius, where l ? 5, n ? 3 and 0 ? ? < 1. In addition, for the hypergraphs among the set of the connected linear k-uniform hypergraphs on n vertices without the Berge?{Bs,K2,t}, we derive two upper bounds for their spectral radius and ?-spectral radius, where n, k ? 3, s ? 2, 1 ? t ? 1/2 (6k2 ? 15k + 10)(s ? 1) + 1, and 0 ? ? < 1.
In this paper, we provide a unified generalization of Cusa-Huygens and Mitrinovic-Adamovic inequalities with the parameters (alpha, beta, lambda), and give the conditions of (alpha, beta, lambda) such that the inequality and its reverse hold on (0, pi/2). As applications, several one-parameter generalizations of Cusa-Huygens and Mitrinovic-Adamovic inequalities will be presented, one of which solves an open problem posed by C.-P Chen and C. Mortici.
