Hankel, Toeplitz, and Hermitian-Toeplitz Determinants for Certain Close-to-convex Functions

Let f be analytic in $$\mathbb {D}=\{z\in \mathbb {C}:|z|<1\}$$ D = { z ∈ C : | z | < 1 } , and be given by $$f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$$ f ( z ) = z + ∑ n = 2 ∞ a n z n . We give sharp bounds for the second Hankel determinant, some Toeplitz, and some Hermitian-Toeplitz determinants of functions in the class of Ozaki close-to-convex functions, together with a sharp bound for the Zalcman functional $$J_{2,3}(f).$$ J 2 , 3 ( f ) .

Finite groups with a nilpotency condition

Let [Formula: see text] and [Formula: see text] be positive integer numbers. In this paper, we study [Formula: see text], the class of all groups [Formula: see text] that for all subsets [Formula: see text] and [Formula: see text] of [Formula: see text] containing [Formula: see text] and [Formula: see text] elements, respectively, there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] is nilpotent, which introduced by Zarrin in 2012. We improve some results of Zarrin and find some sharp bounds for [Formula: see text] and [Formula: see text] such that [Formula: see text] implies that [Formula: see text] is nilpotent. Also we will characterize all finite [Formula: see text]-groups in [Formula: see text], which [Formula: see text].

Fractional Metric Dimension of Generalized Sunlet Networks

Let N = V N , E N be a connected network with vertex V N and edge set E N ⊆ V N , E N . For any two vertices a and b , the distance d a , b is the length of the shortest path between them. The local resolving neighbourhood (LRN) set for any edge e = ab of N is a set of all those vertices whose distance varies from the end vertices a and b of the edge e . A real-valued function Φ from V N to 0,1 is called a local resolving function (LRF) if the sum of all the labels of the elements of each LRN set remains greater or equal to 1. Thus, the local fractional metric dimension (LFMD) of a connected network N is dim lf N = min Φ : Φ is minimal LRF of N . In this study, LFMD of various types of sunlet-related networks such as sunlet network ( S m ), middle sunlet network ( MS m ), and total sunlet network ( TS m ) are studied in the form of exact values and sharp bounds under certain conditions. Furthermore, the unboundedness and boundedness of all the obtained results of LFMD of the sunlet networks are also checked.

Second Hankel Determinant of Logarithmic Coefficients of Convex and Starlike Functions of Order Alpha

In the present paper, we found sharp bounds of the second Hankel determinant of logarithmic coefficients of starlike and convex functions of order $$\alpha $$ α .

MOMENTS AND HYBRID SUBCONVEXITY FOR SYMMETRIC-SQUARE L-FUNCTIONS

We establish sharp bounds for the second moment of symmetric-square L-functions attached to Hecke Maass cusp forms $u_j$ with spectral parameter $t_j$ , where the second moment is a sum over $t_j$ in a short interval. At the central point $s=1/2$ of the L-function, our interval is smaller than previous known results. More specifically, for $\left \lvert t_j\right \rvert $ of size T, our interval is of size $T^{1/5}$ , whereas the previous best was $T^{1/3}$ , from work of Lam. A little higher up on the critical line, our second moment yields a subconvexity bound for the symmetric-square L-function. More specifically, we get subconvexity at $s=1/2+it$ provided $\left \lvert t_j\right \rvert ^{6/7+\delta }\le \lvert t\rvert \le (2-\delta )\left \lvert t_j\right \rvert $ for any fixed $\delta>0$ . Since $\lvert t\rvert $ can be taken significantly smaller than $\left \lvert t_j\right \rvert $ , this may be viewed as an approximation to the notorious subconvexity problem for the symmetric-square L-function in the spectral aspect at $s=1/2$ .

Hankel determinants for starlike and convex functions associated with sigmoid functions

This paper is concerned with Hankel determinants for starlike and convex functions related to modified sigmoid functions. Sharp bounds are given for second and third Hankel determinants.

Computing LF-Metric Dimension of Generalized Gear Networks

The parameter of distance in the theory of networks plays a key role to study the different structural properties of the understudy networks or graphs such as symmetry, assortative, connectivity, and clustering. For the purpose, with the help of the parameter of distance, various types of metric dimensions have been defined to find the locations of machines (or robots) with respect to the minimum consumption of time, the shortest distance among the destinations, and the lesser number of utilized nodes as places of the objects. In this article, the latest derived form of metric dimension called as LF-metric dimension is studied, and various results for the generalized gear networks are obtained in the form of exact values and sharp bounds under certain conditions. The LF-metric dimension of some particular cases of generalized gear networks (called as generalized wheel networks) is also illustrated. Moreover, the bounded and unboundedness of the LF-metric dimension for all obtained results is also presented.

Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

In this paper, we consider Z r \mathbb{Z}^{r} -graded modules on the Cl ⁡ ( X ) \operatorname{Cl}(X) -graded Cox ring C ⁢ [ x 1 , … , x r ] \mathbb{C}[x_{1},\dotsc,x_{r}] of a smooth complete toric variety 𝑋. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive Z s + r + 2 \mathbb{Z}^{s+r+2} -graded modules over any non-standard bigraded polynomial ring C ⁢ [ x 0 , … , x s , y 0 , … , y r ] \mathbb{C}[x_{0},\dotsc,x_{s},\allowbreak y_{0},\dotsc,y_{r}] . In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial.