scholarly journals On the characterization of Jensen m-convex polynomials

2017 ◽  
Vol 3 (2) ◽  
pp. 140-148
Author(s):  
Teodoro Lara ◽  
Nelson Merentes ◽  
Roy Quintero ◽  
Edgar Rosales
Keyword(s):  
Convex Functions ◽  
Polynomial Functions ◽  
Real Polynomial ◽  
Real Numbers ◽  
The Real ◽  
Class Of Functions

AbstractThe main objective of this research is to characterize all the real polynomial functions of degree less than the fourth which are Jensen m-convex on the set of non-negative real numbers. In the first section, it is established for that class of functions what conditions must satisfy a particular polynomial in order to be starshaped on the same set. Finally, both kinds of results are combined in order to find examples of either Jensen m-convex functions which are not starshaped or viceversa.

scholarly journals What is invexity?

1986 ◽  
Vol 28 (1) ◽  
pp. 1-9 ◽  
Author(s):  
A. Ben-Israel ◽  
B. Mond
Keyword(s):  
Mathematical Programming ◽  
Convex Functions ◽  
Constraint Qualification ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Slater Constraint Qualification ◽  
The Relationship ◽  
Unconstrained Problems ◽  
Class Of Functions

AbstractRecently it was shown that many results in Mathematical Programming involving convex functions actually hold for a wider class of functions, called invex. Here a simple characterization of invexity is given for both constrained and unconstrained problems. The relationship between invexity and other generalizations of convexity is illustrated. Finally, it is shown that invexity can be substituted for convexity in the saddle point problem and in the Slater constraint qualification.

scholarly journals Universality probability of a prefix-free machine

2012 ◽  
Vol 370 (1971) ◽  
pp. 3488-3511 ◽  
Author(s):  
George Barmpalias ◽  
David L. Dowe
Keyword(s):  
Turing Degree ◽  
Real Numbers ◽  
Arithmetical Hierarchy ◽  
Halting Problem ◽  
The Real ◽  
The Third ◽  
Free Machine

We study the notion of universality probability of a universal prefix-free machine, as introduced by C. S. Wallace. We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in the arithmetical hierarchy of complexity. Furthermore, we give a computational characterization of the real numbers that are universality probabilities of universal prefix-free machines.

scholarly journals On Locally Uniformly Differentiable Functions on a Complete Non-Archimedean Ordered Field Extension of the Real Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Khodr Shamseddine ◽  
Todd Sierens
Keyword(s):  
Inverse Function ◽  
Field Extension ◽  
Polynomial Functions ◽  
Inverse Function Theorem ◽  
Real Numbers ◽  
Intermediate Value Theorem ◽  
The Real ◽  
Ordered Field ◽  
Differentiable Functions ◽  
Intermediate Value

We study the properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In particular, we show that locally uniformly differentiable functions are C1, they include all polynomial functions, and they are closed under addition, multiplication, and composition. Then we formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions.

scholarly journals On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
J. M. Sepulcre
Keyword(s):  
Important Class ◽  
Exponential Polynomial ◽  
Critical Interval ◽  
Real Numbers ◽  
Positive Real ◽  
Exponential Polynomials ◽  
The Real ◽  
Linearly Independent ◽  
Pointwise Characterization

We provide the proof of a practical pointwise characterization of the setRPdefined by the closure set of the real projections of the zeros of an exponential polynomialP(z)=∑j=1ncjewjzwith real frequencieswjlinearly independent over the rationals. As a consequence, we give a complete description of the setRPand prove its invariance with respect to the moduli of thecj′s, which allows us to determine exactly the gaps ofRPand the extremes of the critical interval ofP(z)by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.

scholarly journals A Generalized Class of Close-to-Convex Functions

2020 ◽  
Vol 44 (4) ◽  
pp. 533-538
Author(s):  
PARDEEP KAUR ◽  
SUKHWINDER SINGH BILLING
Keyword(s):  
Real Number ◽  
Unit Disk ◽  
Convex Functions ◽  
Starlike Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Real Numbers ◽  
Special Cases ◽  
Class Of Functions

Let ℋαϕ(β) denote the class of functions f, analytic in the open unit disk ???? which satisfy the condition ( ( ) ) zf-′(z-)- zf-′′(z-) ℜ (1 − α) + α 1 + ′ > β, z ∈ ????, ϕ(z ) f (z ) where α, β are pre-assigned real numbers and ϕ(z) is a starlike function. The special cases of the class ℋαϕ(β) have been studied in literature by different authors. In 2007, Singh et al. [?] studied the class ℋαz(β) and they established that functions in ℋαz(β) are univalent for all real numbers α, β satisfying the condition α ≤ β < 1 and the result is sharp in the sense that constant β cannot be replaced by a real number smaller than α. Singh et al. [?] in 2005, proved that for 0 < α < 1 functions in class ℋαz(α) are univalent. In 1975, Al-Amiri and Reade [?] showed that functions in class ℋαz(0) are univalent for all α ≤ 0 and also for α = 1 in ????. In the present paper, we prove that members of the class ℋαϕ(β) are close-to-convex and hence univalent for real numbers α, β and for a starlike function ϕ satisfying the condition β + α − 1 < αℜ( ) zϕ′(z) ϕ(z)≤ β < 1.

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