A class of functions whose sublevel sets are absolute retracts

In this work a class of functions is studied, which are built with the help of significant bits sequences on the ring ℤ2n. This class is built with use of a function ψ: ℤ2n → ℤ2. In public literature there are works in which ψ is a linear function. Here we will use a non-linear ψ function for this set. It is known that the period of a polynomial F in the ring ℤ2n is equal to T(mod 2)2α, where α∈ , n01- . The polynomials for which it is true that T(F) = T(F mod 2), in other words α = 0, are called marked polynomials. For our class we are going to use a polynomial with a maximum period as the characteristic polyomial. In the present work we show the bounds of the given class: non-linearity, the weight of the functions, the Hamming distance between functions. The Hamming distance between these functions and functions of other known classes is also given.

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Some new results on the subexponential class

Journal of Applied Probability ◽

10.2307/3214395 ◽

1989 ◽

Vol 26 (4) ◽

pp. 892-897 ◽

Cited By ~ 10

Author(s):

Emily S. Murphree

Keyword(s):

Distribution Function ◽

Sufficient Conditions ◽

Necessary Condition ◽

Subexponential Class ◽

Class Of Functions

A distribution function F on (0,∞) belongs to the subexponential class if the ratio of 1 – F(2)(x) to 1 – F(x) converges to 2 as x →∞. A necessary condition for membership in is used to prove that a certain class of functions previously thought to be contained in has members outside of . Sufficient conditions on the tail of F are found which ensure F belongs to ; these conditions generalize previously published conditions.

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On the Boundary Dieudonné–Pick Lemma

Mathematics ◽

10.3390/math9101108 ◽

2021 ◽

Vol 9 (10) ◽

pp. 1108

Author(s):

Olga Kudryavtseva ◽

Aleksei Solodov

Keyword(s):

Fixed Point ◽

Interior Point ◽

Sharp Estimate ◽

General Boundary ◽

Extremal Functions ◽

Angular Derivative ◽

Additional Information ◽

Carathéodory Theorem ◽

Class Of Functions ◽

Boundary Fixed Point

The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

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Coefficient inequalities related with typically real functions

Mathematica Slovaca ◽

10.1515/ms-2017-0396 ◽

2020 ◽

Vol 70 (4) ◽

pp. 829-838

Author(s):

Saqib Hussain ◽

Shahid Khan ◽

Khalida Inayat Noor ◽

Mohsan Raza

Keyword(s):

Unit Disk ◽

Real Functions ◽

Coefficient Inequalities ◽

Typically Real ◽

Class Of Functions ◽

Typically Real Functions

AbstractIn this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.

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Norm inequalities involving a special class of functions for sector matrices

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02383-z ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Davood Afraz ◽

Rahmatollah Lashkaripour ◽

Mojtaba Bakherad

Keyword(s):

Special Class ◽

Norm Inequalities ◽

Class Of Functions

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Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

Journal of Inequalities and Applications ◽

10.1186/s13660-020-02488-5 ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Dafang Zhao ◽

Muhammad Aamir Ali ◽

Artion Kashuri ◽

Hüseyin Budak ◽

Mehmet Zeki Sarikaya

Keyword(s):

Convex Functions ◽

Fractional Integrals ◽

Special Cases ◽

Definition Of ◽

The Given ◽

Class Of Functions ◽

Interval Valued ◽

New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

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Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications

Fractal and Fractional ◽

10.3390/fractalfract5030080 ◽

2021 ◽

Vol 5 (3) ◽

pp. 80

Author(s):

Hari Mohan Srivastava ◽

Artion Kashuri ◽

Pshtiwan Othman Mohammed ◽

Dumitru Baleanu ◽

Y. S. Hamed

Keyword(s):

Integral Inequalities ◽

Fractional Integrals ◽

Auxiliary Result ◽

Wright Function ◽

Nonconvex Functions ◽

Special Cases ◽

Type Integral ◽

Generic Class ◽

Two Parameters ◽

Class Of Functions

In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efficiency of our main results, an application for error estimation is obtained as well.

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On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

Georgian Mathematical Journal ◽

10.1515/gmj.1998.101 ◽

1998 ◽

Vol 5 (2) ◽

pp. 101-106

Author(s):

L. Ephremidze

Keyword(s):

Hilbert Transform ◽

Measure Space ◽

Integrable Function ◽

Finite Measure ◽

Ergodic Transformation ◽

Finite Measure Space ◽

Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

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On Krylov’s estimates for optional semimartingales

Random Operators and Stochastic Equations ◽

10.1515/rose-2021-2059 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

M. Abdelghani ◽

A. Melnikov ◽

A. Pak

Keyword(s):

Sobolev Space ◽

Measurable Function ◽

Diffusion Processes ◽

Stochastic Integrals ◽

Controlled Diffusion ◽

Convergence Of Solutions ◽

Change Of Variables ◽

General Class Of Functions ◽

Class Of Functions ◽

Controlled Diffusion Processes

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

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Third Hankel determinants for two classes of analytic functions with real coefficients

Forum Mathematicum ◽

10.1515/forum-2021-0014 ◽

2021 ◽

Vol 33 (4) ◽

pp. 973-986

Author(s):

Young Jae Sim ◽

Paweł Zaprawa

Keyword(s):

Analytic Functions ◽

Univalent Functions ◽

Upper Bounds ◽

Hankel Determinant ◽

Lower And Upper Bounds ◽

Hankel Determinants ◽

The Third ◽

Typically Real ◽

Class Of Functions ◽

Typically Real Functions

Abstract In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

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