On fractional linear bounds for probability generating functions

1986 ◽  
Vol 23 (04) ◽  
pp. 904-913
Author(s):  
Claude Lefevre ◽  
Marc Hallin ◽  
Prakash Narayan
Keyword(s):  
Lower Bounds ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Linear Functions ◽  
Upper And Lower Bounds ◽  
Finite Variance ◽  
Infinitely Divisible Distributions ◽  
Slight Improvement ◽  
Infinitely Divisible ◽  
The Family

The best upper and lower bounds for any probability generating function with meanmand finite variance are derived within the family of fractional linear functions with meanm. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions.

On fractional linear bounds for probability generating functions

1986 ◽  
Vol 23 (4) ◽  
pp. 904-913 ◽  
Author(s):  
Claude Lefevre ◽  
Marc Hallin ◽  
Prakash Narayan
Keyword(s):  
Lower Bounds ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Linear Functions ◽  
Upper And Lower Bounds ◽  
Finite Variance ◽  
Infinitely Divisible Distributions ◽  
Slight Improvement ◽  
Infinitely Divisible ◽  
The Family

The best upper and lower bounds for any probability generating function with mean m and finite variance are derived within the family of fractional linear functions with mean m. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions.

On fractional linear bounds for probability generating functions

1986 ◽  
Vol 23 (04) ◽  
pp. 904-913 ◽  
Author(s):  
Claude Lefevre ◽  
Marc Hallin ◽  
Prakash Narayan
Keyword(s):  
Lower Bounds ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Linear Functions ◽  
Upper And Lower Bounds ◽  
Finite Variance ◽  
Infinitely Divisible Distributions ◽  
Slight Improvement ◽  
Infinitely Divisible ◽  
The Family

The best upper and lower bounds for any probability generating function with mean m and finite variance are derived within the family of fractional linear functions with mean m. These are often intractable and simpler bounds, more useful for practical purposes, are then constructed. Direct applications in branching and epidemic theories are briefly presented; a slight improvement of the bounds is obtained for infinitely divisible distributions.

Alphabetic points in compositions and words

2021 ◽  
Vol 31 (4) ◽  
pp. 241-250
Author(s):  
Margaret Archibald ◽  
Aubrey Blecher ◽  
Arnold Knopfmacher
Keyword(s):  
Lower Bounds ◽  
Generating Functions ◽  
Upper And Lower Bounds ◽  
A Value

Abstract We use generating functions to account for alphabetic points (or the lack thereof) in compositions and words. An alphabetic point is a value j such that all the values to its left are not larger than j and all the values to its right are not smaller than j. We also provide the asymptotics for compositions and words which have no alphabetic points, as the size tends to infinity. This is achieved by the construction of upper and lower bounds which converge to each other, and in the latter case by probabilistic arguments.

scholarly journals A Note on the Symmetric Powers of the Standard Representation of $S_n$

10.37236/1484 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
D. Savitt ◽  
R. P. Stanley
Keyword(s):  
Lower Bounds ◽  
Generating Functions ◽  
Upper And Lower Bounds ◽  
Dimensional Representation ◽  
Standard Representation ◽  
Symmetric Powers

In this paper, we prove that the dimension of the space span- ned by the characters of the symmetric powers of the standard $n$-dimensional representation of $S_n$ is asymptotic to $n^2 / 2$. This is proved by using generating functions to obtain formulas for upper and lower bounds, both asymptotic to $n^2/2$, for this dimension. In particular, for $n \ge 7$, these characters do not span the full space of class functions on $S_n$.

scholarly journals Euclidean linear invariance and uniform local convexity

1992 ◽  
Vol 52 (3) ◽  
pp. 401-418 ◽  
Author(s):  
Wancang Ma ◽  
David Minda
Keyword(s):  
Lower Bounds ◽  
Convex Subset ◽  
Convex Functions ◽  
Convex Set ◽  
Holomorphic Functions ◽  
Geometric Interpretation ◽  
Upper And Lower Bounds ◽  
Local Convexity ◽  
Related Matter ◽  
The Family

AbstractLet S(p) be the family of holomorphic functions f defined on the unit disk D, normalized by f(0) = f1(0) – 1 = 0 and univalent in every hyperbolic disk of radius p. Let C(p) be the subfamily consisting of those functions which are convex univalent in every hyperbolic disk of radius p. For p = ∞ these become the classical families S and C of normalized univalent and convex functions, respectively. These families are linearly invariant in the sense of Pommerenke; a natural problem is to calculate the order of these linearly invariant families. More precisely, we give a geometrie proof that C(p) is the universal linearly invariant family of all normalized locally schlicht functions of order at most coth(2p). This gives a purely geometric interpretation for the order of a linearly invariant family. In a related matter, we characterize those locally schlicht functions which map each hyperbolically k-convex subset of D onto a euclidean convex set. Finally, we give upper and lower bounds on the order of the linearly invariant family S(p) and prove that this class is not equal to the universal linearly invariant family of any order.

scholarly journals GENERATING FUNCTIONS OF CAUCHY–STIELTJES TYPE FOR ORTHOGONAL POLYNOMIALS

2009 ◽  
Vol 12 (01) ◽  
pp. 91-98 ◽  
Author(s):  
MAREK BOŻEJKO ◽  
NIZAR DEMNI
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Functions ◽  
Short Proof ◽  
Probabilistic Interpretation ◽  
Infinitely Divisible ◽  
The Family ◽  
Multiplicative Renormalization ◽  
Renormalization Method ◽  
Stieltjes Type

We give a free probabilistic interpretation of the multiplicative renormalization method. As a byproduct, we give a short proof of the Asai–Kubo–Kuo problem on the characterization of the family of measures for which this method applies with h(x) = (1 - x)-1 which turns out to be the free Meixner family. We also give a representation of the Voiculescu transform of all free Meixner laws (even in the non-freely infinitely divisible case).

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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